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Instructions for use Title Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by p-Laplacian Author(s) Giga, Yoshikazu; Tsubouchi, Shuntaro Citation Hokkaido University Preprint Series in Mathematics, 1137, 1-29 Issue Date 2021-08-30 DOI 10.14943/99357 Doc URL http://hdl.handle.net/2115/82542 Type bulletin (article) File Information ConDerConvex.pdf Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

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Instructions for use

Title Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by p-Laplacian

Author(s) Giga, Yoshikazu; Tsubouchi, Shuntaro

Citation Hokkaido University Preprint Series in Mathematics, 1137, 1-29

Issue Date 2021-08-30

DOI 10.14943/99357

Doc URL http://hdl.handle.net/2115/82542

Type bulletin (article)

File Information ConDerConvex.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Continuity of derivatives of a convex solution to a perturbedone-Laplace equation by ๐‘-Laplacian

Yoshikazu Gigaโˆ— and Shuntaro Tsubouchi โ€ 

August 23, 2021

Abstract

We consider a one-Laplace equation perturbed by ๐‘-Laplacian with 1 < ๐‘ <โˆž. We prove that a weak solutionis continuously differentiable (๐ถ1) if it is convex. Note that similar result fails to hold for the unperturbedone-Laplace equation. The main difficulty is to show ๐ถ1-regularity of the solution at the boundary of a facetwhere the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limitis a constant function by establishing a Liouville-type result, which is proved by showing a strong maximumprinciple. Our argument is rather elementary since we assume that the solution is convex. A few generalizationis also discussed.

Keywords ๐ถ1-regularity, one-Laplace equation, strong maximum principle

1 IntroductionWe consider a one-Laplace equation perturbed by ๐‘-Laplacian of the form

๐ฟ๐‘,๐‘๐‘ข = ๐‘“ in ฮฉ (1.1)

with๐ฟ๐‘,๐‘๐‘ข := โˆ’๐‘ฮ”1๐‘ขโˆ’ฮ”๐‘๐‘ข,

whereฮ”1๐‘ข := div (โˆ‡๐‘ข/|โˆ‡๐‘ข |) , ฮ”๐‘๐‘ข = div

(|โˆ‡๐‘ข |๐‘โˆ’2โˆ‡๐‘ข

)in a domain ฮฉ in R๐‘›, โˆ‡๐‘ข = (๐œ•๐‘ฅ1๐‘ข, . . . , ๐œ•๐‘ฅ๐‘›๐‘ข) with ๐œ•๐‘ฅ ๐‘—๐‘ข = ๐œ•๐‘ข/๐œ•๐‘ฅ ๐‘— for a function ๐‘ข = ๐‘ข(๐‘ฅ1, . . . , ๐‘ฅ๐‘›), and div๐‘‹ =๐‘›โˆ‘๐‘–=1๐œ•๐‘ฅ๐‘–๐‘‹๐‘– for a vector field ๐‘‹ = (๐‘‹1, . . . , ๐‘‹๐‘›). The constants ๐‘ > 0 and ๐‘ โˆˆ (1,โˆž) are given and fixed. It has been a

long-standing open problem whether its weak solution is๐ถ1 up to a facet, the place where the gradient โˆ‡๐‘ข vanishes,even if ๐‘“ is smooth. This is a non-trivial question since a weak solution to the unperturbed one-Laplace equation,i.e., โˆ’ฮ”1๐‘ข = ๐‘“ may not be ๐ถ1. This is because the ellipticity degenerates in the direction of โˆ‡๐‘ข for ฮ”1๐‘ข. Our goalin this paper is to solve this open problem under the assumption that a solution is convex.

1.1 Main theorems and our strategyThroughout the paper, we assume ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› < ๐‘ž โ‰ค โˆž), i.e., | ๐‘“ |๐‘ž is locally integrable in ฮฉ. Our main result is

Theorem 1 (๐ถ1-regularity theorem). Let ๐‘ข be a convex weak solution to (1.1) with ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› < ๐‘ž โ‰ค โˆž). Then๐‘ข is in ๐ถ1 (ฮฉ).

โˆ—Graduate School of Mathematical Sciences, The University of Tokyo, Japan. Email: [email protected]โ€ Graduate School of Mathematical Sciences, The University of Tokyo, Japan. Email: [email protected]

1

Difficulty on proving regularity on gradients of solutions to (1.1) can be explained from a viewpoint of ellipticityratio. We set a convex function ๐ธ : R๐‘› โ†’ [0,โˆž) by

๐ธ (๐‘ง) B ๐‘ |๐‘ง | + |๐‘ง |๐‘๐‘

for ๐‘ง โˆˆ R๐‘›.

We rewrite (1.1) byโˆ’div(โˆ‡๐‘ง๐ธ (โˆ‡๐‘ข)) = ๐‘“ in ฮฉ. (1.2)

By differentiating (1.2) by ๐‘ฅ๐‘– (๐‘– โˆˆ {1, . . . , ๐‘› }), we get

โˆ’div(โˆ‡2๐‘ง๐ธ (โˆ‡๐‘ข)โˆ‡๐œ•๐‘ฅ๐‘–๐‘ข

)= ๐œ•๐‘ฅ๐‘– ๐‘“ . (1.3)

By elementary calculations, ellipticity ratio of the Hessian โˆ‡2๐‘ง๐ธ at ๐‘ง0 โˆˆ R๐‘› \ {0} is given by(

ellipticity ratio of โˆ‡2๐‘ง๐ธ (๐‘ง0)

)B

(the largest eigenvalue of โˆ‡2๐‘ง๐ธ (๐‘ง0))

(the lowest eigenvalue of โˆ‡2๐‘ง๐ธ (๐‘ง0))

=max(๐‘โˆ’1, 1) + ๐‘ |๐‘ง0 |1โˆ’๐‘

min(๐‘โˆ’1, 1) .

Since the exponent 1โˆ’ ๐‘ is negative, the ellipticity ratio of โˆ‡2๐‘ง๐ธ (๐‘ง0) blows up as ๐‘ง0 โ†’ 0. From this we can observe

that the equation (1.2) becomes non-uniformly elliptic near the facet. It should be noted that our problem issubstantially different from the (๐‘, ๐‘ž)-growth problem, since for (๐‘, ๐‘ž)-growth equations, non-uniform ellipticityappears as a norm of a gradient blows up [26, Section 6.2]. Although regularity of minimizers of double phasefunctionals, including

H(๐‘ข) Bโˆซ๐ธ๐‘ (โˆ‡๐‘ข) ๐‘‘๐‘ฅ +

โˆซ๐‘Ž(๐‘ฅ)๐ธ๐‘ž (โˆ‡๐‘ข) ๐‘‘๐‘ฅ with 1 < ๐‘ โ‰ค ๐‘ž <โˆž, ๐‘Ž(๐‘ฅ) โ‰ฅ 0

were discussed in scalar and even in vectorial cases by Colombo and Mingione [6, 7], their results do not recover our๐ถ1-regularity results. This is basically derived from the fact that, unlike โˆ‡2

๐‘ง๐ธ๐‘ with 1 < ๐‘ <โˆž, the Hessian matrixโˆ‡2๐‘ง๐ธ1 (๐‘ง0) (๐‘ง0 โ‰  0) always takes 0 as its eigenvalue. In other words, ellipticity of the operator ฮ”1๐‘ข degenerates in

the direction of โˆ‡๐‘ข, which seems to be difficult to handle analytically.On the other hand, the ellipticity ratio of โˆ‡2

๐‘ง๐ธ (๐‘ง0) is uniformly bounded over |๐‘ง0 | > ๐›ฟ for each fixed ๐›ฟ > 0. Inthis sense we may regard the equation (1.3) as locally uniformly elliptic outside the facet. To show Lipschitz bound,we do not need to study over the facet. In fact, local Lipschitz continuity of solutions to (1.1) are already establishedin [32]; see also [33] for a weaker result. To study continuity of derivatives, we have to study regularity up to thefacet. Thus, it seems to be impossible to apply standard arguments based on De Giorgiโ€“Nashโ€“Moser theory. Inthis paper, we would like to show continuity of derivatives of convex solutions by elementary arguments based onconvex analysis.

Let us give a basic strategy to prove Theorem 1. Since the problem is local, we may assume that ฮฉ is convex,or even a ball. By ๐ถ1-regularity criterion for a convex function, to show ๐‘ข is ๐ถ1 at ๐‘ฅ โˆˆ ฮฉ it suffices to prove that

the subdifferential ๐œ•๐‘ข(๐‘ฅ) at ๐‘ฅ โˆˆ ฮฉ is a singleton; (1.4)

see [1, Appendix D], [30, ยง25] and Remark 1 for more detail. Here the subdifferential of ๐‘ข at ๐‘ฅ0 โˆˆ ฮฉ is defined by

๐œ•๐‘ข(๐‘ฅ0) B {๐‘ง โˆˆ R๐‘› | ๐‘ข(๐‘ฅ) โ‰ฅ ๐‘ข(๐‘ฅ0) + โŸจ๐‘ง | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ for all ๐‘ฅ โˆˆ ฮฉ}.

Here โŸจ ยท | ยท โŸฉ stands for the standard inner product in R๐‘›. For a convex function ๐‘ข : ฮฉโ†’ R, we can simply expressthe facet of ๐‘ข as

๐น B {๐‘ฅ โˆˆ ฮฉ | ๐œ•๐‘ข(๐‘ฅ) โˆ‹ 0} = {๐‘ฅ โˆˆ ฮฉ | ๐‘ข(๐‘ฅ) โ‰ค ๐‘ข(๐‘ฆ) for all ๐‘ฆ โˆˆ ฮฉ}.By definition it is clear that the facet ๐น is non-empty if and only if a minimum of ๐‘ข in ฮฉ exists. By convexity of ๐‘ข,we can easily check that ๐น โŠ‚ ฮฉ is a relatively closed convex set in ฮฉ. We also define an open set

๐ท B ฮฉ \๐น = {๐‘ฅ โˆˆ ฮฉ | ๐‘ข(๐‘ฆ) < ๐‘ข(๐‘ฅ) for some ๐‘ฆ โˆˆ ฮฉ}.

Our strategy to show (1.4) depends on whether ๐‘ฅ is inside ๐น or not.

2

Remark 1. [Some properties on differentiability of convex functions] Let ๐‘ฃ a real-valued convex function in aconvex domain ฮฉ โŠ‚ R๐‘›, then following property holds.

1. ๐‘ฃ is locally Lipschitz continuous in ฮฉ, and therefore ๐‘ฃ is a.e. differentiable in ฮฉ by Rademacherโ€™s theorem([1, Theorem 1.19], see also [9, Theorem 3.1 and 3.2] and [30, Theorem 25.5]).

2. For ๐‘ฅ โˆˆ ฮฉ, ๐‘ฃ is differentiable at ๐‘ฅ if and only if the subdifferential set ๐œ•๐‘ฃ(๐‘ฅ) is a singleton. Moreover, if ๐‘ฅ โˆˆ ฮฉsatisfies either of these equivalent conditions, then we have ๐œ•๐‘ฃ(๐‘ฅ) = {โˆ‡๐‘ฃ(๐‘ฅ)} ([1, Proposition D.5], see also[30, Theorem 25.1]). In particular, Rademacherโ€™s theorem implies that ๐œ•๐‘ฃ(๐‘ฅ) = {โˆ‡๐‘ฃ(๐‘ฅ)} for a.e. ๐‘ฅ โˆˆ ฮฉ.

3. ๐‘ฃ โˆˆ ๐ถ1 (ฮฉ) if and only if ๐œ•๐‘ฃ is single-valued ([1, Remark D.3 (iii)], see also [30, Theorem 25.5]).

Throughout the paper we use these well-known results without proofs.

We first discuss the case ๐‘ฅ โˆˆ ๐ท. Our goal is to show directly that ๐‘ข is ๐ถ1, ๐›ผ near a neighborhood of ๐‘ฅ andtherefore ๐œ•๐‘ข(๐‘ฅ) = {โˆ‡๐‘ข(๐‘ฅ)} โ‰  {0} for all ๐‘ฅ โˆˆ ๐ท. This strategy roughly consists of three steps. Among them the firststep, a kind of separation of ๐‘ฅ โˆˆ ๐ท from the facet ๐น, plays an important role. Precisely speaking, we first find aneighborhood ๐ต๐‘Ÿ (๐‘ฅ) โŠ‚ ๐ท, an open ball centered at ๐‘ฅ with its radius ๐‘Ÿ > 0, such that

๐œ•๐œˆ๐‘ข โ‰ฅ ๐œ‡ > 0 a.e. in ๐ต๐‘Ÿ (๐‘ฅ) (1.5)

for some direction ๐œˆ and some constant ๐œ‡ > 0. In order to justify (1.5), we fully make use of convexity of ๐‘ข (Lemma8 in Section A), not elliptic regularity theory. Then with the aid of local Lipschitz continuity of ๐‘ข, the inclusion๐ต๐‘Ÿ (๐‘ฅ) โŠ‚ {0 < ๐œ‡ โ‰ค ๐œ•๐œˆ๐‘ข โ‰ค |โˆ‡๐‘ข | โ‰ค ๐‘€} holds for some finite positive constant ๐‘€ . Secondly, this inclusion allows us tocheck that ๐‘ข admits local๐‘Š2, 2-regularity in ๐ต๐‘Ÿ (๐‘ฅ) by the standard difference quotient method. Therefore we are ableto obtain the equation (1.3) in the distributional sense. Finally, we appeal to the classical De Giorgiโ€“Nashโ€“Mosertheory to obtain local ๐ถ1, ๐›ผ-regularity at ๐‘ฅ โˆˆ ๐ท, since the equation (1.3) is uniformly elliptic in ๐ต๐‘Ÿ (๐‘ฅ). Here theconstant ๐›ผ โˆˆ (0, 1) we have obtained may depend on the location of ๐‘ฅ โˆˆ ๐ท through ellipticity, so ๐›ผ may tend to zeroas ๐‘ฅ tends to the facet.

It takes much efforts to prove that ๐œ•๐‘ข(๐‘ฅ) = {0} for all ๐‘ฅ โˆˆ ๐น. Our strategy for justifying this roughly consistsof three parts; a blow-argument for solutions, a strong maximum principle, and a Liouville-type theorem. Here wedescribe each individual step.

We first make a blow-argument. Precisely speaking, for a given convex solution ๐‘ข : ฮฉโ†’ R and a point ๐‘ฅ0 โˆˆ ฮฉ,we set a sequence of rescaled functions {๐‘ข๐‘Ž}๐‘Ž>0 defined by

๐‘ข๐‘Ž (๐‘ฅ) B๐‘ข(๐‘Ž(๐‘ฅโˆ’ ๐‘ฅ0) + ๐‘ฅ0) โˆ’๐‘ข(๐‘ฅ0)

๐‘Ž.

We show that ๐‘ข๐‘Ž locally uniformly converges to some convex function ๐‘ข0 : R๐‘›โ†’R, which satisfies ๐œ•๐‘ข(๐‘ฅ0) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ0)by construction. Moreover, we prove that ๐‘ข0 satisfies ๐ฟ๐‘, ๐‘๐‘ข0 = 0 in R๐‘› in the distributional sense. There we willface to justify a.e. convergence of gradients, and this is elementarily shown by regarding gradients in the classicalsense as subgradients (Lemma 9 in the appendices).

Next we prove that if ๐‘ฅ0 โˆˆ ๐น, then the convex weak solution ๐‘ข0 constructed as above satisfies ๐œ•๐‘ข0 (๐‘ฅ0) = {0}.Moreover, we are going to prove that ๐‘ข0 is constant (a Liouville-type theorem). For this purpose we establish themaximum principle.

Theorem 2 (Strong maximum principle). Let ๐‘ข be a convex weak solution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in a convex domain ฮฉ โŠ‚ R๐‘›and ๐น โŠ‚ ฮฉ be the facet of ๐‘ข. Then ๐‘ข is affine in each connected component of the open set ๐ท B ฮฉ\๐น. In particular,if ๐น = โˆ…, then ๐‘ข is affine in ฮฉ.

It should be noted that this result is a kind of strong maximum principle in the sense that

๐‘ข โ‰ฅ ๐‘Ž in ๐ท0 and ๐‘ข(๐‘ฅ0) = ๐‘Ž(๐‘ฅ0) for ๐‘ฅ0 โˆˆ ๐ท0 imply that ๐‘ข โ‰ก ๐‘Ž in ๐ท0, (1.6)

where ๐‘Ž(๐‘ฅ) B ๐‘ข(๐‘ฅ0) + โŸจโˆ‡๐‘ข(๐‘ฅ0) | ๐‘ฅ โˆ’ ๐‘ฅ0โŸฉ and ๐ท0 is a connected component of ๐ท. The affine function ๐‘Ž clearlysatisfies ๐ฟ๐‘, ๐‘๐‘Ž = 0 in the classical sense.

In order to justify (1.6), we will face three problems. The first is a justification of the comparison principle, thesecond is regularity of ๐‘ข, and the third is a construction of suitable barrier subsolutions, all of which are essentiallyneeded in the classical proof of E. Hopfโ€™s strong maximum principle [20]. In order to overcome these obstacles,

3

we appeal to both classical and distributional approaches, and restrict our analysis only over regular points. Fordetails, see Section 1.2.

Even though our strong maximum principle is somewhat weakened in the sense that this holds only on eachconnected component of ๐ท โŠ‚ ฮฉ, we are able to show the following Liouville-type theorem.

Theorem 3 (Liouville-type theorem). Let ๐‘ข be a convex weak solution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in R๐‘›. Then ๐น โŠ‚ R๐‘›, the facetof ๐‘ข, satisfies either ๐น = โˆ… or ๐น = R๐‘›. In particular, ๐‘ข satisfies either of the followings.

1. If ๐‘ข attains its minimum in R๐‘›, then ๐‘ข is constant.

2. If ๐‘ข does not attains its minimum in R๐‘›, then ๐‘ข is a non-constant affine function in R๐‘›.

In the proof of the Liouville-type theorem, our strong maximum principle plays an important role. Preciselyspeaking, if a convex solution in the total space does not satisfy โˆ… โŠŠ ๐น โŠŠ R๐‘›, then Theorem 2 and the supportinghyperplane theorem from convex analysis help us to determine the shape of convex solutions. In particular, theconvex solution can be classified into three types of piecewise-linear functions of one-variable. These non-smoothpiecewise-linear functions are, however, no longer weak solutions, which we will prove by some explicit calculations.

By applying the Liouville-type theorem and our blow-argument, we are able to show that subgradients at pointsof the facet are always 0, i.e., ๐œ•๐‘ข(๐‘ฅ) = {0} for all ๐‘ฅ โˆˆ ๐น, and we complete the proof of the ๐ถ1-regularity theorem.Note that the statements in Theorem 2 and 3 should not hold for unperturbed one-Laplace equation โˆ’ฮ”1๐‘ข = ๐‘“ , sinceany absolutely continuous non-decreasing function of one variable ๐‘ข = ๐‘ข(๐‘ฅ1) satisfies โˆ’ฮ”1๐‘ข = 0.

Finally we mention that we are able to refine our strategy, and obtain ๐ถ1-regularity of convex solutions to moregeneral equations. We replace the one-Laplacian ฮ”1 by another operator which is derived from a general convexfunctional of degree 1. This generalization requires us to modify some of our arguments, including a blow-upargument and the Liouville-type theorem. For further details, see Section 1.4 and Section 6.2.

1.2 Literature overview on maximum principlesWe briefly introduce maximum principles related to the paper. We also describe our strategy to establish the strongmaximum principle.

Maximum principles, including comparison principles and strong maximum principles, have been discussed bymany mathematicians in various settings. In the classical settings, E. Hopf proved a variety of maximum principleson elliptic partial differential equations of second order, by elementary arguments based on constructions ofauxiliary functions. E. Hopfโ€™s strong maximum principle is one of the well-known results on maximum principles.In Hopfโ€™s proof of the strong maximum principle [20], he defined an auxiliary function

โ„Ž(๐‘ฅ) B ๐‘’โˆ’๐›ผ |๐‘ฅโˆ’๐‘ฅโˆ— |2 โˆ’ ๐‘’โˆ’๐›ผ๐‘…2

for ๐‘ฅ โˆˆ R๐‘›, (1.7)

which becomes a classical subsolution in a fixed open annulus ๐ธ๐‘… = ๐ธ๐‘… (๐‘ฅโˆ—) B ๐ต๐‘… (๐‘ฅโˆ—) \ ๐ต๐‘…/2 (๐‘ฅโˆ—) for sufficientlylarge ๐›ผ > 0. An alternative function

โ„Ž(๐‘ฅ) B |๐‘ฅโˆ’ ๐‘ฅโˆ— |โˆ’๐›ผ โˆ’๐‘…โˆ’๐›ผ for ๐‘ฅ โˆˆ R๐‘› \ {๐‘ฅโˆ—} (1.8)

is given in [29, Chapter 2.8]. E. Hopfโ€™s classical results on maximum principles are extensively contained in [17,Chapter 3], [28, Chapter 2] and [29, Chapter 2].

The materials [17, Chapter 8โ€“9] and [29, Chapter 3โ€“6] provide proofs of maximum principles, including strongmaximum principles, even for distributional solutions. Among them, [29, Theorem 5.4.1] deals with a justificationof the strong maximum principle for distributional supersolutons to certain quasilinear elliptic equations withdivergence structures,

i.e., โˆ’div(๐ด(๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ))) = 0,

which covers the ๐‘-Laplace equation with 1 < ๐‘ <โˆž. Even in the distributional schemes, the proof of the maximumprinciple [29, Theorem 5.4.1] is partially similar to E. Hopfโ€™s classical one, in the sense that it is completed bycalculating directional derivatives of auxiliary functions. The significant difference is, however, the constructionof spherically symmetric subsolutions of ๐ถ1 class, which is given in [29, Chapter 4], is based on Lerayโ€“Schauderโ€™sfixed point theorem [17, Theorem 11.6]. Also it should be noted that the proofs of comparison principles [29,Theorem 2.4.1 and 3.4.1] are just based on strict monotonicity of the mapping ๐ด(๐‘ฅ, ยท ) : R๐‘› โ†’ R๐‘›, whereas Hopfโ€™sproof appeals to direct constructions of auxiliary functions.

4

With our literature overview in mind, we describe our strategy for showing (1.6). A justification of comparisonprinciples is easily obtained in the distributional schemes (see [29, Chapter 3] as a related material). However,the remaining two obstacles, the differentiability of ๐‘ข and the construction of subsolutions, cannot be resolvedaffirmatively by just imitating arguments given in [29, Chapter 4โ€“5]. In the first place, it should be mentionedthat convex weak solutions we treat in this paper are assumed to have only local Lipschitz regularity, whereassupersolutions treated in [29, Chapter 5] are required to be in ๐ถ1. We recall that ๐ถ1-regularity of convex weaksolutions can be guaranteed in ๐ท โŠ‚ ฮฉ (the outside of the facet) by the classical De Giorgiโ€“Nashโ€“Moser theory, andthis result enables us to overcome the problem whether ๐‘ข is differentiable at certain points. This is the reason whyTheorem 2 need to restrict on ๐ท. Although the construction of distributional subsolutions is generally discussed in[29, Chapter 4], we do not appeal to this. Instead, we directly construct a function ๐‘ฃ = ๐›ฝโ„Ž+ ๐‘Ž in R๐‘› \ {๐‘ฅโˆ—}, where๐›ฝ > 0 is a constant and โ„Ž is defined as in (1.7) or (1.8). We will determine the constants ๐›ผ, ๐›ฝ > 0 so precisely that ๐‘ฃsatisfies ๐ฟ๐‘, ๐‘๐‘ฃ โ‰ค 0 in the classical sense over a fixed open annulus ๐ธ๐‘… = ๐ธ๐‘… (๐‘ฅโˆ—). We also make |โˆ‡๐‘ฃ | very close to|โˆ‡๐‘Ž | โ‰ก |โˆ‡๐‘ข(๐‘ฅ0) | > 0 over ๐ธ๐‘…, so that โˆ‡๐‘ฃ no longer degenerates there. By direct calculation of ๐ฟ๐‘, ๐‘๐‘ฃ, we explicitlyconstruct classical subsolutions to ๐ฟ๐‘, ๐‘๐‘ข = 0 in ๐ธ๐‘…. Finally we are able to deduce (1.6) by an indirect proof.

Another type of definitions of subsolutions and supersolutions to (1.1) in the distributional schemes can befound in F. Krรผgelโ€™s thesis in 2013 [25]. The significant difference is that Krรผgel did not regard the term โˆ‡๐‘ข/|โˆ‡๐‘ข | asa subgradient vector field. Since monotonicity of ๐œ• | ยท | is not used at all, it seems that Krรผgelโ€™s proof of comparisonprinciple [25, Theorem 4.8] needs further explanation. For details, see Remark 3.

1.3 Mathematical models and previous researchesOur problem is derived from a minimizing problem of a certain energy functional, which involves the total variationenergy. The equation (1.1) is deduced from the following Eulerโ€“Lagrange equation;

๐‘“ =๐›ฟ๐บ

๐›ฟ๐‘ข, where ๐บ (๐‘ข) B ๐‘

โˆซฮฉ|โˆ‡๐‘ข | ๐‘‘๐‘ฅ + 1

๐‘

โˆซฮฉ|โˆ‡๐‘ข |๐‘ ๐‘‘๐‘ฅ.

The energy functional ๐บ often appears in fields of materials science and fluid mechanics.In [31], Spohn modeled the relaxation dynamics of a crystal surface below the roughening temperature. On โ„Ž

describing the height of the crystal for a two-dimensional domain ฮฉ is modeled as

โ„Ž๐‘ก +div ๐‘— = 0

with ๐‘— = โˆ’โˆ‡๐œ‡, where ๐œ‡ is a chemical potential. In [31], its evolution is given as

๐œ‡ =๐›ฟฮฆ๐›ฟโ„Ž

with ฮฆ(โ„Ž) =โˆซฮฉ|โˆ‡โ„Ž| ๐‘‘๐‘ฅ + ๐œ…

โˆซฮฉ|โˆ‡โ„Ž|3 ๐‘‘๐‘ฅ

with ๐œ… > 0. This ฮฆ is essentially the same as ๐บ with ๐‘ = 3. Then, the resulting evolution equation for โ„Ž is of theform

๐‘โ„Ž๐‘ก = ฮ”๐ฟ๐‘, 3โ„Ž with ๐‘ =13๐œ….

This equation can be defined as a limit of step motion, which is microscopic in the direction of height [23]; seealso [27]. The initial value problem of this equation can be solved based on the theory of maximal monotoneoperators [12] under the periodic boundary condition. Subdifferentials describing the evolution are characterizedby Kashima [21], [22]. Its evolution speed is calculated by [21] for one dimensional setting and by [22] for radialsetting. It is known that the solution stops in finite time [13], [14]. In [27], numerical calculation based on stepmotion is calculated. If one considers a stationary solution, โ„Ž must satisfies

ฮ”๐ฟ๐‘, 3โ„Ž = 0.

If ๐ฟ๐‘, 3โ„Ž is a constant, our Theorem 1 implies that the height function โ„Ž is ๐ถ1 provided that โ„Ž is convex.For a second order problem,

i.e., ๐‘โ„Ž๐‘ก = ๐ฟ๐‘, ๐‘โ„Ž,

its analytic formulation goes back to [4], [8, Chapter VI] for ๐‘ = 2, and its numerical analysis is given in [19]. Forthe fourth order problem, its numerical study is more recent. The reader is referred to papers by [15], [16], [24].

5

Another important mathematical model for the equation (1.1) is found in fluid mechanics. Especially for ๐‘ = 2and ๐‘› = 2, the energy functional ๐บ appears when modeling stationary laminar incompressible flows of a materialcalled Bingham fluid, which is a typical non Newtonian fluid. Bingham fluid reflects the effect of plasticitycorresponding to ฮ”1๐‘ข as well as that of viscosity corresponding ฮ”2๐‘ข = ฮ”๐‘ข in (1.1). Let us consider a parallelstationary flow with velocity ๐‘ˆ = (0,0, ๐‘ข(๐‘ฅ1, ๐‘ฅ2)) in a cylinder ฮฉร—R. Of course, this is incompressible flow, i.e.,div๐‘ˆ = 0. If this flow is the classical Newtonian fluid, then the Navierโ€“Stokes equations become (1.1) inฮฉwith ๐‘ = 0and ๐‘“ = โˆ’๐œ•๐‘ฅ3๐œ‹, where ๐œ‹ denotes the pressure. In the case that plasticity effects appears, one obtains (1.1), following[8, Chapter VI, Section 1]. There it is also mentioned that since the velocity is assumed to be uni-directional, theexternal force term in (1.1) is considered as constant in this laminar flow model. The significant difference is thatmotion of the Bingham fluid is blocked if the stress of the Bingham fluid exceeds a certain threshold. This physicalphenomenon is essentially explained by the nonlinear term ๐‘ฮ”1๐‘ข, which reflects rigidity of the Bingham fluid. Formore details, see [8, Chapter VI] and the references therein.

On continuity of derivatives for solutions, less is known even for the second order elliptic case. AlthoughKrรผgel gave an observation that solutions can be continuously differentiable [25, Theorem 1.2] on the boundary ofa facet, mathematical justifications of ๐ถ1-regularity have not been well-understood. Our main result (Theorem 1)mathematically establishes continuity of gradient for convex solutions.

1.4 Organization of the paperWe outline the contents of the paper.

Section 2 establishes ๐ถ1, ๐›ผ-regularity at regular points of convex weak solutions (Lemma 1). In order to applyDe Giorgiโ€“Nashโ€“Moser theory, we will need to justify local ๐‘Š2, 2-regularity by the difference quotient method.The key lemma, which is proved by convex analysis, is contained in the appendices (Lemma 8).

Section 3 provides a blow-up argument for convex weak solutions. The aim of Section 3 is to prove that๐‘ข0 : R๐‘› โ†’ R, a limit of rescaled solutions, satisfies ๐ฟ๐‘, ๐‘๐‘ข0 = 0 in the weak sense over the whole space R๐‘›(Proposition 1). To assure this, we will make use of an elementary result on a.e. convergence of gradients, whichis given in the appendices (Lemma 9).

Section 4 is devoted to justifications of maximum principles for the equation ๐ฟ๐‘, ๐‘๐‘ข = 0. We first givedefinitions of sub- and supersolution in the weak sense. Section 4.1 provides a justification of the comparisonprinciple (Proposition 2). Section 4.2 establishes an existence result of classical barrier subsolutions in an openannulus (Lemma 2). Applying these results in Section 4.1โ€“4.2, we prove the strong maximum principle outside thefacet (Theorem 2).

In Section 5, we will show the Liouville-type theorem (Theorem 3) by making use of Theorem 2, and completethe proof of our main theorem (Theorem 1).

Finally in Section 6, we discuss a few generalization of the operators ฮ”1 and ฮ”๐‘ . Since the general strategyfor the proof is the same, we only indicate modification of our arguments. Among them, we especially treatwith a Liouville-type theorem and a blow-up argument, since these proofs require basic facts of a general convexfunctional which is positively homogeneous of degree 1. These well-known facts are contained in the appendicesfor completeness.

2 Regularity outside the facetIn Section 2, we would like to show that ๐‘ข is ๐ถ1 at any ๐‘ฅ โˆˆ ๐ท, and therefore (1.4) holds for all ๐‘ฅ โˆˆ ๐ท. This resultwill be used in the proof of the strong maximum principle (Theorem 2).

We first give a precise definition of weak solutions to ๐ฟ๐‘, ๐‘๐‘ข = ๐‘“ in a convex domain ฮฉ โŠ‚ R๐‘›, which is notnecessarily bounded.

Definition 1. Let ฮฉ โŠ‚ R๐‘› be a domain, which is not necessarily bounded, and ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› < ๐‘ž โ‰ค โˆž). We saythat a function ๐‘ข โˆˆ๐‘Š1, ๐‘

loc (ฮฉ) is a weak solution to (1.1), when for any bounded Lipschitz domain ๐œ” โ‹ ฮฉ, there existsa vector field ๐‘ โˆˆ ๐ฟโˆž (๐œ”, R๐‘›) such that the pair (๐‘ข, ๐‘) โˆˆ๐‘Š1, ๐‘ (๐œ”) ร— ๐ฟโˆž (๐œ”, R๐‘›) satisfies

๐‘

โˆซ๐œ”โŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซ๐œ”

โŸจ|โˆ‡๐‘ข |๐‘โˆ’2โˆ‡๐‘ข

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ = โˆซ๐œ”๐‘“ ๐œ™ ๐‘‘๐‘ฅ (2.1)

6

for all ๐œ™ โˆˆ๐‘Š1, ๐‘0 (๐œ”), and

๐‘ (๐‘ฅ) โˆˆ ๐œ• | ยท | (โˆ‡๐‘ข(๐‘ฅ)) (2.2)

for a.e. ๐‘ฅ โˆˆ ๐œ”. For such pair (๐‘ข, ๐‘), we say that (๐‘ข, ๐‘) satisfies ๐ฟ๐‘, ๐‘๐‘ข = ๐‘“ in ๐‘Šโˆ’1, ๐‘โ€ฒ (๐œ”) or simply say that ๐‘ขsatisfies ๐ฟ๐‘, ๐‘๐‘ข = ๐‘“ in๐‘Šโˆ’1, ๐‘โ€ฒ (๐œ”). Here ๐‘โ€ฒ โˆˆ (1,โˆž) denotes the Hรถlder conjugate exponent of ๐‘ โˆˆ (1,โˆž).

The aim of Section 2 is to show Lemma 1 below.

Lemma 1. Let ๐‘ข be a convex weak solution to (1.1) in a convex domain ฮฉ โŠ‚ R๐‘›, and ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› < ๐‘ž โ‰ค โˆž). If๐‘ฅ0 โˆˆ ๐ท, then we can take a small radius ๐‘Ÿ0 > 0, a unit vector ๐œˆ0 โˆˆ R๐‘›, and a small number ๐œ‡ > 0 such that

๐ต๐‘Ÿ0 (๐‘ฅ0) โŠ‚ ๐ท and โŸจโˆ‡๐‘ข(๐‘ฅ) | ๐œˆ0โŸฉ โ‰ฅ ๐œ‡ for a.e. ๐‘ฅ โˆˆ ๐ต๐‘Ÿ0 (๐‘ฅ0), (2.3)

and there exists a small number ๐›ผ = ๐›ผ(๐œ‡) โˆˆ (0, 1) such that ๐‘ข โˆˆ ๐ถ1, ๐›ผ (๐ต๐‘Ÿ0/2 (๐‘ฅ0)). In particular, ๐‘ข is ๐ถ1 in ๐ท, and๐œ•๐‘ข(๐‘ฅ) = {โˆ‡๐‘ข(๐‘ฅ)} โ‰  {0} for all ๐‘ฅ โˆˆ ๐ท.

Before proving Lemma 1, we introduce difference quotients. For given ๐‘” : ฮฉโ†’R๐‘š (๐‘š โˆˆN), ๐‘— โˆˆ {1, . . . , ๐‘› }, โ„Ž โˆˆR \ {0}, we define

ฮ” ๐‘— , โ„Ž๐‘”(๐‘ฅ) B๐‘”(๐‘ฅ + โ„Ž๐‘’ ๐‘— ) โˆ’๐‘”(๐‘ฅ)

โ„Žโˆˆ R๐‘š for ๐‘ฅ โˆˆ ฮฉ with ๐‘ฅ + โ„Ž๐‘’ ๐‘— โˆˆ ฮฉ,

where ๐‘’ ๐‘— โˆˆ R๐‘› denotes the unit vector in the direction of the ๐‘ฅ ๐‘— -axis.In the proof of Lemma 1, we will use Lemma 7โ€“8 without proofs. For precise proofs, see Section A.

Proof. For each fixed ๐‘ฅ0 โˆˆ ๐ท, we may take and fix ๐‘ฅ1 โˆˆ ฮฉ such that ๐‘ข(๐‘ฅ0) > ๐‘ข(๐‘ฅ1). We set 3๐›ฟ0 B ๐‘ข(๐‘ฅ0) โˆ’๐‘ข(๐‘ฅ1) >0, ๐‘‘0 B |๐‘ฅ0 โˆ’ ๐‘ฅ1 | > 0 and ๐œˆ0 B ๐‘‘โˆ’1

0 (๐‘ฅ0 โˆ’ ๐‘ฅ1). By ๐‘ข โˆˆ ๐ถ (ฮฉ), we may take a sufficiently small ๐‘Ÿ0 > 0 such that

๐‘ข(๐‘ฆ0) โˆ’๐‘ข(๐‘ฆ1) โ‰ฅ ๐›ฟ0 > 0 for all ๐‘ฆ0 โˆˆ ๐ต๐‘Ÿ0 (๐‘ฅ0), ๐‘ฆ1 โˆˆ ๐ต๐‘Ÿ0 (๐‘ฅ1). (2.4)

From (2.4), the inclusion ๐ต๐‘Ÿ0 (๐‘ฅ0) โŠ‚ ๐ท clearly holds. (2.4) also allows us to check that for all ๐‘ฆ0 โˆˆ ๐ต๐‘Ÿ0 (๐‘ฅ0), ๐‘ง0 โˆˆ๐œ•๐‘ข(๐‘ฆ0),

โŸจ๐‘ง0 | ๐œˆ0โŸฉ โ‰ฅ๐‘ข(๐‘ฆ0) โˆ’๐‘ข(๐‘ฆ0 โˆ’ ๐‘‘๐œˆ0)

๐‘‘0โ‰ฅ ๐›ฟ0

๐‘‘0C ๐œ‡0 > 0. (2.5)

For the first inequality in (2.5), we have used Lemma 8, which is basically derived from convexity of ๐‘ข. Recall that๐œ•๐‘ข(๐‘ฅ) = {โˆ‡๐‘ข(๐‘ฅ)} for a.e. ๐‘ฅ โˆˆ ฮฉ, and hence we are able to recover (2.3) from (2.5).

In order to obtain ๐ถ1-regularity in ๐ท, we will appeal to the classical De Giorgiโ€“Nashโ€“Moser theory. Forpreliminaries, we check that the operator ๐ฟ๐‘, ๐‘๐‘ข assures uniform ellipticity in ๐ต๐‘Ÿ0 (๐‘ฅ0). Local Lipschitz continuityof ๐‘ข implies that there exists a sufficiently large number ๐‘€0 โˆˆ (0,โˆž) such that

ess sup๐ต๐‘Ÿ0 (๐‘ฅ0)

|โˆ‡๐‘ข | โ‰ค ๐‘€0 and |๐‘ข(๐‘ฅ) โˆ’๐‘ข(๐‘ฆ) | โ‰ค ๐‘€0 |๐‘ฅโˆ’ ๐‘ฆ | for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐ต๐‘Ÿ0 (๐‘ฅ0). (2.6)

For notational simplicity, we write subdomains by

๐‘ˆ1 B ๐ต๐‘Ÿ0 (๐‘ฅ0) โ‹‘ ๐‘ˆ2 B ๐ต15๐‘Ÿ0/16 (๐‘ฅ0) โ‹‘ ๐‘ˆ3 B ๐ต7๐‘Ÿ0/8 (๐‘ฅ0) โ‹‘ ๐‘ˆ4 B ๐ต3๐‘Ÿ0/4 (๐‘ฅ0) โ‹‘ ๐‘ˆ5 B ๐ต๐‘Ÿ0/2 (๐‘ฅ0).

It should be noted that ๐ธ (๐‘ง) B ๐‘ |๐‘ง | + |๐‘ง |๐‘/๐‘ (๐‘ง โˆˆ R๐‘›) satisfies ๐ธ โˆˆ ๐ถโˆž (R๐‘› \ {0}), and there exists two constants0 < ๐œ†(๐‘, ๐œ‡0, ๐‘€0) โ‰ค ฮ›(๐‘, ๐‘, ๐œ‡0, ๐‘€0) <โˆž such that

๐œ† |๐œ |2 โ‰คโŸจโˆ‡2๐‘ง๐ธ (๐‘ง0)๐œ

๏ฟฝ๏ฟฝ ๐œโŸฉ (2.7)โŸจโˆ‡2๐‘ง๐ธ (๐‘ง0)๐œ

๏ฟฝ๏ฟฝ ๐œ”โŸฉ โ‰ค ฮ›|๐œ | |๐œ” | (2.8)

for all ๐‘ง0, ๐œ , ๐œ” โˆˆ R๐‘› with ๐œ‡0 โ‰ค |๐‘ง0 | โ‰ค ๐‘€0. We can explicitly determine 0 < ๐œ† โ‰ค ฮ› <โˆž by{๐œ†(๐‘, ๐œ‡0, ๐‘€0) B min๐œ‡0โ‰ค๐‘กโ‰ค๐‘€0

(min{1, ๐‘โˆ’1 }๐‘ก ๐‘โˆ’2) ,

ฮ›(๐‘, ๐‘, ๐œ‡0, ๐‘€0) B max๐œ‡0โ‰ค๐‘กโ‰ค๐‘€0

(๐‘๐‘กโˆ’1 +max{1, ๐‘โˆ’1 }๐‘ก ๐‘โˆ’2)

Now we check that ๐‘ข โˆˆ๐‘Š2, 2 (๐‘ˆ4) by the difference quotient method. We refer the reader to [18, Theorem 8.1]as a related result. By [18, Lemma 8.2], it suffices to check that

sup{โˆซ๐‘ˆ4

|โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) |2 ๐‘‘๐‘ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ„Ž โˆˆ R, 0 < |โ„Ž| < ๐‘Ÿ0

16

}<โˆž for each ๐‘— โˆˆ {1, . . . , ๐‘› }. (2.9)

7

Since ๐‘ข โˆˆ๐‘Š1, ๐‘ (๐‘ˆ1) satisfies ๐ฟ๐‘, ๐‘๐‘ข = ๐‘“ in๐‘Šโˆ’1, ๐‘โ€ฒ (๐‘ˆ1), we obtainโˆซ๐‘ˆ1

โŸจโˆ‡๐‘ง๐ธ (โˆ‡๐‘ข) | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ =โˆซ๐‘ˆ1

๐‘“ ๐œ™ ๐‘‘๐‘ฅ (2.10)

for all ๐œ™ โˆˆ ๐‘Š1, ๐‘0 (๐‘ˆ1). Here we note that โˆ‡๐‘ข no longer degenerates in ๐‘ˆ1 by (2.3). We fix a cutoff function

๐œ‚ โˆˆ ๐ถ1๐‘ (๐‘ˆ3) such that

0 โ‰ค ๐œ‚ โ‰ค 1 in๐‘ˆ3, ๐œ‚ โ‰ก 1 in๐‘ˆ4, |โˆ‡๐œ‚ | โ‰ค๐‘

๐‘Ÿ0(2.11)

for some constant ๐‘ > 0. For each fixed ๐‘— โˆˆ {1, . . . , ๐‘› }, โ„Ž โˆˆ R with 0 < |โ„Ž| < ๐‘Ÿ0/16, we test ๐œ™ B ฮ” ๐‘— ,โˆ’โ„Ž (๐œ‚2ฮ” ๐‘— , โ„Ž๐‘ข)into (2.10). We note that ๐œ™ โˆˆ ๐‘Š1,โˆž (๐‘ˆ1) โŠ‚ ๐‘Š1, ๐‘ (๐‘ˆ1) by (2.8), and this is compactly supported in ๐‘ˆ2. Hence๐œ™ โˆˆ๐‘Š1, ๐‘

0 (๐‘ˆ2) is an admissible test function. By testing ๐œ™, we have

0 =โˆซ๐‘ˆ2

โŸจฮ” ๐‘— , โ„Ž (โˆ‡๐‘ง๐ธ (โˆ‡๐‘ข(๐‘ฅ)))

๏ฟฝ๏ฟฝ ๐œ‚2โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) +2๐œ‚ฮ” ๐‘— , โ„Ž๐‘ขโˆ‡๐œ‚โŸฉโˆ’โˆซ๐‘ˆ2

๐‘“ฮ”โˆ’ ๐‘— , โ„Ž (๐œ‚2ฮ” ๐‘— , โ„Ž๐‘ข) ๐‘‘๐‘ฅ

=โˆซ๐‘ˆ2

๐œ‚2โŸจ๐ดโ„Ž (๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ))โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) ๏ฟฝ๏ฟฝ โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข)โŸฉ ๐‘‘๐‘ฅ+2

โˆซ๐‘ˆ2

๐œ‚ฮ” ๐‘— , โ„Ž๐‘ขโŸจ๐ดโ„Ž (๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ))โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข)

๏ฟฝ๏ฟฝ โˆ‡๐œ‚โŸฉ ๐‘‘๐‘ฅโˆ’โˆซ๐‘ˆ2

๐‘“ฮ”โˆ’ ๐‘— , โ„Ž (๐œ‚2ฮ” ๐‘— , โ„Ž๐‘ข) ๐‘‘๐‘ฅ

C ๐ผ1 + ๐ผ2 โˆ’ ๐ผ3. (2.12)

Here ๐ดโ„Ž = ๐ดโ„Ž (๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ)) denotes a matrix-valued function in๐‘ˆ2 given by

๐ดโ„Ž (๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ)) Bโˆซ 1

0โˆ‡2๐‘ง๐ธ ((1โˆ’ ๐‘ก)โˆ‡๐‘ข(๐‘ฅ) + ๐‘กโˆ‡๐‘ข(๐‘ฅ + โ„Ž๐‘’ ๐‘— )) ๐‘‘๐‘ก.

We note that with the aid of (2.3)โ€“(2.6), we obtain

๐œ‡0 โ‰ค |(1โˆ’ ๐‘ก)โˆ‡๐‘ข(๐‘ฅ) + ๐‘กโˆ‡๐‘ข(๐‘ฅ + โ„Ž๐‘’ ๐‘— ) |โ‰ค ๐‘€0

for a.e. ๐‘ฅ โˆˆ๐‘ˆ2 and for all 0 โ‰ค ๐‘ก โ‰ค 1. Combining this result with (2.7)โ€“(2.8), we conclude that ๐ดโ„Ž satisfies

๐œ† |๐œ |2 โ‰ค โŸจ๐ดโ„Ž (๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ))๐œ | ๐œโŸฉ (2.13)

โŸจ๐ดโ„Ž (๐‘ฅ, โˆ‡๐‘ข(๐‘ฅ))๐œ | ๐œ”โŸฉ โ‰ค ฮ›|๐œ | |๐œ” | (2.14)

for all ๐œ, ๐œ” โˆˆ R๐‘› and for a.e. ๐‘ฅ โˆˆ๐‘ˆ2. We set an integral

๐ฝ Bโˆซ๐‘ˆ2

๐œ‚2 |โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) |2 ๐‘‘๐‘ฅ.

By (2.13), it is clear that ๐ผ1 โ‰ฅ ๐œ†๐ฝ. By Youngโ€™s inequality and applying a Poincarรฉ-type inequality (Lemma 7) to๐œ‚2ฮ” ๐‘— , โ„Ž๐‘ข โˆˆ๐‘Š1, 2

0 (๐‘ˆ2), we obtain for any ๐œ€ > 0,

|๐ผ3 | โ‰ค14๐œ€

โˆฅ ๐‘“ โˆฅ2๐ฟ2 (๐‘ˆ2) + ๐œ€

โˆซ๐‘ˆ2

|โˆ‡(๐œ‚2ฮ” ๐‘— , โ„Ž๐‘ข) |2 ๐‘‘๐‘ฅ

โ‰ค 14๐œ€

โˆฅ ๐‘“ โˆฅ2๐ฟ2 (๐‘ˆ2) +4๐œ€

โˆซ๐‘ˆ2

|ฮ” ๐‘— , โ„Ž๐‘ข |2 |โˆ‡๐œ‚ |2 ๐‘‘๐‘ฅ +2๐œ€โˆซ๐‘ˆ2

๐œ‚2 |โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) |2 ๐‘‘๐‘ฅ.

Here we have invoked the property 0 โ‰ค ๐œ‚ โ‰ค 1 in ๐‘ˆ2. We fix ๐œ€ B ๐œ†/6 > 0. By (2.14) and Youngโ€™s inequality, wehave

|๐ผ2 | โ‰ค 2ฮ›โˆซ๐‘ˆ2

๐œ‚ |โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) | ยท |ฮ” ๐‘— , โ„Ž๐‘ข | |โˆ‡๐œ‚ | ๐‘‘๐‘ฅ

โ‰ค ๐œ†

3๐ฝ + 3ฮ›2

๐œ†

โˆซ๐‘ˆ2

|ฮ” ๐‘— , โ„Ž๐‘ข |2 |โˆ‡๐œ‚ |2 ๐‘‘๐‘ฅ.

8

It follows from (2.6) that โˆฅฮ” ๐‘— , โ„Ž๐‘ขโˆฅ๐ฟโˆž (๐‘ˆ2) โ‰ค ๐‘€0. Therefore we obtain from (2.12),โˆซ๐‘ˆ4

|โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) |2 ๐‘‘๐‘ฅ โ‰ค ๐ฝ =โˆซ๐‘ˆ2

๐œ‚2 |โˆ‡(ฮ” ๐‘— , โ„Ž๐‘ข) |2 ๐‘‘๐‘ฅ โ‰ค ๐ถ (๐œ†, ฮ›)(๐‘€2

0 โˆฅโˆ‡๐œ‚โˆฅ2๐ฟ2 (๐‘ˆ2) + โˆฅ ๐‘“ โˆฅ2

๐ฟ2 (๐‘ˆ2)

).

The estimate (2.9) follows from this, and therefore ๐‘ข โˆˆ๐‘Š2, 2 (๐‘ˆ4).For each ๐œ“ โˆˆ ๐ถโˆž

๐‘ (๐‘ˆ4), we test ๐œ•๐‘ฅ ๐‘—๐œ“ โˆˆ ๐ถโˆž๐‘ (๐‘ˆ4) into (2.10). Integrating by parts, we obtainโˆซ

๐‘ˆ4

โŸจโˆ‡2๐‘ง๐ธ (โˆ‡๐‘ข)โˆ‡๐œ•๐‘ฅ ๐‘—๐‘ข

๏ฟฝ๏ฟฝ โˆ‡๐œ“โŸฉ ๐‘‘๐‘ฅ = โˆ’โˆซ๐‘ˆ4

๐‘“ ๐œ•๐‘ฅ ๐‘—๐œ“ ๐‘‘๐‘ฅ (2.15)

for all ๐œ“ โˆˆ ๐ถโˆž๐‘ (๐‘ˆ4). Noting that ๐‘“ โˆˆ ๐ฟ๐‘ž (๐‘ˆ4) โŠ‚ ๐ฟ2 (๐‘ˆ4), ๐œ•๐‘ฅ ๐‘—๐‘ข โˆˆ ๐‘Š1, 2 (๐‘ˆ4), and (2.7)โ€“(2.8), we may extend ๐œ“ โˆˆ

๐‘Š1, 20 (๐‘ˆ4) by a density argument. The conditions (2.7)โ€“(2.8) imply that โˆ‡2

๐‘ง๐ธ (โˆ‡๐‘ข) is uniformly elliptic over ๐‘ˆ1.Hence by [17, Theorem 8.22], there exists๐›ผ =๐›ผ(๐œ†, ฮ›, ๐‘›, ๐‘ž) โˆˆ (0, 1) such that ๐œ•๐‘ฅ ๐‘—๐‘ข โˆˆ๐ถ๐›ผ (๐‘ˆ5) for each ๐‘— โˆˆ {1, . . . , ๐‘› }.This regularity result implies ๐œ•๐‘ข(๐‘ฅ) = {โˆ‡๐‘ข(๐‘ฅ)} โ‰  {0} for all ๐‘ฅ โˆˆ ๐ท. โ–ก

3 A blow-up argumentIn order to show that (1.4) holds true even for ๐‘ฅ โˆˆ ๐น, we first make a blow-argument and construct a convex weaksolution in the whole space R๐‘›, in the sense of Definition 1.

Proposition 1. Let ฮฉ โŠ‚ R๐‘› be a convex domain, and ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› < ๐‘ž โ‰ค โˆž). Assume that ๐‘ข is a convex weaksolution to (1.1), and ๐‘ฅ0 โˆˆ ฮฉ. Then there exists a convex function ๐‘ข0 : R๐‘› โ†’ R such that

1. ๐‘ข0 is a weak solution to ๐ฟ๐‘, ๐‘๐‘ข0 = 0 in R๐‘›.

2. The inclusion ๐œ•๐‘ข(๐‘ฅ0) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ0) holds. That is, if ๐‘ โˆˆ ๐œ•๐‘ข(๐‘ฅ0), then we have

๐‘ข0 (๐‘ฅ) โ‰ฅ ๐‘ข0 (๐‘ฅ0) + โŸจ๐‘ | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ for all ๐‘ฅ โˆˆ R๐‘›.

In particular, if ๐‘ฅ0 โˆˆ ๐น, then the facet of ๐‘ข0 is non-empty.

Proof. Without loss of generality, we may assume that ๐‘ฅ0 = 0 and ๐‘ข(๐‘ฅ0) = 0. First we fix a closed ball ๐ต๐‘… (0) =๐ต๐‘… โŠ‚ ฮฉ. We note that ๐‘ข โˆˆ Lip(๐ต๐‘…) since ๐‘ข is convex. Hence there exists a sufficiently large number ๐‘€ โˆˆ (0,โˆž)such that

ess sup๐ต๐‘…

|โˆ‡๐‘ข | โ‰ค ๐‘€ and |๐‘ข(๐‘ฅ) โˆ’๐‘ข(๐‘ฆ) | โ‰ค ๐‘€ |๐‘ฅโˆ’ ๐‘ฆ | for all ๐‘ฅ, ๐‘ฆ โˆˆ ๐ต๐‘… .

We take and fix a vector field ๐‘ โˆˆ ๐ฟโˆž (๐ต๐‘…, R๐‘›) such that the pair (๐‘ข, ๐‘) โˆˆ ๐‘Š1, ๐‘ (๐ต๐‘…) ร— ๐ฟโˆž (๐ต๐‘…, R๐‘›) satisfies๐ฟ๐‘, ๐‘๐‘ข = ๐‘“ in๐‘Šโˆ’1, ๐‘โ€ฒ (๐ต๐‘…). For each ๐‘Ž > 0, we define a rescaled convex function ๐‘ข๐‘Ž : ๐ต๐‘…/๐‘Ž โ†’R and a dilated vectorfield ๐‘๐‘Ž โˆˆ ๐ฟโˆž (๐ต๐‘…/๐‘Ž, R๐‘›) by

๐‘ข๐‘Ž (๐‘ฅ) B๐‘ข(๐‘Ž๐‘ฅ)๐‘Ž

, ๐‘๐‘Ž (๐‘ฅ) B ๐‘ (๐‘Ž๐‘ฅ) for ๐‘ฅ โˆˆ ๐ต๐‘…/๐‘Ž .

We also set ๐‘“๐‘Ž โˆˆ ๐ฟ๐‘ž (๐ต๐‘…/๐‘Ž) by๐‘“๐‘Ž (๐‘ฅ) B ๐‘Ž ๐‘“ (๐‘Ž๐‘ฅ) for ๐‘ฅ โˆˆ ๐ต๐‘…/๐‘Ž .

Then it is easy to check that the pair (๐‘ข๐‘Ž, ๐‘๐‘Ž) โˆˆ๐‘Š1,โˆž (๐ต๐‘…/๐‘Ž)ร—๐ฟโˆž (๐ต๐‘…/๐‘Ž, R๐‘›) satisfies ๐ฟ๐‘, ๐‘๐‘ข๐‘Ž = ๐‘“๐‘Ž in๐‘Šโˆ’1, ๐‘โ€ฒ (๐ต๐‘…/๐‘Ž).By definition of ๐‘ข๐‘Ž, we clearly have

sup๐ต๐‘…/๐‘Ž

|๐‘ข๐‘Ž | โ‰ค ๐‘€ <โˆž, โˆฅโˆ‡๐‘ข๐‘Žโˆฅ๐ฟโˆž (๐ต๐‘…/๐‘Ž) โ‰ค ๐‘€ <โˆž for all ๐‘Ž > 0. (3.1)

Hence by the Arzelร โ€“Ascoli theorem and a diagonal argument, we can take a decreasing sequence {๐‘Ž๐‘ }โˆž๐‘=1 โŠ‚(0,โˆž), such that ๐‘Ž๐‘ โ†’ 0 as ๐‘ โ†’โˆž, and

๐‘ข๐‘Ž๐‘ โ†’ ๐‘ข0 locally uniformly in R๐‘›. (3.2)

9

for some function ๐‘ข0 : R๐‘› โ†’ R. Clearly ๐‘ข0 is convex in R๐‘›, and the inclusion ๐œ•๐‘ข(๐‘ฅ0) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ0) holds true by theconstruction of rescaled functions ๐‘ข๐‘Ž. If ๐‘ฅ0 โˆˆ ๐น, then we have {0} โŠ‚ ๐œ•๐‘ข(๐‘ฅ0) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ0) and therefore ๐‘ฅ0 lies in thefacet of ๐‘ข0. We are left to show that ๐‘ข0 is a weak solution to ๐ฟ๐‘, ๐‘๐‘ข0 = 0 in R๐‘›. Before proving this, we note thatfrom (3.1)โ€“(3.2) and Lemma 9, it follows that

โˆ‡๐‘ข๐‘Ž๐‘ (๐‘ฅ) โ†’ โˆ‡๐‘ข0 (๐‘ฅ) and |โˆ‡๐‘ข0 (๐‘ฅ) | โ‰ค ๐‘€ for a.e. ๐‘ฅ โˆˆ R๐‘› (3.3)

as ๐‘ โ†’โˆž. We arbitrarily fix an open ball ๐ต๐‘Ÿ โŠ‚ R๐‘›. Note that the inclusion ๐ต๐‘Ÿ โŠ‚ ๐ต๐‘…/๐‘Ž holds for all 0 < ๐‘Ž < ๐‘…/๐‘Ÿ .Hence we easily realize that a family of pairs {(๐‘ข๐‘Ž, ๐‘๐‘Ž)}0<๐‘Ž<๐‘…/๐‘Ÿ โŠ‚๐‘Š1,โˆž (๐ต๐‘Ÿ ) ร— ๐ฟโˆž (๐ต๐‘Ÿ , R๐‘›) satisfies

๐‘๐‘Ž (๐‘ฅ) โˆˆ ๐œ• | ยท | (โˆ‡๐‘ข๐‘Ž (๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ ๐ต๐‘Ÿ , (3.4)

๐‘

โˆซ๐ต๐‘Ÿ

โŸจ๐‘๐‘Ž | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +โˆซ๐ต๐‘Ÿ

โŸจ|โˆ‡๐‘ข๐‘Ž |๐‘โˆ’2โˆ‡๐‘ข๐‘Ž

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ = โˆซ๐ต๐‘Ÿ

๐‘“๐‘Ž๐œ™๐‘‘๐‘ฅ for all ๐œ™ โˆˆ๐‘Š1, ๐‘0 (๐ต๐‘Ÿ ). (3.5)

By definition of ๐‘“๐‘Ž, we get โˆฅ ๐‘“๐‘Žโˆฅ๐ฟ๐‘ž (๐ต๐‘Ÿ ) = ๐‘Ž1โˆ’๐‘›/๐‘ž โˆฅ ๐‘“ โˆฅ๐ฟ๐‘ž (๐ต๐‘Ž๐‘Ÿ ) โ‰ค ๐‘Ž1โˆ’๐‘›/๐‘ž โˆฅ ๐‘“ โˆฅ๐ฟ๐‘ž (๐ต๐‘…) for all 0 < ๐‘Ž < ๐‘…/๐‘Ÿ . Hence by the

continuous embedding ๐ฟ๐‘ž (๐ต๐‘Ÿ ) โ†ฉโ†’๐‘Šโˆ’1, ๐‘โ€ฒ (๐ต๐‘Ÿ ), we obtain

๐‘“๐‘Ž๐‘ โ†’ 0 in๐‘Šโˆ’1, ๐‘โ€ฒ (๐ต๐‘Ÿ ) as ๐‘ โ†’โˆž. (3.6)

By (3.1) and (3.3), we can apply Lebesgueโ€™s dominated convergence theorem and get

|โˆ‡๐‘ข๐‘Ž๐‘ |๐‘โˆ’2โˆ‡๐‘ข๐‘Ž๐‘ โ†’ |โˆ‡๐‘ข0 |๐‘โˆ’2โˆ‡๐‘ข0 in ๐ฟ ๐‘โ€ฒ (๐ต๐‘Ÿ , R๐‘›) as ๐‘ โ†’โˆž. (3.7)

It is clear that โˆฅ๐‘๐‘Žโˆฅ๐ฟโˆž (๐ต๐‘Ÿ ,R๐‘›) โ‰ค 1 for all 0 < ๐‘Ž < ๐‘…/๐‘Ÿ . Hence by [5, Corollary 3.30], up to a subsequence, we mayassume that

๐‘๐‘Ž๐‘โˆ—โ‡€ ๐‘0, ๐‘Ÿ in ๐ฟโˆž (๐ต๐‘Ÿ , R๐‘›) as ๐‘ โ†’โˆž (3.8)

for some ๐‘0, ๐‘Ÿ โˆˆ ๐ฟโˆž (๐ต๐‘Ÿ , R๐‘›). By lower-semicontinuity of the norm with respect to the weakโˆ— topology and(3.3)โ€“(3.4), we get

โˆฅ๐‘0, ๐‘Ÿ โˆฅ๐ฟโˆž (๐ต๐‘Ÿ ,R๐‘›) โ‰ค 1, ๐‘0, ๐‘Ÿ (๐‘ฅ) =โˆ‡๐‘ข0 (๐‘ฅ)|โˆ‡๐‘ข0 (๐‘ฅ) |

for a.e. ๐‘ฅ โˆˆ ๐ต๐‘Ÿ with โˆ‡๐‘ข0 (๐‘ฅ) โ‰  0,

which implies that๐‘0, ๐‘Ÿ (๐‘ฅ) โˆˆ ๐œ• | ยท | (โˆ‡๐‘ข0 (๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ ๐ต๐‘Ÿ . (3.9)

Letting ๐‘Ž = ๐‘Ž๐‘ in (3.5) and ๐‘ โ†’โˆž, we obtain

๐‘

โˆซ๐ต๐‘Ÿ

โŸจ๐‘0, ๐‘Ÿ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +โˆซ๐ต๐‘Ÿ

โŸจ|โˆ‡๐‘ข0 |๐‘โˆ’2โˆ‡๐‘ข0

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ = 0 for all ๐œ™ โˆˆ๐‘Š1, ๐‘0 (๐ต๐‘Ÿ ) (3.10)

by (3.5)โ€“(3.8). Since ๐ต๐‘Ÿ โŠ‚ R๐‘› is arbitrary, (3.9)โ€“(3.10) means that ๐‘ข0 is a weak solution to ๐ฟ๐‘, ๐‘๐‘ข0 = 0 in R๐‘›, inthe sense of Definition 1. โ–ก

4 Maximum principlesIn Section 4, we justify maximum principles for the equation ๐ฟ๐‘, ๐‘๐‘ข = 0.

We first define subsolutions and supersolutions in the weak sense.

Definition 2. Letฮฉ โŠ‚ R๐‘› be a bounded domain. A pair (๐‘ข, ๐‘) โˆˆ๐‘Š1, ๐‘ (ฮฉ) ร—๐ฟโˆž (ฮฉ, R๐‘›) is called a weak subsolutionto ๐ฟ๐‘, ๐‘๐‘ข = 0 in ฮฉ, if it satisfies

๐‘

โˆซฮฉโŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซฮฉ

โŸจ|โˆ‡๐‘ข |๐‘โˆ’2โˆ‡๐‘ข

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ โ‰ค 0 (4.1)

for all 0 โ‰ค ๐œ™ โˆˆ ๐ถโˆž๐‘ (ฮฉ), and

๐‘ (๐‘ฅ) โˆˆ ๐œ• | ยท | (โˆ‡๐‘ข(๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ ฮฉ. (4.2)

10

Similarly we call a pair (๐‘ข, ๐‘) โˆˆ๐‘Š1, ๐‘ (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) a weak supersolution ๐ฟ๐‘, ๐‘๐‘ข = 0 in ฮฉ, if it satisfies (4.2)and

๐‘

โˆซฮฉโŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซฮฉ

โŸจ|โˆ‡๐‘ข |๐‘โˆ’2โˆ‡๐‘ข

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ โ‰ฅ 0 (4.3)

for all 0 โ‰ค ๐œ™ โˆˆ ๐ถโˆž๐‘ (ฮฉ). For ๐‘ข โˆˆ๐‘Š1, ๐‘ (ฮฉ), we simply say that ๐‘ข is respectively a subsolution and a supersolution to

๐ฟ๐‘, ๐‘๐‘ข = 0 in the weak sense if there is ๐‘ โˆˆ ๐ฟโˆž (ฮฉ, R๐‘›) such that the pair (๐‘ข, ๐‘) is a weak subsolution and a weaksupersolution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in ฮฉ.

Remark 2. We describe some remarks on our definitions of weak solutions, subsolutions and supersolutions.

1. By an approximation argument, we may extend the test function class of (4.1) to

๐ท+ (ฮฉ) B {๐œ™ โˆˆ๐‘Š1, ๐‘ (ฮฉ) | ๐œ™ โ‰ฅ 0 a.e. in ฮฉ, supp๐œ™ โŠ‚ ฮฉ}.

Indeed, for ๐œ™ โˆˆ ๐ท+ (ฮฉ) and 0 < ๐œ€ < dist(supp๐œ™, ๐œ•ฮฉ), the function,

๐œ™๐œ€ (๐‘ฅ) =โˆซฮฉ๐œ™(๐‘ฅโˆ’ ๐‘ฆ)๐œŒ๐œ€ (๐‘ฆ) ๐‘‘๐‘ฆ for ๐‘ฅ โˆˆ ฮฉ

satisfies 0 โ‰ค ๐œ™๐œ€ โˆˆ ๐ถโˆž๐‘ (ฮฉ). Here for 0 < ๐œ€ <โˆž, 0 โ‰ค ๐œŒ๐œ€ โˆˆ ๐ถโˆž

๐‘ (๐ต๐œ€ (0)) denotes a standard mollifier so that

0 โ‰ค ๐œŒ โˆˆ ๐ถโˆž๐‘ (๐ต1), โˆฅ๐œŒโˆฅ๐ฟ1 (R๐‘›) = 1, ๐œŒ๐œ€ (๐‘ฅ) B ๐œ€โˆ’๐‘›๐œŒ(๐‘ฅ/๐œ€) for ๐‘ฅ โˆˆ R๐‘›.

By testing ๐œ™๐œ€ into (4.1) for sufficiently small ๐œ€ > 0 and letting ๐œ€โ†’ 0, we conclude that if the pair (๐‘ข, ๐‘)satisfies (4.1) for all 0 โ‰ค ๐œ™ โˆˆ ๐ถโˆž

๐‘ (ฮฉ), then (4.1) holds for all ๐œ™ โˆˆ ๐ท+ (ฮฉ). A similar result is also valid for(4.3).

2. By Definition 1โ€“2, if a pair (๐‘ข, ๐‘) โˆˆ ๐‘Š1, ๐‘ (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) satisfies ๐ฟ๐‘, ๐‘๐‘ข = 0 in ๐‘Šโˆ’1, ๐‘โ€ฒ (ฮฉ), then ๐‘ข isclearly both a subsolution and a supersolution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in ฮฉ in the weak sense. Conversely, if a pair(๐‘ข, ๐‘) โˆˆ๐‘Š1, ๐‘ (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) is both a weak subsolution and a weak supersolution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in ฮฉ, thenthe pair (๐‘ข, ๐‘) satisfies ๐ฟ๐‘, ๐‘๐‘ข = 0 in๐‘Šโˆ’1, ๐‘โ€ฒ (ฮฉ). Indeed, by the previous remark we have already known thatthe pair (๐‘ข, ๐‘) satisfies (4.1) and (4.3) for all ๐œ™ โˆˆ ๐ท+ (ฮฉ), which clearly yields

๐‘

โˆซฮฉโŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซฮฉ

โŸจ|โˆ‡๐‘ข |๐‘โˆ’2โˆ‡๐‘ข

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ = 0 (4.4)

for all ๐œ™ โˆˆ ๐ท+ (ฮฉ). We decompose arbitrary ๐œ™ โˆˆ ๐ถโˆž๐‘ (ฮฉ) by ๐œ™ = ๐œ™+ โˆ’ ๐œ™โˆ’, where ๐œ™+ B max{๐œ™, 0 }, ๐œ™โˆ’ B

max{โˆ’๐œ™, 0 } โˆˆ ๐ท+ (ฮฉ). By testing ๐œ™+, ๐œ™โˆ’ โˆˆ ๐ท+ (ฮฉ) into (4.4), we conclude that (4.4) holds for all ๐œ™ โˆˆ๐ถโˆž๐‘ (ฮฉ).

By density of ๐ถโˆž๐‘ (ฮฉ) โŠ‚๐‘Š1, ๐‘

0 (ฮฉ), it is clear that (4.4) is valid for all ๐œ™ โˆˆ๐‘Š1, ๐‘0 (ฮฉ).

3. For a bounded domain ฮฉ โŠ‚ R๐‘›, let ๐‘ข โˆˆ ๐ถ2 (ฮฉ) satisfy the following two conditions (4.5)โ€“(4.6);

โˆ‡๐‘ข(๐‘ฅ) โ‰  0 for all ๐‘ฅ โˆˆ ฮฉ, (4.5)

(๐ฟ๐‘, ๐‘๐‘ข)(๐‘ฅ) = โˆ’(๐‘ฮ”1๐‘ข +ฮ”๐‘๐‘ข)(๐‘ฅ) โ‰ค 0 for all ๐‘ฅ โˆˆ ฮฉ. (4.6)

Then for any fixed 0 โ‰ค ๐œ™ โˆˆ ๐ถโˆž๐‘ (ฮฉ), we have

0 โ‰ฅโˆซฮฉ(๐ฟ๐‘, ๐‘๐‘ข)๐œ™๐‘‘๐‘ฅ = ๐‘

โˆซฮฉ

โŸจโˆ‡๐‘ข|โˆ‡๐‘ข |

๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +โˆซฮฉ

โŸจ|โˆ‡๐‘ข |๐‘โˆ’2โˆ‡๐‘ข

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ,with the aid of integration by parts and (4.6). We also note that

๐œ• | ยท | (โˆ‡๐‘ข(๐‘ฅ)) ={โˆ‡๐‘ข(๐‘ฅ)|โˆ‡๐‘ข(๐‘ฅ) |

}for all ๐‘ฅ โˆˆ ฮฉ

by (4.5). Therefore the pair (๐‘ข, โˆ‡๐‘ข/|โˆ‡๐‘ข |) โˆˆ ๐‘Š1, ๐‘ (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) satisfies (4.1)โ€“(4.2). For such ๐‘ข, wesimply say that ๐‘ข satisfies ๐ฟ๐‘, ๐‘๐‘ข โ‰ค 0 in ฮฉ in the classical sense.

11

4.1 Comparison principleWe justify the comparison principle, i.e., for any subsolution ๐‘ขโˆ’ and supersolution ๐‘ข+,

๐‘ขโˆ’ โ‰ค ๐‘ข+ on ๐œ•ฮฉ implies that ๐‘ขโˆ’ โ‰ค ๐‘ข+ in ฮฉ,

under the condition that ๐‘ข+ and ๐‘ขโˆ’ admits continuity properties in ฮฉ.

Proposition 2. Let ฮฉ โŠ‚ R๐‘› be a bounded domain. Assume that ๐‘ข+, ๐‘ขโˆ’ โˆˆ ๐ถ (ฮฉ) โˆฉ๐‘Š1, ๐‘ (ฮฉ) is a subsolution and asupersolution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in the weak sense respectively. If ๐‘ข+, ๐‘ขโˆ’ satisfies

๐‘ขโˆ’ (๐‘ฅ) โ‰ค ๐‘ข+ (๐‘ฅ) for all ๐‘ฅ โˆˆ ๐œ•ฮฉ, (4.7)

then ๐‘ขโˆ’ โ‰ค ๐‘ข+ in ฮฉ.

Before proving Proposition 2, we recall that the mapping ๐ด : R๐‘› โˆ‹ ๐‘ง โ†ฆโ†’ |๐‘ง |๐‘โˆ’2๐‘ง โˆˆ R๐‘› satisfies strict monotonicity,

i.e., โŸจ๐ด(๐‘ง2) โˆ’ ๐ด(๐‘ง1) | ๐‘ง2 โˆ’ ๐‘ง1โŸฉ > 0 for all ๐‘ง1, ๐‘ง2 โˆˆ R๐‘› with ๐‘ง1 โ‰  ๐‘ง2. (4.8)

Proof. We take arbitrary ๐›ฟ > 0. By ๐‘ข+, ๐‘ขโˆ’ โˆˆ ๐ถ (ฮฉ) and (4.7), we can take a subdomain ฮฉโ€ฒ โ‹ ฮฉ such that ๐‘ขโˆ’ โ‰ค๐‘ข+ + ๐›ฟ in ฮฉ \ฮฉโ€ฒ. This implies that the support of the truncated non-negative function ๐‘ค ๐›ฟ B (๐‘ข+โˆ’๐‘ขโˆ’ + ๐›ฟ)โˆ’ โˆˆ๐‘Š1, ๐‘ (ฮฉ) is contained in ฮฉโ€ฒ โŠŠ ฮฉ and therefore ๐‘ค ๐›ฟ โˆˆ ๐ท+ (ฮฉ). Let ๐‘+, ๐‘โˆ’ โˆˆ ๐ฟโˆž (ฮฉ, R๐‘›) be vector fields such that(๐‘ข+, ๐‘+), (๐‘ขโˆ’, ๐‘โˆ’) satisfies (4.1)โ€“(4.2), (4.2)โ€“(4.3) respectively. As in Remark 2, we may test ๐‘ค ๐›ฟ in (4.1) and (4.3).Note that โˆ‡๐‘ค ๐›ฟ = โˆ’๐œ’๐›ฟโˆ‡(๐‘ข+โˆ’๐‘ขโˆ’), where ๐œ’๐›ฟ denotes the characteristic function of ๐ด๐›ฟ B {๐‘ฅ โˆˆ ฮฉ | ๐‘ข+ + ๐›ฟ โ‰ค ๐‘ขโˆ’}.Hence, we have

0 โ‰ค โˆ’๐‘โˆซ๐ด๐›ฟ

โŸจ๐‘+โˆ’ ๐‘โˆ’

๏ฟฝ๏ฟฝ โˆ‡๐‘ข+โˆ’โˆ‡๐‘ขโˆ’โŸฉ๐‘‘๐‘ฅโˆ’

โˆซ๐ด๐›ฟ

โŸจ|โˆ‡๐‘ข+ |๐‘โˆ’2โˆ‡๐‘ข+โˆ’ |โˆ‡๐‘ขโˆ’ |๐‘โˆ’2โˆ‡๐‘ขโˆ’

๏ฟฝ๏ฟฝ โˆ‡๐‘ข+โˆ’โˆ‡๐‘ขโˆ’โŸฉ๐‘‘๐‘ฅ

โ‰ค โˆ’โˆซ๐ด๐›ฟ

โŸจ|โˆ‡๐‘ข+ |๐‘โˆ’2โˆ‡๐‘ข+โˆ’ |โˆ‡๐‘ขโˆ’ |๐‘โˆ’2โˆ‡๐‘ขโˆ’

๏ฟฝ๏ฟฝ โˆ‡๐‘ข+โˆ’โˆ‡๐‘ขโˆ’โŸฉ๐‘‘๐‘ฅ.

Here we have invoked (4.2) and monotonicity of the subdifferential operator ๐œ• | ยท |. From (4.8) we can easily checkthat โˆ‡๐‘ข+ = โˆ‡๐‘ขโˆ’ in ๐ด๐›ฟ , and therefore ๐‘ค ๐›ฟ = 0 in ๐‘Š1, ๐‘

0 (ฮฉ). This means that ๐‘ขโˆ’ โ‰ค ๐‘ข+ + ๐›ฟ a.e. in ฮฉ. By regularityassumptions ๐‘ข+, ๐‘ขโˆ’ โˆˆ๐ถ (ฮฉ), we conclude that ๐‘ขโˆ’ โ‰ค ๐‘ข++๐›ฟ inฮฉ. Since ๐›ฟ > 0 is arbitrary, this completes the proof. โ–ก

Remark 3. In 2013, Krรผgel gave another type of definitions of weak subsolutions and weak supersolutions to๐ฟ๐‘, ๐‘ = ๐‘Ž, where ๐‘Ž โˆˆ R is a constant. In Krรผgelโ€™s definition [25, Definition 4.6], a function ๐‘ขโˆ’ โˆˆ๐‘Š1, ๐‘ (ฮฉ) is calleda subsolution to ๐ฟ๐‘, ๐‘ = ๐‘Ž if ๐‘ขโˆ’ satisfiesโˆซ

๐ทโˆ’

โŸจโˆ‡๐‘ขโˆ’|โˆ‡๐‘ขโˆ’ |

๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +โˆซ๐นโˆ’

|โˆ‡๐œ™| ๐‘‘๐‘ฅ +โŸจ|โˆ‡๐‘ขโˆ’ |๐‘โˆ’2โˆ‡๐‘ขโˆ’

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ โ‰ค โˆซฮฉ๐‘Ž๐œ™ ๐‘‘๐‘ฅ (4.9)

for all ๐œ™ โˆˆ ๐ท+ (ฮฉ). Here ๐นโˆ’ B {๐‘ฅ โˆˆ ฮฉ | โˆ‡๐‘ขโˆ’ (๐‘ฅ) = 0}, ๐ทโˆ’ B ฮฉ\๐นโˆ’. Similarly a function ๐‘ข+ โˆˆ๐‘Š1, ๐‘ (ฮฉ) is called asupersolution to ๐ฟ๐‘, ๐‘ = ๐‘Ž if ๐‘ข+ satisfiesโˆซ

๐ท+

โŸจโˆ‡๐‘ข+|โˆ‡๐‘ข+ |

๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +โˆซ๐น+

|โˆ‡๐œ™| ๐‘‘๐‘ฅ +โŸจ|โˆ‡๐‘ข+ |๐‘โˆ’2โˆ‡๐‘ข+

๏ฟฝ๏ฟฝ โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ โ‰ฅ โˆซฮฉ๐‘Ž๐œ™ ๐‘‘๐‘ฅ (4.10)

for all ๐œ™ โˆˆ ๐ท+ (ฮฉ). Here ๐น+ B {๐‘ฅ โˆˆ ฮฉ | โˆ‡๐‘ข+ (๐‘ฅ) = 0}, ๐ท+ B ฮฉ \๐น+.The comparison principle discussed by Krรผgel [25, Theorem 4.8] states that

(๐‘ขโˆ’โˆ’๐‘ข+)+ โˆˆ ๐ท+ (ฮฉ) implies ๐‘ขโˆ’ โ‰ค ๐‘ข+ a.e. in ฮฉ. (4.11)

By testing (๐‘ขโˆ’โˆ’๐‘ข+)+ โˆˆ ๐ท+ (ฮฉ) into (4.9)(4.10) and substracting the two inequalities, Krรผgel claims that โˆ‡๐‘ขโˆ’ = โˆ‡๐‘ข+over ฮฉโ€ฒ B {๐‘ฅ โˆˆ ฮฉ | ๐‘ขโˆ’ (๐‘ฅ) โ‰ฅ ๐‘ข+ (๐‘ฅ)} and hence ๐‘ขโˆ’ = ๐‘ข+ a.e. in ฮฉโ€ฒ. Despite Krรผgelโ€™s comment that integrals over ๐นโˆ’

and ๐น+ cancel out, however, it seems unclear whetherโˆซ๐นโˆ’

|โˆ‡(๐‘ขโˆ’โˆ’๐‘ข+)+ | ๐‘‘๐‘ฅ =โˆซ๐น+

|โˆ‡(๐‘ขโˆ’โˆ’๐‘ข+)+ | ๐‘‘๐‘ฅ (4.12)

is valid. This problem is essentially due to the fact that Krรผgel did not appeal to monotonicity of the subdifferentialoperator ๐œ• | ยท | and did not regard the term โˆ‡๐‘ข/|โˆ‡๐‘ข | as an ๐ฟโˆž-vector field satisfying the property (4.2). In our proofof the comparison principle (Proposition 2), we make use of monotonicity of the operator ๐œ• | ยท |. Compared to ourargument based on monotonicity, the equality (4.12) itself seems to be too strong to hold true.

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4.2 Construction of classical subsolutionsIn Section 4.2, we construct a classical subsolution to ๐ฟ๐‘, ๐‘๐‘ข = 0 in an open annulus.

Lemma 2. Let ๐‘ โˆˆ R๐‘› \ {0}, ๐‘š > 0. Then for each fixed open ball ๐ต๐‘… (๐‘ฅโˆ—) โŠ‚ R๐‘›, there exists a function โ„Ž โˆˆ๐ถโˆž (R๐‘› \ {๐‘ฅโˆ—}) such that

โ„Ž = 0 on ๐œ•๐ต๐‘… (๐‘ฅโˆ—), 0 โ‰ค โ„Ž โ‰ค ๐‘š on ๐ธ๐‘… (๐‘ฅโˆ—), (4.13)

๐œ•๐œˆโ„Ž < 0 on ๐œ•๐ต๐‘… (๐‘ฅโˆ—), (4.14)

|โˆ‡โ„Ž| โ‰ค |๐‘ |2

in ๐ธ๐‘… (๐‘ฅโˆ—), (4.15)

๐‘ฃ(๐‘ฅ) B โ„Ž(๐‘ฅ) + โŸจ๐‘ | ๐‘ฅโŸฉ satisfies ๐ฟ๐‘, ๐‘๐‘ฃ โ‰ค 0 in ๐ธ๐‘… (๐‘ฅโˆ—), in the classical sense. (4.16)

Here ๐ธ๐‘… (๐‘ฅโˆ—) B ๐ต๐‘… (๐‘ฅโˆ—) \ ๐ต๐‘…/2 (๐‘ฅโˆ—) is an open annulus, and ๐œˆ in (4.14) denotes the exterior unit vector normal to๐ต๐‘… (๐‘ฅโˆ—).

Before proving Lemma 2, we fix some notations on matrices. For a given ๐‘›ร—๐‘› matrix ๐ด, we write tr(๐ด) as thetrace of ๐ด. We denote 1๐‘› by the ๐‘›ร—๐‘› unit matrix. For column vectors ๐‘ฅ = (๐‘ฅ๐‘–)๐‘– , ๐‘ฆ = (๐‘ฆ๐‘–)๐‘– โˆˆ R๐‘›, we define a tensor๐‘ฅ โŠ— ๐‘ฆ, which is regarded as a real-valued ๐‘›ร—๐‘› matrix

๐‘ฅ โŠ— ๐‘ฆ B (๐‘ฅ๐‘–๐‘ฆ ๐‘— )๐‘–, ๐‘— =ยฉยญยญยซ๐‘ฅ1๐‘ฆ1 ยท ยท ยท ๐‘ฅ1๐‘ฆ๐‘›...

. . ....

๐‘ฅ๐‘›๐‘ฆ1 ยท ยท ยท ๐‘ฅ๐‘›๐‘ฆ๐‘›

ยชยฎยฎยฌ .Assume that โ„Ž satisfies (4.15). Then the triangle inequality implies that

0 <12|๐‘ | โ‰ค |โˆ‡๐‘ฃ | โ‰ค 3

2|๐‘ | in ๐ธ๐‘… (๐‘ฅโˆ—). (4.17)

The estimate (4.17) allows us to calculate ๐ฟ๐‘, ๐‘๐‘ฃ in the classical sense over ๐ธ๐‘… (๐‘ฅโˆ—). By direct calculations we have

โˆ’๐ฟ๐‘, ๐‘๐‘ฃ = +div(โˆ‡๐‘ง๐ธ (โˆ‡๐‘ฃ)) =๐‘›โˆ‘

๐‘–, ๐‘—=1๐œ•๐‘ง๐‘– ๐‘ง ๐‘—๐ธ (โˆ‡๐‘ฃ)๐œ•๐‘ฅ๐‘– ๐‘ฅ ๐‘— ๐‘ฃ = tr

(โˆ‡2๐‘ง๐ธ (โˆ‡๐‘ฃ)โˆ‡2โ„Ž

)in ๐ธ๐‘… (๐‘ฅโˆ—).

We note that โˆ‡2๐‘ฃ = โˆ‡2โ„Ž by definition. Here we recall a well-known result on Pucciโ€™s extremal operators. For givenconstants 0 < ๐œ† โ‰ค ฮ› <โˆž and a fixed ๐‘›ร—๐‘› symmetric matrix ๐‘€ , we define

Mโˆ’ (๐‘€, ๐œ†, ฮ›) B ๐œ†โˆ‘๐œ†๐‘–>0

๐œ†๐‘– +ฮ›โˆ‘๐œ†๐‘–<0

๐œ†๐‘– ,

where ๐œ†๐‘– โˆˆ R are the eigenvalues of ๐‘€ . The following formula is a well-known result [1, Remark 5.36] ;

Mโˆ’ (๐‘€, ๐œ†, ฮ›) = inf{tr(๐ด๐‘€)

๏ฟฝ๏ฟฝ ๐ด โˆˆ A๐œ†,ฮ›},

where A๐œ†,ฮ› denotes the set of all symmetric matrices whose eigenvalues all belong to the closed interval [๐œ†, ฮ›].By (4.17) ๐ฟ๐‘, ๐‘๐‘ฃ is an uniformly elliptic operator in ๐ธ๐‘… (๐‘ฅโˆ—). This enables us to find constants 0 < ๐œ† โ‰ค ฮ› < โˆž,depending on 0 < ๐‘ < โˆž, 1 < ๐‘ < โˆž, |๐‘ | > 0, such that โˆ‡2

๐‘ง๐ธ (โˆ‡๐‘ฃ) โˆˆ [๐œ†, ฮ›] in ๐ธ๐‘… (๐‘ฅโˆ—). Combining these results, itsuffices to show that

Mโˆ’(โˆ‡2โ„Ž(๐‘ฅ), ๐œ†, ฮ›

)= ๐œ†

โˆ‘๐œ†๐‘–>0

๐œ†๐‘– (๐‘ฅ) +ฮ›โˆ‘๐œ†๐‘–<0

๐œ†๐‘– (๐‘ฅ) > 0 for all ๐‘ฅ โˆˆ ๐ธ๐‘… (๐‘ฅโˆ—), (4.18)

where ๐œ†๐‘– (๐‘ฅ) โˆˆ R denotes the eigenvalues of โˆ‡2โ„Ž(๐‘ฅ).Now we construct classical subsolutions. Our first construction is a modification of that by E. Hopf [20].

13

Proof. Without loss of generality we may assume ๐‘ฅโˆ— = 0. We define

โ„Ž(๐‘ฅ) B ๐‘’โˆ’๐›ผ |๐‘ฅ |2 โˆ’ ๐‘’โˆ’๐›ผ๐‘…2

for ๐‘ฅ โˆˆ R๐‘›. (4.19)

Here ๐›ผ = ๐›ผ(๐‘, ๐‘›, ๐‘, |๐‘ |, ๐‘…) > 0 is a sufficiently large constant to be chosen later. It is clear that 0 โ‰ค โ„Ž(๐‘ฅ) โ‰ค๐‘’โˆ’๐›ผ๐‘…

2/4 โˆ’ ๐‘’โˆ’๐›ผ๐‘…2 in ๐ธ๐‘… (0). We first let ๐›ผ > 0 be so large that

๐‘š๐‘’๐›ผ๐‘…2 โ‰ฅ ๐‘’3๐›ผ๐‘…2/4 โˆ’1. (4.20)

From (4.20), we can easily check (4.13). By direct calculation we get

โˆ‡โ„Ž(๐‘ฅ) = โˆ’2๐›ผ๐‘’โˆ’๐›ผ |๐‘ฅ |2๐‘ฅ, and โˆ‡2โ„Ž(๐‘ฅ) = โˆ’2๐›ผ๐‘’โˆ’๐›ผ |๐‘ฅ |

21๐‘› +4๐›ผ2๐‘’โˆ’๐›ผ |๐‘ฅ |

2๐‘ฅ โŠ— ๐‘ฅ for each ๐‘ฅ โˆˆ R๐‘›.

From this result, (4.14) is clear. Also, we have

|โˆ‡โ„Ž(๐‘ฅ) | โ‰ค 2๐›ผ๐‘…๐‘’โˆ’๐›ผ๐‘…2/4 for all ๐‘ฅ โˆˆ ๐ธ๐‘… (0).

Let ๐›ผ > 0 be so large that

๐›ผ๐‘’โˆ’๐›ผ๐‘…2/4 โ‰ค |๐‘ |

4๐‘…, (4.21)

then we can check that โ„Ž satisfies (4.15). Now we prove (4.16) to complete the proof. For ๐‘ฅ โ‰  0, the eigenvalues ofโˆ‡2โ„Ž(๐‘ฅ) are given by{

๐œ† โˆฅ (๐‘ฅ) B 4๐›ผ2 |๐‘ฅ |2๐‘’โˆ’๐›ผ |๐‘ฅ |2 โˆ’2๐›ผ๐‘’โˆ’๐›ผ |๐‘ฅ |2 ,๐œ†โŠฅ (๐‘ฅ) B โˆ’2๐›ผ๐‘’โˆ’๐›ผ |๐‘ฅ |2 ,

and the geometric multiplicities are{

1,๐‘›โˆ’1.

Assume that ๐›ผ satisfies๐›ผ >

2๐‘…2 , (4.22)

so that ๐œ† โˆฅ > 0 > ๐œ†โŠฅ in ๐ธ๐‘… (0). Therefore we get

Mโˆ’(โˆ‡2โ„Ž(๐‘ฅ), ๐œ†, ฮ›

)= ๐œ†๐œ† โˆฅ (๐‘ฅ) + (๐‘›โˆ’1)ฮ›๐œ†โŠฅ (๐‘ฅ) = 2๐›ผ๐‘’โˆ’๐›ผ |๐‘ฅ |

2 [๐œ†(2๐›ผ |๐‘ฅ |2 โˆ’1) โˆ’ (๐‘›โˆ’1)ฮ›

]โ‰ฅ 2๐›ผ๐‘’โˆ’๐›ผ |๐‘ฅ |

2[๐œ†

(๐‘…2

2๐›ผโˆ’1

)โˆ’ (๐‘›โˆ’1)ฮ›

].

We can take sufficiently large ๐›ผ = ๐›ผ( |๐‘ |, ๐‘š, ๐‘›, ๐‘…, ๐œ†, ฮ›) > 0 so that ๐›ผ satisfies (4.18) and (4.20)โ€“(4.22). For suchconstant ๐›ผ > 0, the function ๐‘ฃ defined as in (4.19) satisfies (4.13)โ€“(4.16). โ–ก

It is possible to construct an alternative function โ„Ž โˆˆ ๐ถโˆž (R๐‘› \ {๐‘ฅ0}) which satisfies (4.13)โ€“(4.16). We giveanother proof of Lemma 2, which is derived from [29, Chapter 2.8].

Proof. Without loss of generality we may assume ๐‘ฅโˆ— = 0. We define

โ„Ž(๐‘ฅ) B ๐›ฝ[|๐‘ฅ |โˆ’๐›ผ โˆ’๐‘…โˆ’๐›ผ] for ๐‘ฅ โˆˆ R๐‘› \ {0}. (4.23)

We will later determine positive constants ๐›ผ, ๐›ฝ > 0, depending on ๐‘, ๐‘š, ๐‘›, ๐‘, |๐‘ |, ๐‘…. It is clear that 0 โ‰ค โ„Ž(๐‘ฅ) โ‰ค๐›ฝ๐‘…โˆ’๐›ผ (2๐›ผ โˆ’1) in ๐ธ๐‘… (0). We first let ๐›ผ, ๐›ฝ > 0 satisfy

๐›ฝ โ‰ค ๐‘š๐‘…๐›ผ

2๐›ผ โˆ’1. (4.24)

Then โ„Ž satisfies (4.13). By direct calculation we get

โˆ‡โ„Ž(๐‘ฅ) = โˆ’ ๐›ผ๐›ฝ๐‘ฅ

|๐‘ฅ |๐›ผ+2 , and โˆ‡2โ„Ž(๐‘ฅ) = ๐›ผ๐›ฝ

|๐‘ฅ |๐›ผ+2

[(๐›ผ+2) ๐‘ฅ โŠ— ๐‘ฅ

|๐‘ฅ |2โˆ’1๐‘›

]

14

for each ๐‘ฅ โˆˆ R๐‘› \ {0}. The estimate (4.14) is clear by this result. Also, we have

|โˆ‡โ„Ž(๐‘ฅ) | โ‰ค ๐›ผ๐›ฝ

(๐‘…/2)๐›ผ+1 for all ๐‘ฅ โˆˆ ๐ธ๐‘… (0).

Let ๐›ผ, ๐›ฝ > 0 satisfy

๐›ฝ โ‰ค |๐‘ | (๐‘…/2)๐›ผ+1

2๐›ผ, (4.25)

then we can check that โ„Ž satisfies (4.15). Now we prove (4.16) to complete the proof. For ๐‘ฅ โ‰  0, the eigenvalues ofโˆ‡2โ„Ž(๐‘ฅ) are given by{

๐œ† โˆฅ (๐‘ฅ) B (๐›ผ+1)๐›ผ๐›ฝ |๐‘ฅ |โˆ’๐›ผโˆ’2,๐œ†โŠฅ (๐‘ฅ) B โˆ’๐›ผ๐›ฝ |๐‘ฅ |โˆ’๐›ผโˆ’2,

and the geometric multiplicities are{

1,๐‘›โˆ’1.

It is clear that ๐œ† โˆฅ > 0 > ๐œ†โŠฅ in R๐‘› \ {0}, and therefore

Mโˆ’(โˆ‡2โ„Ž(๐‘ฅ), ๐œ†, ฮ›

)= ๐›ผ๐›ฝ |๐‘ฅ |โˆ’๐›ผโˆ’2 [(๐›ผ+1)๐œ†โˆ’ (๐‘›โˆ’1)ฮ›]

for all ๐‘ฅ โˆˆ ๐ธ๐‘… (0). We take and fix sufficiently large ๐›ผ = ๐›ผ(๐‘›, ๐œ†, ฮ›) > 0 so that ๐›ผ satisfies (4.18). For such ๐›ผ > 0,we choose sufficiently small ๐›ฝ = ๐›ฝ( |๐‘ |, ๐‘…, ๐›ผ) > 0 so that ๐›ฝ satisfies (4.24)โ€“(4.25). Then the function โ„Ž defined asin (4.23) satisfies (4.13)โ€“(4.16). โ–ก

4.3 Strong maximum principleWe prove the strong maximum principle (Theorem 2).

Proof. Let ๐ท0 โŠ‚ ๐ท be a connected component of the open set ๐ท, and ๐‘ฅ0 โˆˆ ๐ท0. Without loss of generality wemay assume that ๐‘ฅ0 = 0 and ๐‘ข(0) = 0. By Lemma 1, it is clear that ๐œ•๐‘ข(0) = {โˆ‡๐‘ข(0)} โ‰  {0}. We set a vector๐‘ B โˆ‡๐‘ข(0) โˆˆ R๐‘› \ {0} and a relatively closed set

ฮฃ B {๐‘ฅ โˆˆ ๐ท0 | ๐‘ข(๐‘ฅ) = โŸจ๐‘ | ๐‘ฅโŸฉ}.

and we will prove that ฮฃ = ๐ท0. It is also clear that 0 โˆˆ ฮฃ and hence ฮฃ โ‰  โˆ…. Suppose for contradiction that ฮฃ โŠŠ ๐ท0.Then it follows that ๐œ•ฮฃ โˆฉ๐ท0 โ‰  โˆ…, since ๐ท0 is connected. We may take and fix a point ๐‘ฅโˆ— โˆˆ ๐ท0 \ฮฃ such thatdist(๐‘ฅโˆ—, ฮฃ) < dist(๐‘ฅโˆ—, ๐œ•๐ท0). By extending a closed ball centered at ๐‘ฅโˆ— until it hits ฮฃ, we can take a point ๐‘ฆโˆ— โˆˆ ๐ท0and a closed ball ๐ต๐‘… (๐‘ฅโˆ—) โŠ‚ ๐ท0 such that ๐‘ฆโˆ— โˆˆ ๐œ•๐ต๐‘… (๐‘ฅโˆ—) โˆฉฮฃ and ๐‘ข(๐‘ฅ) > โŸจ๐‘ | ๐‘ฅโŸฉ for all ๐‘ฅ โˆˆ ๐ต๐‘… (๐‘ฅโˆ—). We note that

0 = min๐‘ฅโˆˆ๐œ•๐ต๐‘… (๐‘ฅโˆ—)

(๐‘ข(๐‘ฅ) โˆ’ โŸจ๐‘ | ๐‘ฅโŸฉ), achieved at ๐‘ฆโˆ— โˆˆ ๐œ•๐ต๐‘… (๐‘ฅโˆ—),๐‘š B min

๐‘ฅโˆˆ๐œ•๐ต๐‘…/2 (๐‘ฅโˆ—)(๐‘ข(๐‘ฅ) โˆ’ โŸจ๐‘ | ๐‘ฅโŸฉ) > 0, (4.26)

by construction of ๐ต๐‘… (๐‘ฅโˆ—). Let โ„Ž โˆˆ ๐ถโˆž (R๐‘› \ {๐‘ฅโˆ—}) be an auxiliary function as in Lemma 2. Then from (4.26) itis easy to check that ๐‘ฃ B โ„Ž + โŸจ๐‘ | ๐‘ฅโŸฉ satisfies ๐‘ฃ โ‰ค ๐‘ข on ๐œ•๐ธ๐‘… (๐‘ฅโˆ—), in the sense of (4.7). By Proposition 2, we have๐‘ฃ โ‰ค ๐‘ข on ๐ธ๐‘… (๐‘ฅโˆ—). Hence 0 โ‰ค ๐‘ขโˆ’ โŸจ๐‘ | ๐‘ฅโŸฉ โˆ’ โ„Ž in ๐ธ๐‘… (๐‘ฅโˆ—). This inequality becomes equality at ๐‘ฆโˆ— โˆˆ ๐œ•๐ธ๐‘… (๐‘ฅโˆ—) by (4.13)and (4.26). Therefore the function ๐‘ข(๐‘ฅ) โˆ’ โŸจ๐‘ | ๐‘ฅโŸฉ โˆ’ โ„Ž(๐‘ฅ) (๐‘ฅ โˆˆ ๐ธ๐‘… (๐‘ฅโˆ—)) takes its minimum at ๐‘ฆโˆ— โˆˆ ๐œ•๐ต๐‘… (๐‘ฅโˆ—). Also by๐‘ฆโˆ— โˆˆ ฮฃ and the subgradient inequality

๐‘ข(๐‘ฅ) โ‰ฅ โŸจ๐‘ | ๐‘ฅโŸฉ for all ๐‘ฅ โˆˆ ฮฉ,

it is clear that the function ๐‘ค(๐‘ฅ) B ๐‘ข(๐‘ฅ) โˆ’ โŸจ๐‘ | ๐‘ฅโŸฉ (๐‘ฅ โˆˆ ๐ท0) takes its minimum 0 at ๐‘ฆโˆ— โˆˆ ๐ท0. We note that๐‘ค, ๐‘คโˆ’ โ„Ž โˆˆ ๐ถ1 (๐ท0) by Lemma 1. By calculating classical partial derivatives at ๐‘ฆโˆ— in the direction ๐œˆ0 B (๐‘ฆโˆ—โˆ’๐‘ฅโˆ—)/๐‘…,we obtain

0 โ‰ฅ ๐œ•๐œˆ0 (๐‘คโˆ’ โ„Ž) (๐‘ฆโˆ—) = โˆ’๐œ•๐œˆ0โ„Ž(๐‘ฆโˆ—) > 0.

This is a contradiction, and therefore ฮฃ = ๐ท0. โ–ก

5 Proofs of main theoremsIn Section 5, we give proofs of the Liouville-type theorem (Theorem 3) and the ๐ถ1-regularity theorem (Thorem 1).

15

5.1 Liouville-type theoremFor a preparation, we prove Lemma 3 below.

Lemma 3. Let ๐‘ข be a real-valued convex function in R๐‘›. Assume that ๐‘ข satisfies the following,

1. The facet of ๐‘ข, ๐น โŠ‚ R๐‘›, satisfies โˆ… โŠŠ ๐น โŠŠ R๐‘›.

2. ๐‘ข attains its minimum 0.

3. ๐‘ข is affine in each connected component of ๐ท B R๐‘› \๐น.

Then up to a rotation and a shift translation, ๐‘ข can be expressed as either of the following three types ofpiecewise-linear functions.

๐‘ข(๐‘ฅ) = max{ ๐‘ก1๐‘ฅ1, 0 } for all ๐‘ฅ โˆˆ R๐‘›, (5.1)

๐‘ข(๐‘ฅ) = max{ ๐‘ก1๐‘ฅ1, โˆ’๐‘ก2๐‘ฅ1 } for all ๐‘ฅ โˆˆ R๐‘›, (5.2)

๐‘ข(๐‘ฅ) = max{ ๐‘ก1๐‘ฅ1, 0, โˆ’๐‘ก2 (๐‘ฅ1 + ๐‘™0) } for all ๐‘ฅ โˆˆ R๐‘›. (5.3)

Here ๐‘ก1, ๐‘ก2, ๐‘™ > 0 are constants.

Before starting the proof of Lemma 3, we introduce notations on affine hyperplanes. For ๐‘ โˆˆ R๐‘› \ {0} and๐‘ฅ0 โˆˆ R๐‘›, we define

๐ป๐‘, ๐‘ฅ0 B {๐‘ฅ โˆˆ R๐‘› | โŸจ๐‘ | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ = 0},๐ปโˆ’๐‘, ๐‘ฅ0 B {๐‘ฅ โˆˆ R๐‘› | โŸจ๐‘ | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ < 0},

๐ป+๐‘, ๐‘ฅ0 B {๐‘ฅ โˆˆ R๐‘› | โŸจ๐‘ | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ > 0}.

In order to prove the Liouville-type theorem, we will make use of the supporting hyperplane theorem, which statesthat for any non-empty closed convex set ๐ถ โŠ‚ R๐‘› and ๐‘ฅ0 โˆˆ ๐œ•๐ถ, there exists ๐‘ โˆˆ R๐‘› \ {0} such that

sup๐‘ฅโˆˆ๐ถ

โŸจ๐‘ | ๐‘ฅโŸฉ โ‰ค โŸจ๐‘ | ๐‘ฅ0โŸฉ, and in particular ๐ป+๐‘, ๐‘ฅ0 โŠ‚ R

๐‘› \๐ถ.

For such ๐‘ โˆˆ R๐‘› \ {0}, a hyperplane ๐ป๐‘, ๐‘ฅ0 is often called a supporting hyperplane for ๐ถ at the boundary point ๐‘ฅ0.For the proof of the supporting hyperplane theorem, see [3, Proposition 1.5.1].

Proof. Since R๐‘› is connected and ๐น โŠ‚ R๐‘› is a closed convex set, it follows that ๐œ•๐น โ‰  โˆ…. Without loss of generalitywe may assume that 0 โˆˆ ๐œ•๐น and ๐‘ข(0) = 0.

By the supporting hyperplane theorem, we can take and fix a supporting hyperplane for ๐น at the boundary point0, which we write ๐ป๐‘, 0 โŠ‚ R๐‘›. By rotation, we may assume that ๐‘ = ๐‘’1. Let ๐ท1 be the connected component of๐ท which contains ๐ป+

๐‘’1 , 0 โŠ‚ R๐‘› \๐น = ๐ท. By the assumption 3 and ๐‘ข(0) = 0, it follows that there exists ๐‘ โˆˆ R๐‘› \ {0}such that ๐‘ข(๐‘ฅ) = โŸจ๐‘ | ๐‘ฅโŸฉ for all ๐‘ฅ โˆˆ ๐ท1. We should note that ๐ป๐‘, 0 = ๐ป๐‘’1 , 0 and hence ๐‘ = ๐‘ก1๐‘’1 for some ๐‘ก1 โˆˆ (0,โˆž),since otherwise it follows that ๐ป+

๐‘’1 , 0 โˆฉ๐ปโˆ’๐‘, 0 โ‰  โˆ… and 0 โ‰ค ๐‘ข(๐‘ฅ0) = โŸจ๐‘ | ๐‘ฅ0โŸฉ < 0 for any ๐‘ฅ0 โˆˆ ๐ป+

๐‘’1 , 0 โˆฉ๐ปโˆ’๐‘, 0. The result

๐ป๐‘, 0 = ๐ป๐‘’1 , 0 also implies that ๐ป๐‘’1 , 0 โŠ‚ ๐œ•๐น โŠ‚ ๐น โŠ‚ {๐‘ฅ โˆˆ R๐‘› | ๐‘ฅ1 โ‰ค 0} = ๐ปโˆ’๐‘’1 , 0 โˆช๐ป๐‘’1 , 0. Now we will deduce three

possible representations of ๐‘ข.If ๐œ•๐น = ๐ป๐‘’1 , 0, then we have either ๐น = ๐ปโˆ’

๐‘’1 , 0 โˆช๐ป๐‘’1 , 0 or ๐น = ๐ป๐‘’1 , 0, since the open set ๐ปโˆ’๐‘’1 , 0 = {๐‘ฅ โˆˆ R๐‘› | ๐‘ฅ1 < 0}

is connected. For the first case, ๐‘ข is clearly expressed by (5.1). For the second case, it is clear that ๐ท consists of twoconnected components ๐ท1 = ๐ป+

๐‘’1 , 0 and ๐ท2 = ๐ปโˆ’๐‘’1 , 0. Again by the condition 3 and similar arguments to the above,

we can determine ๐‘ข |๐ท2 as ๐‘ข(๐‘ฅ) = โŸจโˆ’๐‘ก2๐‘’1 | ๐‘ฅโŸฉ for all ๐‘ฅ โˆˆ ๐ท2. Here ๐‘ก2 โˆˆ (0,โˆž) is a constant. Hence we obtain (5.2).For the case ๐ป๐‘’1 , 0 โŠŠ ๐œ•๐น, we take and fix ๐‘ง0 โˆˆ ๐œ•๐น \๐ป๐‘’1 , 0 and a supporting hyperplane for ๐น at ๐‘ง0, which we write by๐ป๐‘โ€ฒ, ๐‘ง0 . Let ๐ท2 be the connected component of ๐ท which contains ๐ป+

๐‘โ€ฒ, ๐‘ง0โŠ‚ ๐ท. By the assumption 3 and ๐‘ข(๐‘ง0) = 0,

it follows that there exists ๐‘โ€ฒโ€ฒ โˆˆ R๐‘› \ {0} such that ๐‘ข(๐‘ฅ) = โŸจ๐‘โ€ฒโ€ฒ | ๐‘ฅโˆ’ ๐‘ง0โŸฉ for all ๐‘ฅ โˆˆ ๐ท2. Completely similarly to thearguments above for showing that ๐ป๐‘, 0 = ๐ป๐‘’1 , 0, we can easily notice that ๐ป๐‘โ€ฒโ€ฒ, ๐‘ง0 = ๐ป๐‘โ€ฒ, ๐‘ง0 and hence ๐‘โ€ฒโ€ฒ = ๐‘ก โ€ฒ1๐‘

โ€ฒ forsome constant ๐‘ก โ€ฒ1 โˆˆ (0,โˆž). Moreover, we also realize that ๐‘โ€ฒ = ๐‘กโˆ—๐‘’1 for some ๐‘กโˆ— โˆˆ R\ {0}. Otherwise it follows thatthe two hyperplanes ๐ป๐‘’1 , 0 and ๐ป๐‘โ€ฒ, ๐‘ง0 cross, and hence we get ๐ท1 = ๐ท2 and ๐ป+

๐‘’1 , 0 โˆฉ๐ปโˆ’๐‘โ€ฒ, ๐‘ง0

โ‰  โˆ…, which implies thatthere exists a point ๐‘ฅ0 โˆˆ ๐ท such that ๐‘ข(๐‘ฅ0) < 0. This is clearly a contradiction. This result and convexity of ๐‘ข implythat ๐ท consists of two connected components ๐ท1 = ๐ป+

๐‘’1 , 0 and ๐ท2 = ๐ป+โˆ’๐‘’1 , ๐‘ง0 , and that ๐น = {๐‘ฅ โˆˆ R๐‘› | โˆ’๐‘™0 โ‰ค ๐‘ฅ1 โ‰ค 0}.

Here ๐‘™0 B dist(๐ป๐‘’1 , 0, ๐ปโˆ’๐‘’1 , ๐‘ง0 ) > 0. Finally we obtain the last possible expression (5.3). ๐‘ข can be expressed byeither of (5.1)โ€“(5.3). โ–ก

16

Now we give the proof of Theorem 3.

Proof. Assume by contradiction that ๐น, the facet of ๐‘ข, would satisfy โˆ… โŠŠ ๐น โŠŠ R๐‘›. Without loss of generality, wemay assume that ๐‘ข attains its minimum 0. By the strong maximum principle (Theorem 2), the convex weak solution๐‘ข is affine in each connected component of ๐ท B R๐‘› \๐น. Therefore we are able to apply Lemma 3. By rotation andtranslation, ๐‘ข can be expressed as (5.1)โ€“(5.3). Now we prove that ๐‘ข is no longer a weak solution to ๐ฟ๐‘, ๐‘๐‘ข = 0 inR๐‘›. We set open cubes ๐‘„ โ€ฒ B (โˆ’1, 1)๐‘›โˆ’1 โŠ‚ R๐‘›โˆ’1 and ๐‘„ B (โˆ’๐‘‘, ๐‘‘) ร—๐‘„ โ€ฒ โŠ‚ R๐‘›, where ๐‘‘ > 0 is to be chosen later.We claim that ๐‘ข does not satisfy ๐ฟ๐‘, ๐‘๐‘ข = 0 in๐‘Šโˆ’1, ๐‘โ€ฒ (๐‘„). Assume by contradiction that there exists a vector field๐‘ โˆˆ ๐ฟโˆž (๐‘„, R๐‘›) such that the pair (๐‘ข, ๐‘) โˆˆ๐‘Š1, ๐‘ (๐‘„) ร— ๐ฟโˆž (๐‘„, R๐‘›) satisfies ๐ฟ๐‘, ๐‘๐‘ข = 0 in๐‘Šโˆ’1, ๐‘โ€ฒ (๐‘„).

For the first case (5.1), we have

|๐‘ (๐‘ฅ) | โ‰ค 1 for a.e. ๐‘ฅ โˆˆ ๐‘„, and ๐‘ (๐‘ฅ) = ๐‘’1 for a.e. ๐‘ฅ โˆˆ ๐‘„๐‘Ÿ B (0, ๐‘‘) ร—๐‘„ โ€ฒ โŠ‚ R๐‘›. (5.4)

by definition of ๐‘ . We also set another open cube ๐‘„๐‘™ B (โˆ’๐‘‘, 0) ร—๐‘„ โ€ฒ โŠ‚ R๐‘›. We take and fix non-negative functions๐œ™1 โˆˆ ๐ถ1

๐‘ ((โˆ’๐‘‘, ๐‘‘)), ๐œ™2 โˆˆ ๐ถ1๐‘ (๐‘„ โ€ฒ) such that

๐œ™โ€ฒ1 โ‰ฅ 0 in (โˆ’๐‘‘, 0), max(โˆ’๐‘‘, ๐‘‘)

๐œ™1 = ๐œ™1 (0) > 0, and ๐œ™2 . 0. (5.5)

We define an admissible test function ๐œ™ โˆˆ ๐ถ1๐‘ (๐‘„) by ๐œ™(๐‘ฅ1, ๐‘ฅ

โ€ฒ) B ๐œ™1 (๐‘ฅ1)๐œ™2 (๐‘ฅ โ€ฒ) for (๐‘ฅ1, ๐‘ฅโ€ฒ) โˆˆ (โˆ’๐‘‘, ๐‘‘) ร—๐‘„ โ€ฒ =๐‘„. Test

๐œ™ โˆˆ ๐ถ1๐‘ (๐‘„) into ๐ฟ๐‘, ๐‘๐‘ข = 0 in ๐‘Šโˆ’1, ๐‘โ€ฒ (๐‘„), and divide the integration over ๐‘„ into that over ๐‘„๐‘™ and ๐‘„๐‘Ÿ . Then (5.4)

implies that

0 = ๐‘โˆซ๐‘„๐‘™

โŸจ๐‘ + |0|๐‘โˆ’20 | โˆ‡(๐œ™1๐œ™2)โŸฉ ๐‘‘๐‘ฅ +โˆซ๐‘„๐‘Ÿ

โŸจ(๐‘ + ๐‘ก ๐‘โˆ’1

1 )๐‘’1

๏ฟฝ๏ฟฝ๏ฟฝ โˆ‡(๐œ™1๐œ™2)โŸฉ๐‘‘๐‘ฅ

โ‰ค ๐‘โˆซ๐‘„๐‘™

๐œ™โ€ฒ1๐œ™2 ๐‘‘๐‘ฅ + ๐‘โˆซ๐‘„๐‘™

๐œ™1 |โˆ‡๐œ™2 | ๐‘‘๐‘ฅ

+ ๐‘๐œ™1 (0)โˆซ๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ)โŸจ๐‘’1 | โˆ’๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ+ ๐‘ก ๐‘โˆ’1

1 ๐œ™1 (0)โˆซ๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ)โŸจ๐‘’1 | โˆ’๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ

C ๐ผ1 + ๐ผ2 + ๐ผ3 + ๐ผ4.Here we have applied the Gaussโ€“Green theorem to the integration over ๐‘„๐‘Ÿ , and the Cauchyโ€“Schwarz inequality tothe integration over ๐‘„๐‘™ . For the integrations ๐ผ1 and ๐ผ2, we make use of Fubiniโ€™s theorem and (5.5). Then we have

๐ผ1 =โˆซ๐‘„โ€ฒ

(โˆซ 0

โˆ’๐‘‘๐œ™โ€ฒ1 (๐‘ฅ1)๐‘‘๐‘ฅ1

)๐œ™2 (๐‘ฅ โ€ฒ)๐‘‘๐‘ฅ โ€ฒ = ๐‘๐œ™1 (0)

โˆซ๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ)๐‘‘๐‘ฅ โ€ฒ = ๐‘๐œ™1 (0)โˆฅ๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ) = โˆ’๐ผ3,

๐ผ2 โ‰ค ๐‘๐œ™1 (0)โˆซ 0

โˆ’๐‘‘๐‘‘๐‘ฅ1

โˆซ๐‘„โ€ฒ|โˆ‡๐œ™2 (๐‘ฅ โ€ฒ) | ๐‘‘๐‘ฅ โ€ฒ = ๐‘๐‘‘๐œ™1 (0)โˆฅโˆ‡๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ) .

Finally we obtain

0 โ‰ค ๐ผ1 + ๐ผ2 + ๐ผ3 + ๐ผ4 โ‰ค ๐ผ2 + ๐ผ4 โ‰ค ๐œ™1 (0)(๐‘๐‘‘โˆฅโˆ‡๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ) โˆ’ ๐‘ก

๐‘โˆ’11 โˆฅ๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ)

). (5.6)

From (5.6), we can easily deduce a contradiction by choosing sufficiently small ๐‘‘ = ๐‘‘ (๐‘, ๐‘, ๐‘ก1, ๐œ™2) > 0. Similarlywe can prove that ๐‘ข defined as in (5.3) does not satisfy ๐ฟ๐‘, ๐‘๐‘ข = 0 in๐‘Šโˆ’1, ๐‘โ€ฒ (๐‘„), since it suffices to restrict ๐‘‘ < ๐‘™0.We consider the remaining case (5.2). We have

๐‘ (๐‘ฅ) ={๐‘’1 for a.e. ๐‘ฅ โˆˆ ๐‘„๐‘Ÿ ,โˆ’๐‘’1 for a.e. ๐‘ฅ โˆˆ ๐‘„๐‘™ .

by definition of ๐‘ . We test the same function ๐œ™ โˆˆ ๐ถ1๐‘ (๐‘„) in ๐ฟ๐‘, ๐‘๐‘ข = 0, then it follows that

0 =โˆซ๐‘„๐‘™

โŸจโˆ’(๐‘ + ๐‘ก ๐‘โˆ’1

2 )๐‘’1

๏ฟฝ๏ฟฝ๏ฟฝ โˆ‡(๐œ™1๐œ™2)โŸฉ๐‘‘๐‘ฅ +

โˆซ๐‘„๐‘Ÿ

โŸจ(๐‘ + ๐‘ก ๐‘โˆ’1

1 )๐‘’1

๏ฟฝ๏ฟฝ๏ฟฝ โˆ‡(๐œ™1๐œ™2)โŸฉ๐‘‘๐‘ฅ

= โˆ’(๐‘ + ๐‘ก ๐‘โˆ’12 )

โˆซ๐‘„โ€ฒ๐œ™1 (0)๐œ™2 (๐‘ฅ โ€ฒ)โŸจ๐‘’1 | ๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ+ (๐‘ + ๐‘ก ๐‘โˆ’1

1 )โˆซ๐‘„โ€ฒ๐œ™1 (0)๐œ™2 (๐‘ฅ โ€ฒ)โŸจ๐‘’1 | โˆ’๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ

= โˆ’๐œ™1 (0)(2๐‘ + ๐‘ก ๐‘โˆ’1

1 + |๐‘ก2 |๐‘โˆ’1) โˆซ

๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ) ๐‘‘๐‘ฅ โ€ฒ < 0,

which is a contradiction. This completes the proof. โ–ก

17

Remark 4. The estimate (5.6) breaks for ๐‘ = 1, since the equation |0|๐‘โˆ’20 = 0 is no longer valid for ๐‘ = 1. Thismeans that we have implicitly used differentiability of the function |๐‘ง |๐‘/๐‘ at 0 โˆˆ R๐‘›. Also it should be noted thatfor the one-variable case, functions as in (5.1), which are in general not in ๐ถ1, are one-harmonic in R.

5.2 ๐ถ1-regularity theoremWe give the proof of Theorem 1.

Proof. We may assume that ฮฉ is convex. By [30, Theorem 25.1 and 25.5] and Lemma 1, it suffices to show that๐œ•๐‘ข(๐‘ฅ0) = {0} for all ๐‘ฅ0 โˆˆ ๐น. Let ๐‘ฅ0 โˆˆ ๐น. We get a convex function ๐‘ข0 : R๐‘› โ†’ R as a blow-up limit as in Proposition1. We note that the facet of ๐‘ข0 is non-empty by Proposition 1. Hence by the Liouville-type theorem (Theorem 3),๐‘ข0 is constant and we obtain ๐œ•๐‘ข0 (๐‘ฅ0) = {0}. Combining these results, we have {0} โŠ‚ ๐œ•๐‘ข(๐‘ฅ0) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ0) = {0} andtherefore ๐œ•๐‘ข(๐‘ฅ0) = {0}. This completes the proof. โ–ก

6 GeneralizationIn Section 6, we would like to discuss ๐ถ1-regularity of convex weak solutions to

๐ฟ๐‘ข B โˆ’div(โˆ‡๐‘งฮจ(โˆ‡๐‘ข)) โˆ’div(โˆ‡๐‘ง๐‘Š (โˆ‡๐‘ข)) = ๐‘“ in ฮฉ โŠ‚ R๐‘›, (6.1)

which covers (1.1). Precisely speaking, throughout Section 6, we make these following assumptions for ฮจ and ๐‘Šon regularity and ellipticity. For regularity, we only require

ฮจ โˆˆ ๐ถ (R๐‘›) โˆฉ๐ถ2 (R๐‘› \ {0}), ๐‘Š โˆˆ ๐ถ1 (R๐‘›) โˆฉ๐ถ2 (R๐‘› \ {0}). (6.2)

For๐‘Š , we assume that for each fixed 0 < ๐œ‡ โ‰ค ๐‘€ <โˆž, there exist constants 0 < ๐›พ < ฮ“ <โˆž such that๐‘Š satisfies

๐›พ |๐œ |2 โ‰คโŸจโˆ‡2๐‘ง๐‘Š (๐‘ง0)๐œ

๏ฟฝ๏ฟฝ ๐œโŸฉ, (6.3)๏ฟฝ๏ฟฝโŸจโˆ‡2๐‘ง๐‘Š (๐‘ง0)๐œ

๏ฟฝ๏ฟฝ ๐œ”โŸฉ๏ฟฝ๏ฟฝ โ‰ค ฮ“ |๐œ | |๐œ” | (6.4)

for all ๐‘ง0, ๐œ , ๐œ” โˆˆ R๐‘› with ๐œ‡ โ‰ค |๐‘ง0 | โ‰ค ๐‘€ . Also, there is no loss of generality in assuming that

โˆ‡๐‘ง๐‘Š (0) = 0. (6.5)

Finally, we assume that ฮจ is positively homogeneous of degree 1. In other words, ฮจ satisfies

ฮจ(๐œ†๐‘ง0) = ๐œ†ฮจ(๐‘ง0) (6.6)

holds for all ๐‘ง0 โˆˆ R๐‘› and ๐œ† > 0. This clearly yields ฮจ(0) = 0.By modifying some of our arguments, we are able to show that

Theorem 4 (๐ถ1-regularity theorem for general equations). Let ฮฉ โŠ‚ R๐‘› be a domain. Assume that ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› <๐‘ž โ‰ค โˆž) and the functionals ฮจ and๐‘Š satisfy (6.2)โ€“(6.5). If ๐‘ข is a convex weak solution to (6.1), then ๐‘ข is in ๐ถ1 (ฮฉ).

If we setฮจ(๐‘ง) B ๐‘ |๐‘ง |, ๐‘Š (๐‘ง) B |๐‘ง |๐‘

๐‘, where 1 < ๐‘ <โˆž,

then the equation (6.1) becomes (1.1). Therefore Theorem 4 generalizes Theorem 1.

6.1 PreliminariesIn Section 6.1, we mention some basic properties of ฮจ and๐‘Š , which are derived from the assumptions (6.2)โ€“(6.5).

For ๐‘Š , by (6.2)โ€“(6.3) and (6.5) it is easy to check that the continuous mapping ๐ด : R๐‘› โˆ‹ ๐‘ง โ†ฆโ†’ โˆ‡๐‘Š (๐‘ง) โˆˆ R๐‘›satisfies strict monotonicity (4.8). In particular, by (6.5) we have

โŸจ๐ด(๐‘ง) | ๐‘งโŸฉ > 0 for all ๐‘ง โˆˆ R๐‘› \ {0}. (6.7)

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For the proof, see Lemma 10 in the appendices.For ฮจ, we first note that ฮจ satisfies the triangle inequality

ฮจ(๐‘ง1 + ๐‘ง2) โ‰ค ฮจ(๐‘ง1) +ฮจ(๐‘ง2) for all ๐‘ง1, ๐‘ง2 โˆˆ R๐‘›. (6.8)

We define a function ฮจฬƒ : R๐‘› โ†’ [0,โˆž] by

ฮจฬƒ(๐œ) B sup{โŸจ๐œ | ๐‘งโŸฉ | ๐‘ง โˆˆ R๐‘›, ฮจ(๐‘ง) โ‰ค 1}.

ฮจฬƒ is the support function for the closed convex set๐ถฮจ B {๐‘ง โˆˆ R๐‘› | ฮจ(๐‘ง) โ‰ค 1}. By definition it is easy to check that ฮจฬƒis convex and lower semicontinuous. Also, if ๐œ โˆˆ R๐‘› satisfies ฮจฬƒ(๐œ) <โˆž, then the following Cauchyโ€“Schwarz-typeinequality holds;

โŸจ๐‘ง | ๐œโŸฉ โ‰ค ฮจ(๐‘ง)ฮจฬƒ(๐œ) for all ๐‘ง โˆˆ R๐‘›. (6.9)

If a convex function ฮจ is positively homogeneous of degree 1, then the subdifferential operator ๐œ•ฮจ is explicitlygiven by

๐œ•ฮจ(๐‘ง) ={๐œ โˆˆ R๐‘›

๏ฟฝ๏ฟฝ ฮจฬƒ(๐œ) โ‰ค 1, ฮจ(๐‘ง) = โŸจ๐‘ง | ๐œโŸฉ}

(6.10)

for all ๐‘ง โˆˆ R๐‘›. In particular, we have the following formula

โŸจโˆ‡๐‘งฮจ(๐‘ง0) | ๐‘ง0โŸฉ = ฮจ(๐‘ง0) for all ๐‘ง0 โˆˆ R๐‘› \ {0}, (6.11)

which is often called Eulerโ€™s identity. Also, assumptions (6.2) and (6.6) imply that

โˆ‡ฮจ(๐œ†๐‘ง0) = โˆ‡ฮจ(๐‘ง0), โˆ‡2ฮจ(๐œ†๐‘ง0) = ๐œ†โˆ’1โˆ‡2ฮจ(๐‘ง0) (6.12)

for all ๐œ† > 0 and ๐‘ง0 โˆˆ R๐‘› \ {0}. Proofs of (6.8)โ€“(6.10) are given in Lemma 11 of the appendices for the readerโ€™sconvenience.

Remark 5. The results (6.11)โ€“(6.12) give us the following basic property for ฮจ.

1. We set a constant๐พ B sup{|โˆ‡๐‘งฮจ(๐‘ง0) | | ๐‘ง0 โˆˆ R๐‘›, |๐‘ง0 | = 1},

which is finite. Then we have ๐œ•ฮจ(๐‘ง0) โŠ‚ ๐ต๐พ (0) for all ๐‘ง0 โˆˆ R๐‘›. For the case ๐‘ง0 โ‰  0, this inclusion is clearby (6.12) and ๐œ•ฮจ(๐‘ง0) = {โˆ‡๐‘งฮจ(๐‘ง0)}. For ๐‘ง0 = 0, we take arbitrary ๐‘ค โˆˆ ๐œ•ฮจ(0) \ {0}. Then by the subgradientinequality, Eulerโ€™s identity (6.11) and the Cauchyโ€“Schwarz inequality, we have

|๐‘ค |2 = โŸจ๐‘ค | ๐‘คโˆ’0โŸฉ +ฮจ(0)โ‰ค ฮจ(๐‘ค) = โŸจโˆ‡๐‘งฮจ(๐‘ค) | ๐‘คโŸฉ โ‰ค ๐พ |๐‘ค |.

This estimate yields the inclusion ๐œ•ฮจ(0) โŠ‚ ๐ต๐พ (0).

2. For ๐‘ง0 โˆˆ R๐‘› \ {0}, the Hessian matrix โˆ‡2๐‘งฮจ(๐‘ง0) satisfies

0 โ‰คโŸจโˆ‡2๐‘งฮจ(๐‘ง0)๐œ

๏ฟฝ๏ฟฝ ๐œโŸฉ, (6.13)๏ฟฝ๏ฟฝโŸจโˆ‡2๐‘งฮจ(๐‘ง0)๐œ

๏ฟฝ๏ฟฝ ๐œ”โŸฉ๏ฟฝ๏ฟฝ โ‰ค ๐ถ

|๐‘ง0 ||๐œ | |๐œ” | (6.14)

for all ๐œ, ๐œ” โˆˆ R๐‘›. Here the finite constant ๐ถ is explicitly given by

๐ถ B sup{๏ฟฝ๏ฟฝโŸจโˆ‡2

๐‘งฮจ(๐‘ค)๐œ๏ฟฝ๏ฟฝ ๐œ”โŸฉ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ ๐‘ง, ๐œ , ๐œ” โˆˆ R๐‘›, |๐‘ค | = |๐œ‚ | = |๐œ” | = 1

}.

Lemma 4 states lower semicontinuity of a functional in the weakโˆ— topology of an ๐ฟโˆž-space. This result is usedin the justification of a blow-up argument for the equation (6.1).

Lemma 4. Let ฮฉ โŠ‚ R๐‘š be a Lebesgue measurable set, and let ฮจ : R๐‘› โ†’ [0,โˆž) be a convex function whichsatisfies (6.6). Assume that a vector field ๐‘ โˆˆ ๐ฟโˆž (ฮฉ, R๐‘›) and a sequence {๐‘๐‘ }๐‘ โŠ‚ ๐ฟโˆž (ฮฉ, R๐‘›) satisfy ๐‘๐‘

โˆ—โ‡€ ๐‘ in

๐ฟโˆž (ฮฉ, R๐‘›). Then we haveess sup๐‘ฅโˆˆฮฉ

ฮจฬƒ(๐‘ (๐‘ฅ)) โ‰ค liminf๐‘โ†’โˆž

ess sup๐‘ฅโˆˆฮฉ

ฮจฬƒ(๐‘๐‘ (๐‘ฅ)), (6.15)

where ฮจฬƒ denotes the support function of the closed convex set ๐ถฮจ B {๐‘ง โˆˆ R๐‘› | ฮจ(๐‘ง) โ‰ค 1}.

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We give an elementary proof of Lemma 4, which is based on a definition of ฮจฬƒ.

Proof. We consider the case ๐ถโˆž B liminf๐‘โ†’โˆž

ฮจฬƒ(๐‘๐‘ ) ๐ฟโˆž (ฮฉ) <โˆž, since otherwise (6.15) is clear. Fix arbitrary ๐œ€ > 0.

Then we may take a subsequence {๐‘๐‘ ๐‘— }โˆž๐‘—=1 such that

ess sup๐‘ฅโˆˆฮฉ

ฮจฬƒ(๐‘๐‘ ๐‘— (๐‘ฅ)) โ‰ค ๐ถโˆž + ๐œ€ <โˆž. (6.16)

Take arbitrary 0 โ‰ค ๐œ™ โˆˆ ๐ฟ1 (ฮฉ) and ๐‘ค โˆˆ ๐ถฮจ. Then with the aid of (6.9), we have

โŸจ๐‘๐‘ ๐‘— (๐‘ฅ) | ๐‘คโŸฉ โ‰ค ๐ถโˆž + ๐œ€

for all ๐‘— โˆˆ N and for a.e. ๐‘ฅ โˆˆ ฮฉ, which yieldsโˆซฮฉ

[๐ถโˆž + ๐œ€โˆ’ โŸจ๐‘๐‘ ๐‘— (๐‘ฅ) | ๐‘คโŸฉ

]๐œ™(๐‘ฅ) ๐‘‘๐‘ฅ โ‰ฅ 0 (6.17)

for all ๐‘— โˆˆ N. Letting ๐‘— โ†’โˆž, we have โˆซฮฉ[๐ถโˆž + ๐œ€โˆ’ โŸจ๐‘ (๐‘ฅ) | ๐‘คโŸฉ]๐œ™(๐‘ฅ) ๐‘‘๐‘ฅ โ‰ฅ 0

by ๐‘๐‘ ๐‘—

โˆ—โ‡€ ๐‘ in ๐ฟโˆž (ฮฉ, R๐‘›). Since 0 โ‰ค ๐œ™ โˆˆ ๐ฟ1 (ฮฉ) is arbitrary, for each ๐‘ค โˆˆ ๐ถฮจ, there exists an L๐‘›-measurable set

๐‘ˆ๐‘ค โŠ‚ ฮฉ, such that L๐‘› (๐‘ˆ๐‘ค) = 0 and

โŸจ๐‘ (๐‘ฅ) | ๐‘คโŸฉ โ‰ค ๐ถโˆž + ๐œ€ for all ๐‘ฅ โˆˆ ฮฉ \๐‘ˆ๐‘ค .

Here we denote L๐‘› by the ๐‘›-dimensional Lebesgue measure. Since ๐ถฮจ โŠ‚ R๐‘› is separable, we may take a countableand dense set ๐ท โŠ‚ ๐ถ๐œ“ . We set an L๐‘›-measurable set

๐‘ˆ Bโ‹ƒ๐‘คโˆˆ๐ท

๐‘ˆ๐‘ค โŠ‚ ฮฉ,

which clearly satisfies L๐‘› (๐‘ˆ) = 0. Then we conclude that

โŸจ๐‘ (๐‘ฅ) | ๐‘คโŸฉ โ‰ค ๐ถโˆž + ๐œ€ for all ๐‘ฅ โˆˆ ฮฉ \๐‘ˆ, ๐‘ค โˆˆ ๐ถฮจ

from density of ๐ท โŠ‚ ๐ถฮจ. Hence by definition of ฮจฬƒ, it is clear that

ฮจฬƒ(๐‘ (๐‘ฅ)) โ‰ค ๐ถโˆž + ๐œ€ for a.e. ๐‘ฅ โˆˆ ฮฉ.

Since ๐œ€ > 0 is arbitrary, this completes the proof of (6.15). โ–ก

6.2 Sketches of the proofsWe first give definitions of weak solutions to (6.1). We also define weak subsolutions, and supersolutions to anequation ๐ฟ๐‘ข = 0 in a bounded domain.

Definition 3. Let ฮฉ โŠ‚ R๐‘› be a domain.

1. Let ๐‘“ โˆˆ ๐ฟ๐‘žloc (ฮฉ) (๐‘› < ๐‘ž โ‰ค โˆž). We say that a function ๐‘ข โˆˆ ๐‘Š1,โˆžloc (ฮฉ) is a weak solution to (6.1), when

for any bounded Lipschitz domain ๐œ” โ‹ ฮฉ, there exists a vector field ๐‘ โˆˆ ๐ฟโˆž (๐œ”, R๐‘›) such that the pair(๐‘ข, ๐‘) โˆˆ๐‘Š1,โˆž (๐œ”) ร— ๐ฟโˆž (๐œ”, R๐‘›) satisfiesโˆซ

๐œ”โŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซ๐œ”โŸจ๐ด(โˆ‡๐‘ข) | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ =

โˆซ๐œ”๐‘“ ๐œ™ ๐‘‘๐‘ฅ (6.18)

for all ๐œ™ โˆˆ๐‘Š1, 10 (๐œ”), and

๐‘ (๐‘ฅ) โˆˆ ๐œ•ฮจ(โˆ‡๐‘ข(๐‘ฅ)) (6.19)

for a.e. ๐‘ฅ โˆˆ ๐œ”. Here ๐ด denotes the continuous mapping ๐ด : R๐‘› โˆ‹ ๐‘ฅ โ†ฆโ†’ โˆ‡๐‘ง๐‘Š (๐‘ฅ) โˆˆ R๐‘›. For such pair (๐‘ข, ๐‘),we say that (๐‘ข, ๐‘) satisfies ๐ฟ๐‘ข = ๐‘“ in๐‘Šโˆ’1,โˆž (๐œ”) or simply say that ๐‘ข satisfies ๐ฟ๐‘ข = ๐‘“ in๐‘Šโˆ’1,โˆž (๐œ”).

20

2. Assume that ฮฉ is bounded. A pair (๐‘ข, ๐‘) โˆˆ๐‘Š1,โˆž (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) is called a weak subsolution to ๐ฟ๐‘ข = 0 inฮฉ, if it satisfies โˆซ

ฮฉโŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซฮฉโŸจ๐ด(โˆ‡๐‘ข) | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ โ‰ค 0 (6.20)

for all 0 โ‰ค ๐œ™ โˆˆ ๐ถโˆž๐‘ (ฮฉ), and

๐‘ (๐‘ฅ) โˆˆ ๐œ•ฮจ(โˆ‡๐‘ข(๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ ฮฉ. (6.21)

Similarly we call a pair (๐‘ข, ๐‘) โˆˆ๐‘Š1, ๐‘ (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) a weak supersolution ๐ฟ๐‘, ๐‘๐‘ข = 0 in ฮฉ, if it satisfies(6.21) and โˆซ

ฮฉโŸจ๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซฮฉโŸจ๐ด(โˆ‡๐‘ข) | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ โ‰ฅ 0

for all 0 โ‰ค ๐œ™ โˆˆ๐ถโˆž๐‘ (ฮฉ). For ๐‘ข โˆˆ๐‘Š1, ๐‘ (ฮฉ), we simply say that ๐‘ข is respectively a subsolution and a supersolution

to ๐ฟ๐‘ข = 0 in the weak sense if there is ๐‘ โˆˆ ๐ฟโˆž (ฮฉ, R๐‘›) such that the pair (๐‘ข, ๐‘) is a weak subsolution and aweak supersolution to ๐ฟ๐‘ข = 0 in ฮฉ.

Remark 6. We describe some remarks on Definition 3.

1. In this paper we treat a convex solution, which clearly satisfies local Lipschitz regularity. Hence it is notrestrictive to assume local or global ๐‘Š1,โˆž-regularity for solutions in Definition 3. Also it should be notedthat if a vector field ๐‘ satisfies (6.19), then ๐‘ is in ๐ฟโˆž by Remark 5. Hence our regularity assumptions of thepair (๐‘ข, ๐‘) involve no loss of generality.

2. Integrals in (6.18) make sense by ๐‘, โˆ‡๐‘ข โˆˆ ๐ฟโˆž (๐œ”, R๐‘›), ๐ด โˆˆ ๐ถ (R๐‘›, R๐‘›), and the continuous embedding๐‘Š1, 1

0 (๐œ”) โ†ฉโ†’ ๐ฟ๐‘žโ€ฒ (๐œ”).

3. For a bounded domain ฮฉ โŠ‚ R๐‘›, let ๐‘ข โˆˆ ๐ถ2 (ฮฉ) satisfy

โˆ‡๐‘ข(๐‘ฅ) โ‰  0 for all ๐‘ฅ โˆˆ ฮฉ, and

๐ฟ๐‘ข(๐‘ฅ) โ‰ค 0 for all ๐‘ฅ โˆˆ ฮฉ.

Then the pair (๐‘ข, โˆ‡๐‘งฮจ(โˆ‡๐‘ข)) โˆˆ๐‘Š1, ๐‘ (ฮฉ) ร— ๐ฟโˆž (ฮฉ, R๐‘›) satisfies (6.20)โ€“(6.21). For such ๐‘ข, we simply say that๐‘ข satisfies ๐ฟ๐‘ข โ‰ค 0 in ฮฉ in the classical sense.

To prove Theorem 4, we may assume that ฮฉ is a bounded convex domain, since our argument is local. Asdescribed in Section 1.1, we would like to prove that a convex solution ๐‘ข to (6.1) satisfies (1.4) for all ๐‘ฅ โˆˆ ฮฉ.

For the case ๐‘ฅ โˆˆ ๐ท, we can show (1.4) by De Giorgiโ€“Nashโ€“Moser theory. This is basically due to the fact thatthe functional

๐ธ (๐‘ง) B ฮจ(๐‘ง) +๐‘Š (๐‘ง) for ๐‘ง โˆˆ R๐‘›

satisfy the following property. For each fixed constants 0 < ๐œ‡ โ‰ค ๐‘€ <โˆž, there exists constants 0 < ๐œ† โ‰ค ฮ› <โˆž suchthat the estimates (2.7)โ€“(2.8) hold for all ๐‘ง0, ๐œ , ๐œ” โˆˆ R๐‘› with ๐œ‡ โ‰ค |๐‘ง0 | โ‰ค ๐‘€ . In other words, the operator ๐ฟ is locallyuniformly elliptic outside a facet, in the sense that for a function ๐‘ฃ the operator ๐ฟ๐‘ฃ becomes uniformly elliptic in aplace where 0 < ๐œ‡ โ‰ค |โˆ‡๐‘ฃ | โ‰ค ๐‘€ <โˆž holds. This ellipticity is an easy consequence of (6.3)โ€“(6.4) and (6.13)โ€“(6.14).Appealing to local uniform ellipticity of the operator ๐ฟ outside the facet and De Giorgiโ€“Nashโ€“Moser theory, weare able to show that a convex solution to ๐ฟ๐‘ข = ๐‘“ is ๐ถ1, ๐›ผ near a neighborhood of each fixed point ๐‘ฅ โˆˆ ๐ท, similarlyto the proof of Lemma 1.

For the case ๐‘ฅ โˆˆ ๐น, we first make a blow-argument to construct a convex function ๐‘ข0 : R๐‘› โ†’ R satisfying๐œ•๐‘ข(๐‘ฅ) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ), and ๐ฟ๐‘ข0 = 0 in R๐‘› in the sense of Definition 3. Next we justify a maximum principle, whichis described as in (1.6), holds on each connected component of ๐ท. This result enables us to apply Lemma 3,and thus similarly in Section 5.1, we are able to prove a Liouville-type theorem. Hence it follows that a convexsolution ๐‘ข0, which is constructed by the previous blow-argument, should be constant. Finally the inclusions{0} โŠ‚ ๐œ•๐‘ข(๐‘ฅ) โŠ‚ ๐œ•๐‘ข0 (๐‘ฅ) โŠ‚ {0} hold, and this completes the proof of (1.4), i.e., ๐œ•๐‘ข(๐‘ฅ) = {0}.

For maximum principles on the equation ๐ฟ๐‘ข = 0, the proofs are almost similar to those in Section 4. Indeed,we first recall that the operator ๐ด : R๐‘› โˆ‹ ๐‘ง0 โ†ฆโ†’ โˆ‡๐‘ง๐‘Š (๐‘ง0) โˆˆ R๐‘› satisfies strict monotonicity (4.8). Combining withmonotonicity of the subdifferential operator ๐œ•ฮจ, we can easily prove a comparison principle as in Proposition 2.Also, similarly to Lemma 2, we can construct classical barrier subsolutions to ๐ฟ๐‘ข = 0 in an open annulus, since

21

the operator ๐ฟ is locally uniformly elliptic outside a facet. These results enable us to prove a maximum principleoutside a facet.

We are left to justify the remaining two problems, a blow-argument and the Liouville-type theorem. To showthem, we have to make use of some basic facts on a convex functional which is homogeneous of degree 1. Thesefundamental results are contained in Section A.3.

For a blow-up argument as in Section 3, we similarly define rescaled solutions. Existence of a limit of theserescaled functions are guaranteed by the Arzelร โ€“Ascoli theorem and a diagonal argument. By proving Lemma 5below, we are able to demonstrate that ๐‘ข0, a limit of rescaled solutions, is a weak solution to ๐ฟ๐‘ข = 0 in R๐‘›, and thisfinishes our blow-up argument.

Lemma 5. Let ๐‘ˆ โŠ‚ R๐‘› be a bounded domain. Assume that sequences of functions {๐‘ข๐‘ }โˆž๐‘=1 โŠ‚ ๐‘Š1,โˆž (๐‘ˆ) and{ ๐‘“๐‘ }โˆž๐‘=1 โŠ‚ ๐ฟ๐‘ž (๐‘ˆ) (๐‘› < ๐‘ž โ‰ค โˆž) satisfy all of the following.

1. For each ๐‘ โˆˆ N, ๐‘ข๐‘ satisfies ๐ฟ๐‘ข๐‘ = ๐‘“๐‘ in๐‘Šโˆ’1,โˆž (๐‘ˆ).

2. There exists a constant ๐‘€ > 0, independent of ๐‘ โˆˆ N, such that

|โˆ‡๐‘ข๐‘ (๐‘ฅ) | โ‰ค ๐‘€ for a.e. ๐‘ฅ โˆˆ๐‘ˆ. (6.22)

3. There exists a function ๐‘ข โˆˆ๐‘Š1,โˆž (๐‘ˆ) such that

โˆ‡๐‘ข๐‘ (๐‘ฅ) โ†’ โˆ‡๐‘ข(๐‘ฅ) for a.e. ๐‘ฅ โˆˆ๐‘ˆ. (6.23)

4. ๐‘“๐‘ strongly converges to 0 in ๐ฟ๐‘ž (๐‘ˆ).

Then ๐‘ข satisfies ๐ฟ๐‘ข = 0 in๐‘Šโˆ’1,โˆž (๐‘ˆ).

Proof. For each ๐‘ โˆˆ N, there exists a vector field ๐‘๐‘ โˆˆ ๐ฟโˆž (๐‘ˆ, R๐‘›) such that

๐‘๐‘ (๐‘ฅ) โˆˆ ๐œ•ฮจ(โˆ‡๐‘ข(๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ๐‘ˆ, (6.24)โˆซ๐‘ˆโŸจ๐‘๐‘ | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซ๐‘ˆโŸจ๐ด(โˆ‡๐‘ข๐‘ ) | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ =

โˆซ๐‘ˆ๐‘“๐‘๐œ™๐‘‘๐‘ฅ for all ๐œ™ โˆˆ๐‘Š1, 1

0 (๐‘ˆ). (6.25)

Combining the assumption ๐‘“๐‘ โ†’ ๐‘“ in ๐ฟ๐‘ž (๐‘ˆ) with the continuous embedding ๐ฟ๐‘ž (๐‘ˆ) โ†ฉโ†’๐‘Šโˆ’1,โˆž (๐‘ˆ), we get

๐‘“๐‘ โ†’ 0 in๐‘Šโˆ’1,โˆž (๐‘ˆ). (6.26)

By ๐ด โˆˆ ๐ถ (R๐‘›, R๐‘›) and (6.22), the vector fields {๐ด(โˆ‡๐‘ข๐‘ )}โˆž๐‘=1 satisfy

๐ด(โˆ‡๐‘ข๐‘ (๐‘ฅ)) โ†’ ๐ด(โˆ‡๐‘ข(๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ๐‘ˆ,

|๐ด๐‘ (โˆ‡๐‘ข๐‘ (๐‘ฅ)) โˆ’ ๐ด(โˆ‡๐‘ข(๐‘ฅ)) | โ‰ค ๐ถ for a.e. ๐‘ฅ โˆˆ๐‘ˆ,

where ๐ถ is independent of ๐‘ โˆˆ N. From these and Lebesgueโ€™s dominated convergence theorem, it follows that

๐ด(โˆ‡๐‘ข๐‘ )โˆ—โ‡€ ๐ด(โˆ‡๐‘ข) in ๐ฟโˆž (๐‘ˆ, R๐‘›). (6.27)

As mentioned in Remark 5โ€“6, the {๐‘๐‘ }โˆž๐‘=1 โŠ‚ ๐ฟโˆž (๐‘ˆ, R๐‘›) is bounded. Hence by [5, Corollary 3.30], we may takea subsequence {๐‘๐‘ ๐‘— }โˆž๐‘—=1 so that

๐‘๐‘ ๐‘—

โˆ—โ‡€ ๐‘ in ๐ฟโˆž (๐‘ˆ, R๐‘›) (6.28)

for some ๐‘ โˆˆ ๐ฟโˆž (๐‘ˆ, R๐‘›). By (6.25)โ€“(6.28) we obtainโˆซ๐‘ˆโŸจ๐‘ | ๐œ™โŸฉ ๐‘‘๐‘ฅ +

โˆซ๐‘ˆโŸจ๐ด(โˆ‡๐‘ข) | โˆ‡๐œ™โŸฉ ๐‘‘๐‘ฅ = 0 for all ๐œ™ โˆˆ๐‘Š1, 1

0 (๐‘ˆ).

Now we are left to prove that๐‘ (๐‘ฅ) โˆˆ ๐œ•ฮจ(โˆ‡๐‘ข(๐‘ฅ)) for a.e. ๐‘ฅ โˆˆ๐‘ˆ.

22

By (6.10), it suffices to show that ๐‘ satisfiesฮจฬƒ(๐‘ (๐‘ฅ)) โ‰ค 1, (6.29)

ฮจ(โˆ‡๐‘ข(๐‘ฅ)) = โŸจ๐‘ | โˆ‡๐‘ข(๐‘ฅ)โŸฉ (6.30)

for a.e. ๐‘ฅ โˆˆ๐‘ˆ. Similarly, it follows that for each ๐‘ โˆˆ N, the vector field ๐‘๐‘ satisfies{ฮจฬƒ(๐‘๐‘ (๐‘ฅ)) โ‰ค 1,

ฮจ(โˆ‡๐‘ข๐‘ (๐‘ฅ)) = โŸจ๐‘๐‘ | โˆ‡๐‘ข(๐‘ฅ)โŸฉ, for a.e. ๐‘ฅ โˆˆ๐‘ˆ.

Hence (6.29) is an easy consequence of Lemma 4. We recall (6.2), and thus ๐œ•ฮจ(๐‘ง0) = {โˆ‡๐‘งฮจ(๐‘ง0)} holds for all๐‘ง0 โˆˆ R๐‘› \ {0}. Combining (6.23), we can check that ๐‘๐‘ (๐‘ฅ) โ†’ ๐‘ (๐‘ฅ) for a.e. ๐‘ฅ โˆˆ ๐ท B {๐‘ฅ โˆˆ๐‘ˆ | โˆ‡๐‘ข(๐‘ฅ) โ‰  0}. Hence(6.30) holds for a.e. ๐‘ฅ โˆˆ ๐ท. Note that (6.30) is clear for ๐‘ฅ โˆˆ๐‘ˆ \๐ท, and this completes the proof. โ–ก

We prove a Liouville-type theorem as in Theorem 3. In other words, for a convex solution to ๐ฟ๐‘ข = 0 in R๐‘›,we show that ๐น, the facet of ๐‘ข, would satisfy either ๐น = โˆ… or ๐น = R๐‘›. Assume by contradiction that ๐น satisfiesโˆ… โŠŠ ๐น โŠŠ R๐‘›. Then by Lemma 3, we may write a convex solution ๐‘ข by either of (5.1)โ€“(5.3). However, Lemma 6below states that ๐‘ข is no longer a weak solution, and this completes our proof.

Lemma 6. Let ๐‘ข be a piecewise-linear function defined as in either of (5.1)โ€“(5.3). Then ๐‘ข is not a weak solutionto ๐ฟ๐‘ข = 0 in R๐‘›.

Proof. As in the proof of Theorem 3, we introduce a constant ๐‘‘ > 0, and set open cubes๐‘„ โ€ฒ โŠ‚ R๐‘›โˆ’1 and๐‘„, ๐‘„๐‘™ , ๐‘„๐‘Ÿ โŠ‚R๐‘›. By choosing sufficiently small ๐‘‘ > 0, we show that ๐‘ข does not satisfy ๐ฟ๐‘ข = 0 in ๐‘Šโˆ’1,โˆž (๐‘„). Assume bycontradiction that there exists a vector field ๐‘ โˆˆ ๐ฟโˆž (๐‘„, R๐‘›) such that the pair (๐‘ข, ๐‘) satisfies ๐ฟ๐‘ข = 0 in๐‘Šโˆ’1,โˆž (๐‘„).

We first show that a function ๐‘ข defined as in (5.1) is not a weak solution. For this case, (6.12) implies that ๐‘satisfies ๐‘ (๐‘ฅ) = โˆ‡๐‘งฮจ(๐‘’1) for a.e. ๐‘ฅ โˆˆ ๐‘„๐‘Ÿ . We take and fix non-negative functions ๐œ™1 โˆˆ ๐ถ1

๐‘ ((โˆ’๐‘‘, ๐‘‘)), ๐œ™2 โˆˆ ๐ถ1๐‘ (๐‘„ โ€ฒ)

such that (5.5) holds, and define ๐œ™ โˆˆ ๐ถ1๐‘ (๐‘„) by ๐œ™(๐‘ฅ1, ๐‘ฅ

โ€ฒ) B ๐œ™1 (๐‘ฅ1)๐œ™2 (๐‘ฅ โ€ฒ) for (๐‘ฅ1, ๐‘ฅโ€ฒ) โˆˆ (โˆ’๐‘‘, ๐‘‘) ร—๐‘„ โ€ฒ = ๐‘„. Testing

๐œ™ into ๐ฟ๐‘ข = 0 in๐‘Šโˆ’1,โˆž (๐‘„), we have

0 =โˆซ๐‘„๐‘™

โŸจ๐‘ + ๐ด(0) | โˆ‡(๐œ™1๐œ™2)โŸฉ ๐‘‘๐‘ฅ +โˆซ๐‘„๐‘Ÿ

โŸจโˆ‡๐‘งฮจ(๐‘’1) + ๐ด(๐‘ก1๐‘’1) | โˆ‡(๐œ™1๐œ™2)โŸฉ ๐‘‘๐‘ฅ

โ‰คโˆซ๐‘„๐‘™

ฮจ(โˆ‡(๐œ™1๐œ™2))ฮจฬƒ(๐‘ (๐‘ฅ)) ๐‘‘๐‘ฅ

+๐œ™1 (0)โˆซ๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ)โŸจโˆ‡๐‘งฮจ(๐‘’1) | โˆ’๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ+๐œ™1 (0)

โˆซ๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ)โŸจ๐ด(๐‘ก1๐‘’1) | โˆ’๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ

C ๐ผ1 + ๐ผ2 + ๐ผ3.

Here we have used the Cauchyโ€“Schwarz-type inequality (6.5) for the integral over๐‘„๐‘™ , and applied the Gaussโ€“Greentheorem to the integration over ๐‘„๐‘Ÿ . For ๐ผ1, we make use of (6.9)โ€“(6.8), Fubiniโ€™s theorem and (5.5). Then we have

๐ผ1 โ‰คโˆซ๐‘„๐‘™

๐œ™1 (๐‘ฅ1)ฮจ(0, โˆ‡๐‘ฅโ€ฒ๐œ™2 (๐‘ฅ โ€ฒ)) ๐‘‘๐‘ฅ +โˆซ๐‘„๐‘™

๐œ™โ€ฒ1 (๐‘ฅ1)๐œ™2 (๐‘ฅ โ€ฒ)ฮจ(๐‘’1) ๐‘‘๐‘ฅ

โ‰ค ๐œ™1 (0)(๐‘‘ ยท โˆฅฮจ(0, โˆ‡๐‘ฅโ€ฒ๐œ™2)โˆฅ๐ฟ1 (๐‘„โ€ฒ) +ฮจ(๐‘’1)โˆฅ๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ)

),

where โˆ‡๐‘ฅโ€ฒ๐œ™2 B (๐œ•๐‘ฅ2๐œ™2, . . . , ๐œ•๐‘ฅ๐‘›๐œ™2). For ๐ผ2, recalling Eulerโ€™s identity (6.11), we get ๐ผ2 = โˆ’๐œ™1 (0)ฮจ(๐‘’1)โˆฅ๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ) .We set a constant ๐œ‡ B โŸจ๐ด(๐‘ก1๐‘’1) | ๐‘’1โŸฉ, which is positive by (6.7). Then we obtain

๐ผ1 + ๐ผ2 + ๐ผ3 โ‰ค ๐œ™1 (0)(๐‘‘ ยท โˆฅฮจ(0, โˆ‡๐‘ฅโ€ฒ๐œ™2)โˆฅ๐ฟ1 (๐‘„โ€ฒ) โˆ’ ๐œ‡โˆฅ๐œ™2โˆฅ๐ฟ1 (๐‘„โ€ฒ)

).

Choosing ๐‘‘ = ๐‘‘ (๐œ‡, ฮจ, ๐œ™2) > 0 sufficiently small, we have 0 โ‰ค ๐ผ1 + ๐ผ2 + ๐ผ3 < 0, which is a contradiction. Similarlywe can deduce that ๐‘ข defined as in (5.3) does not satisfy ๐ฟ๐‘ข = 0 in ๐‘Šโˆ’1,โˆž (๐‘„), since it suffices to restrict ๐‘‘ < ๐‘™0.For the remaining case (5.2), we have already known that

๐‘ (๐‘ฅ) ={

โˆ‡๐‘งฮจ(๐‘’1) for a.e. ๐‘ฅ โˆˆ ๐‘„๐‘Ÿ ,โˆ‡๐‘งฮจ(โˆ’๐‘’1) for a.e. ๐‘ฅ โˆˆ ๐‘„๐‘™

23

by definition of ๐‘ and (6.12). We set two constants ๐œ‡1 B โŸจ๐ด(๐‘ก1๐‘’1) | ๐‘’1โŸฉ, ๐œ‡2 B โŸจ๐ด(โˆ’๐‘ก2๐‘’1) | โˆ’๐‘’1โŸฉ, both of which arepositive by (6.7). Testing the same function ๐œ™ โˆˆ ๐ถ1

๐‘ (๐‘„) into ๐ฟ๐‘ข = 0 in๐‘Šโˆ’1,โˆž (๐‘„), we obtain

0 =โˆซ๐‘„๐‘™

โŸจโˆ‡๐‘งฮจ(โˆ’๐‘’1) + ๐ด(โˆ’๐‘ก2๐‘’1) | โˆ‡(๐œ™1๐œ™2)โŸฉ ๐‘‘๐‘ฅ +โˆซ๐‘„๐‘Ÿ

โŸจโˆ‡๐‘งฮจ(๐‘’1) + ๐ด(๐‘ก1๐‘’1) | โˆ‡(๐œ™1๐œ™2)โŸฉ ๐‘‘๐‘ฅ

=โˆซ๐‘„โ€ฒ๐œ™1 (0)๐œ™2 (๐‘ฅ โ€ฒ)โŸจโˆ‡๐‘งฮจ(โˆ’๐‘’1) + ๐ด(โˆ’๐‘ก2๐‘’1) | ๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ+

โˆซ๐‘„โ€ฒ๐œ™1 (0)๐œ™2 (๐‘ฅ โ€ฒ)โŸจโˆ‡๐‘งฮจ(๐‘’1) + ๐ด(๐‘ก1๐‘’1) | โˆ’๐‘’1โŸฉ ๐‘‘๐‘ฅ โ€ฒ

= โˆ’๐œ™1 (0) (ฮจ(๐‘’1) +ฮจ(โˆ’๐‘’1) + ๐œ‡1 + ๐œ‡2)โˆซ๐‘„โ€ฒ๐œ™2 (๐‘ฅ โ€ฒ) ๐‘‘๐‘ฅ โ€ฒ < 0,

which is a contradiction. Here we have used the Gaussโ€“Green theorem and Eulerโ€™s identity (6.11). This completesthe proof. โ–ก

AcknowledgementThe first author is partly supported by the Japan Society for the Promotion of Science through grants Kiban A (No.19H00639). Challenging Pioneering Research (Kaitaku) (No. 18H05323), Kiban A (No. 17H01091).

A Proofs for a few basic factsIn this section, we give proofs for a few basic facts used in this paper for completeness.

A.1 A Poincarรฉ-type inequalityWe give a precise proof of Lemma 7, a Poincarรฉ-type inequality for difference quotients of functions in๐‘Š1, ๐‘

0 (1 โ‰ค๐‘ <โˆž). This result is used in the proof of Lemma 1. The proof of Lemma 7 is essentially a modification of that ofthe Poincarรฉ inequality for the Sobolev space๐‘Š1, ๐‘

0 [10, Proposition 3.10].

Lemma 7. Let ฮฉ โŠ‚ R๐‘› be a bounded open set and 1 โ‰ค ๐‘ <โˆž. For all ๐‘ข โˆˆ๐‘Š1, ๐‘0 (ฮฉ), ๐‘— โˆˆ {1, . . . , ๐‘› }, โ„Ž โˆˆ R \ {0},

we haveโˆฅฮ” ๐‘— , โ„Ž๐‘ขโˆฅ๐ฟ๐‘ (ฮฉ) โ‰ค โˆฅโˆ‡๐‘ขโˆฅ๐ฟ๐‘ (ฮฉ) . (A.1)

Here ฮ” ๐‘— , โ„Ž๐‘ข is defined by

ฮ” ๐‘— , โ„Ž๐‘ข(๐‘ฅ) B๐‘ข(๐‘ฅ + โ„Ž๐‘’ ๐‘— ) โˆ’๐‘ข(๐‘ฅ)

โ„Žfor ๐‘ฅ โˆˆ ฮฉ.

Before the proof of Lemma 7, we note that ฮ” ๐‘— , โ„Ž๐‘ข(๐‘ฅ) makes sense for a.e. ๐‘ฅ โˆˆ ฮฉ by the zero extension of๐‘ข โˆˆ๐‘Š1, ๐‘

0 (๐‘ˆ). That is, for a given ๐‘ข โˆˆ๐‘Š1, ๐‘0 (๐‘ˆ), we set ๐‘ข โˆˆ๐‘Š1, ๐‘ (R๐‘›) by

๐‘ข(๐‘ฅ) B{๐‘ข(๐‘ฅ) ๐‘ฅ โˆˆ๐‘ˆ,

0 ๐‘ฅ โˆˆ R๐‘› \๐‘ˆ. (A.2)

Proof. We fix ๐‘— โˆˆ {1, . . . , ๐‘› }, โ„Ž โˆˆ R \ {0}. We first note that the operator ฮ” ๐‘— , โ„Ž : ๐‘Š1, ๐‘0 (๐‘ˆ) โ†’ ๐ฟ ๐‘ (๐‘ˆ) is bounded,

since for all ๐‘ข โˆˆ๐‘Š1, ๐‘0 (๐‘ˆ) we have

โˆฅฮ” ๐‘— , โ„Ž๐‘ขโˆฅ๐ฟ๐‘ (๐‘ˆ) โ‰ค1|โ„Ž|

[(โˆซ๐‘ˆ|๐‘ข(๐‘ฅ + โ„Ž) |๐‘ ๐‘‘๐‘ฅ

)1/๐‘+(โˆซ๐‘ˆ|๐‘ข(๐‘ฅ) |๐‘ ๐‘‘๐‘ฅ

)1/๐‘]โ‰ค 2

|โ„Ž| โˆฅ๐‘ขโˆฅ๐ฟ๐‘ (๐‘ˆ) โ‰ค

๐ถ (๐‘, ๐‘ˆ)|โ„Ž| โˆฅโˆ‡๐‘ขโˆฅ๐ฟ๐‘ (๐‘ˆ)

by the Minkowski inequality and the Poincarรฉ inequality. Here ๐‘ข โˆˆ๐‘Š1, ๐‘ (R๐‘›) is defined as in (A.2). Hence by adensity argument, it suffices to check that (A.1) holds true for all ๐‘ข โˆˆ ๐ถโˆž

๐‘ (๐‘ˆ). Let ๐‘ข โˆˆ ๐ถโˆž๐‘ (๐‘ˆ). Then for all ๐‘ฅ โˆˆ๐‘ˆ,

24

we have

|๐‘ข(๐‘ฅ + โ„Ž๐‘’ ๐‘— ) โˆ’๐‘ข(๐‘ฅ) | =๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโˆซ 1

0

โŸจโˆ‡๐‘ข(๐‘ฅ + ๐‘กโ„Ž๐‘’ ๐‘— )

๏ฟฝ๏ฟฝ โ„Ž๐‘’ ๐‘— โŸฉ ๐‘‘๐‘ก๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ‰ค |โ„Ž|

โˆซ 1

0|โˆ‡๐‘ข(๐‘ฅ + ๐‘กโ„Ž๐‘’ ๐‘— ) | ๐‘‘๐‘ก โ‰ค |โ„Ž|

(โˆซ 1

0|โˆ‡๐‘ข(๐‘ฅ + ๐‘กโ„Ž๐‘’ ๐‘— ) |๐‘ ๐‘‘๐‘ก

)1/๐‘

by the Cauchy-Schwarz inequality and Hรถlderโ€™s inequality. From this estimate we get

โˆฅฮ” ๐‘— , โ„Ž๐‘ขโˆฅ ๐‘๐ฟ๐‘ (๐‘ˆ) โ‰คโˆซฮฉ

โˆซ 1

0|โˆ‡๐‘ข(๐‘ฅ + ๐‘กโ„Ž๐‘’ ๐‘— ) |๐‘ ๐‘‘๐‘ก ๐‘‘๐‘ฅ

=โˆซ 1

0

โˆซ๐‘ˆ|โˆ‡๐‘ข(๐‘ฅ + ๐‘กโ„Ž๐‘’ ๐‘— ) |๐‘ ๐‘‘๐‘ฅ๏ธธ ๏ธท๏ธท ๏ธธ

โ‰คโˆฅโˆ‡๐‘ข โˆฅ๐‘๐ฟ๐‘ (๐‘ˆ)

๐‘‘๐‘ก (by Fubiniโ€™s theorem)

โ‰ค โˆฅโˆ‡๐‘ขโˆฅ ๐‘๐ฟ๐‘ (๐‘ˆ) .

Hence we obtain (A.1) for all ๐‘ข โˆˆ ๐ถโˆž๐‘ (๐‘ˆ), and this completes the proof. โ–ก

A.2 Convex analysisLemma 8 is used in the proof of Lemma 1 for a justification of local ๐‘Š2, 2-regularity of a convex weak solutionoutside of the facet.

Lemma 8. Let ๐‘ข be a real-valued convex function in a convex domain ฮฉ โŠ‚ R๐‘›. Assume that ๐‘ฅ1, ๐‘ฅ2 โˆˆ ฮฉ satisfy๐‘ฅ1 โ‰  ๐‘ฅ2, and set ๐‘‘ B |๐‘ฅ2 โˆ’ ๐‘ฅ1 | > 0, ๐œˆ B ๐‘‘โˆ’1 (๐‘ฅ2 โˆ’ ๐‘ฅ1). Then for all ๐‘ง2 โˆˆ ๐œ•๐‘ข(๐‘ฅ2), we have

โŸจ๐‘ง2 | ๐œˆโŸฉ โ‰ฅ ๐‘ข(๐‘ฅ2) โˆ’๐‘ข(๐‘ฅ1)๐‘‘

. (A.3)

Proof. By ๐‘ง2 โˆˆ ๐œ•๐‘ข(๐‘ฅ2), we have a subgradient inequality

๐‘ข(๐‘ฅ) โ‰ฅ ๐‘ข(๐‘ฅ2) + โŸจ๐‘ง2 | ๐‘ฅโˆ’ ๐‘ฅ2โŸฉ

for all ๐‘ฅ โˆˆ ฮฉ. Substituting ๐‘ฅ B ๐‘ฅ1 = ๐‘ฅ2 โˆ’ ๐‘‘๐œˆ โˆˆ ฮฉ, we obtain

๐‘ข(๐‘ฅ1) โ‰ฅ ๐‘ข(๐‘ฅ2) โˆ’ ๐‘‘โŸจ๐‘ง2 | ๐œˆโŸฉ,

which yields (A.3). โ–ก

Remark 7. Instead of subgradient inequalities, we are able to show (A.3) by monotonicity of ๐œ•๐‘ข. For each fixed๐‘ฅ1, ๐‘ฅ2 โˆˆ ฮฉ with ๐‘ฅ1 โ‰  ๐‘ฅ2, we may take and fix ๐‘ฅ3 B ๐‘ฅ1 + ๐‘ก (๐‘ฅ2 โˆ’ ๐‘ฅ1) for some 0 < ๐‘ก < 1 and ๐‘ง3 โˆˆ ๐œ•๐‘ข(๐‘ฅ3) such that

๐‘ข(๐‘ฅ2) โˆ’๐‘ข(๐‘ฅ1) = โŸจ๐‘ง3 | ๐‘ฅ2 โˆ’ ๐‘ฅ1โŸฉ, (A.4)

with the aid of the mean value theorem for non-smooth convex functions [1, Theorem D.6]. ๐‘ฅ2 โˆ’ ๐‘ฅ1 = ๐‘‘๐œˆ is clearby definitions of ๐‘‘, ๐œˆ. Noting ๐‘ฅ2 โˆ’ ๐‘ฅ3 = (1โˆ’ ๐‘ก)๐‘‘๐œˆ, we can check that

โŸจ๐‘ง2 โˆ’ ๐‘ง3 | ๐œˆโŸฉ = 1(1โˆ’ ๐‘ก)๐‘‘ โŸจ๐‘ง2 โˆ’ ๐‘ง3 | ๐‘ฅ2 โˆ’ ๐‘ฅ3โŸฉ โ‰ฅ 0

by monotonicity of ๐œ•๐‘ข. Combining these results with (A.4), we obtain

๐‘ข(๐‘ฅ2) โˆ’๐‘ข(๐‘ฅ1) = ๐‘‘โŸจ๐‘ง3 | ๐œˆโŸฉ โ‰ค ๐‘‘โŸจ๐‘ง2 | ๐œˆโŸฉ,

which yields (A.3).

The following lemma is used in the proof of Proposition 1.

25

Lemma 9. Let๐‘ˆ โŠ‚ R๐‘› be a convex open set, and let {๐‘ข๐‘ }โˆž๐‘=1 be a sequence of real-valued convex functions in๐‘ˆ.Assume that this sequence is uniformly Lipschitz. In other words, there is a constant ๐ฟ > 0 independent of ๐‘ โˆˆ Nsuch that

|๐‘ข๐‘ (๐‘ฅ) โˆ’๐‘ข๐‘ (๐‘ฆ) | โ‰ค ๐ฟ |๐‘ฅโˆ’ ๐‘ฆ | for all ๐‘ฅ, ๐‘ฆ โˆˆ๐‘ˆ. (A.5)

If there exists a function ๐‘ขโˆž : ๐‘ˆโ†’ R such that

๐‘ข๐‘ (๐‘ฅ) โ†’ ๐‘ขโˆž (๐‘ฅ) for all ๐‘ฅ โˆˆ๐‘ˆ, (A.6)

then we have โˆ‡๐‘ข๐‘ (๐‘ฅ) โ†’ โˆ‡๐‘ขโˆž (๐‘ฅ) for a.e. ๐‘ฅ โˆˆ๐‘ˆ.

Remark 8. From (A.5)โ€“(A.6), it is easy to show that ๐‘ขโˆž is also convex, ๐‘ข๐‘ โ†’ ๐‘ขโˆž uniformly in๐‘ˆ, and

|๐‘ขโˆž (๐‘ฅ) โˆ’๐‘ขโˆž (๐‘ฆ) | โ‰ค ๐ฟ |๐‘ฅโˆ’ ๐‘ฆ | for all ๐‘ฅ, ๐‘ฆ โˆˆ๐‘ˆ.

Our proof of Lemma 9 is inspired by [11, Lemma A.3].

Proof. We define L๐‘›-measurable sets

๐‘ƒ๐‘ B {๐‘ฅ โˆˆ๐‘ˆ | ๐‘ข๐‘ is not differentiable at ๐‘ฅ} for ๐‘ โˆˆ Nโˆช {โˆž}.

Clearly ๐‘ƒ๐‘ (๐‘ โˆˆ Nโˆช {โˆž}) satisfies L๐‘› (๐‘ƒ๐‘ ) = 0 by Lipschitz continuity of ๐‘ข๐‘ , and therefore the L๐‘›-measurableset

๐‘ƒ Bโ‹ƒ

๐‘ โˆˆNโˆช{โˆž}๐‘ƒ๐‘ โŠ‚ ๐‘ˆ

also satisfies L๐‘› (๐‘ƒ) = 0. We claim that

โˆ‡๐‘ข๐‘ (๐‘ฅ0) โ†’ โˆ‡๐‘ขโˆž (๐‘ฅ0) for all ๐‘ฅ0 โˆˆ๐‘ˆ \๐‘ƒ. (A.7)

We take and fix arbitrary ๐‘ฅ0 โˆˆ๐‘ˆ \๐‘ƒ. We note that โˆ‡๐‘ข๐‘ (๐‘ฅ0) exists for each ๐‘ โˆˆ N since ๐‘ฅ0 โˆ‰ ๐‘ƒ๐‘ , and we obtain

sup๐‘ โˆˆN

|โˆ‡๐‘ข๐‘ (๐‘ฅ0) | โ‰ค ๐ฟ

with the aid of (A.5). Hence it suffices to check that, if a subsequence {๐‘ข๐‘๐‘˜ }๐‘˜ โŠ‚ {๐‘ข๐‘ }๐‘ satisfies

โˆ‡๐‘ข๐‘๐‘˜ (๐‘ฅ0) โ†’ ๐‘ฃ (๐‘˜ โ†’โˆž) for some ๐‘ฃ โˆˆ R๐‘ , (A.8)

then ๐‘ฃ = โˆ‡๐‘ขโˆž (๐‘ฅ0). Since ๐‘ฅ0 โˆ‰ ๐‘ƒ๐‘๐‘˜ and therefore ๐œ•๐‘ข๐‘๐‘˜ (๐‘ฅ0) = {โˆ‡๐‘ข๐‘๐‘˜ (๐‘ฅ0)} for each ๐‘˜ โˆˆ N, we easily get

๐‘ข๐‘๐‘˜ (๐‘ฅ) โ‰ฅ ๐‘ข๐‘๐‘˜ (๐‘ฅ0) + โŸจโˆ‡๐‘ข๐‘๐‘˜ (๐‘ฅ0) | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ for all ๐‘ฅ โˆˆ๐‘ˆ, ๐‘˜ โˆˆ N.

Letting ๐‘˜ โ†’โˆž, we have๐‘ขโˆž (๐‘ฅ) โ‰ฅ ๐‘ขโˆž (๐‘ฅ0) + โŸจ๐‘ฃ | ๐‘ฅโˆ’ ๐‘ฅ0โŸฉ for all ๐‘ฅ โˆˆ๐‘ˆ

by (A.6) and (A.8). This means that ๐‘ฃ โˆˆ ๐œ•๐‘ขโˆž (๐‘ฅ0). Note again that ๐‘ฅ0 โˆ‰ ๐‘ƒโˆž and therefore ๐œ•๐‘ขโˆž (๐‘ฅ0) = {โˆ‡๐‘ขโˆž (๐‘ฅ0)},which yields ๐‘ฃ = โˆ‡๐‘ขโˆž (๐‘ฅ0). This completes the proof of (A.7). โ–ก

A.3 Convex functionalsWe prove some basic property of convex functionals ฮจ and๐‘Š in Section 6.

Lemma 10. Let ๐‘Š be a convex function which satisfies (6.2)-(6.3) and (6.5). Then the mapping ๐ด : R๐‘› โˆ‹ ๐‘ง โ†ฆโ†’โˆ‡๐‘Š (๐‘ง) โˆˆ R๐‘› satisfies strict monotonicity (4.8).

Proof. We take arbitrary ๐‘ง1, ๐‘ง2 โˆˆ R๐‘› with ๐‘ง1 โ‰  ๐‘ง2 and define a line segment ๐ฟ B {๐‘ง1 + ๐‘ก (๐‘ง2 โˆ’ ๐‘ง1) โˆˆ R๐‘› | 0 โ‰ค ๐‘ก โ‰ค 1}.We first consider the case 0 โˆ‰ ๐ฟ. Then there exist constants 0 < ๐œ‡ โ‰ค ๐‘€ <โˆž such that ๐œ‡ โ‰ค |๐‘ง0 | โ‰ค ๐‘€ holds for all

๐‘ง0 โˆˆ ๐ฟ. Here we can take a constant ๐›พ > 0 such that (6.3) holds for all ๐‘ง0 โˆˆ ๐ฟ. Then by๐‘Š โˆˆ ๐ถ2 (R๐‘› \ {0}), we have

โŸจ๐ด(๐‘ง1) โˆ’ ๐ด(๐‘ง2) | ๐‘ง2 โˆ’ ๐‘ง1โŸฉ =โˆซ 1

0

โŸจโˆ‡2๐‘ง๐‘Š (๐‘ง1 + ๐‘ก (๐‘ง2 โˆ’ ๐‘ง1)) (๐‘ง2 โˆ’ ๐‘ง1)

๏ฟฝ๏ฟฝ ๐‘ง2 โˆ’ ๐‘ง1โŸฉ ๐‘‘๐‘ก โ‰ฅ ๐›พ |๐‘ง2 โˆ’ ๐‘ง1 |2 > 0.

26

To consider the remaining case 0 โˆˆ ๐ฟ, it suffices to show (6.7). Indeed, the assumption 0 โˆˆ ๐ฟ allows us to write๐‘ง1 = โˆ’๐‘™1๐œˆ, ๐‘ง2 = ๐‘™2๐œˆ for some unit vector ๐œˆ and some constants ๐‘™1, ๐‘™2 โ‰ฅ 0. Under this notation, we obtain

โŸจ๐ด(๐‘ง2) โˆ’ ๐ด(๐‘ง1) | ๐‘ง2 โˆ’ ๐‘ง1โŸฉ = โŸจ๐ด(๐‘™2๐œˆ) | (๐‘™1 + ๐‘™2)๐œˆโŸฉ + โŸจ๐ด(โˆ’๐‘™1๐œˆ) | โˆ’(๐‘™1 + ๐‘™2)๐œˆโŸฉ > 0

by (6.7). Here we note that at least one of ๐‘™1, ๐‘™2 is positive since ๐‘™1 + ๐‘™2 = |๐‘ง2 โˆ’ ๐‘ง1 | > 0.We prove (6.7) to complete the proof. Let ๐‘ง โˆˆ R๐‘› \ {0}. Then we obtain

๐‘‘๐‘ BโŸจ๐ด(๐‘ง/2๐‘โˆ’1) โˆ’ ๐ด(๐‘ง/2๐‘ )

๏ฟฝ๏ฟฝ ๐‘งโŸฉ > 0

for each ๐‘ โˆˆ N, since we have already shown (4.8) for the case 0 โˆ‰ ๐ฟ. By definition of ๐‘‘ ๐‘— ( ๐‘— โˆˆ N), it is clear thatโŸจ๐ด(๐‘ง) โˆ’ ๐ด(๐‘ง/2๐‘ )

๏ฟฝ๏ฟฝ ๐‘งโŸฉ = ๐‘‘1 + ยท ยท ยท + ๐‘‘๐‘ โ‰ฅ ๐‘‘1.

Letting ๐‘ โ†’โˆž, we obtain โŸจ๐ด(๐‘ง) | ๐‘งโŸฉ โ‰ฅ ๐‘‘1 > 0 by ๐ด โˆˆ ๐ถ (R๐‘›, R๐‘›). โ–ก

We precisely prove (6.8)โ€“(6.10) in Lemma 11. See also [2, Section 1.3] and [30, ยง13] as related items.

Lemma 11. Let ฮจ : R๐‘› โ†’ [0,โˆž) be a convex function which is positively homogeneous of degree 1.

1. ฮจ satisfies the triangle inequality (6.8).

2. Assume that ๐œ โˆˆ R๐‘› satisfies ฮจฬƒ(๐œ) <โˆž. Then the Cauchyโ€“Schwarz-type inequality (6.9) holds.

3. The subdifferential operator ๐œ•ฮจ is given by (6.10).

Proof. By convexity of ฮจ and (6.6), ฮจ satisfies

ฮจ(๐‘ง1 + ๐‘ง2)2

= ฮจ( ๐‘ง1 + ๐‘ง2

2

)โ‰ค ฮจ(๐‘ง1) +ฮจ(๐‘ง2)

2for all ๐‘ง1, ๐‘ง2 โˆˆ R๐‘›,

which yields (6.8).We next show the Cauchyโ€“Schwarz inequality (6.9). Let ๐‘ง โˆˆ R๐‘›. If ฮจ(๐‘ง) > 0, then we have

โŸจ๐‘ง | ๐œโŸฉ = ฮจ(๐‘ง)โŸจ

๐‘ง

ฮจ(๐‘ง)

๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐œโŸฉ โ‰ค ฮจ(๐‘ง)ฮจฬƒ(๐œ)

by ๐‘ง/ฮจ(๐‘ง) โˆˆ ๐ถฮจ. For the case ฮจ(๐‘ง) = 0, we note that ๐œ†๐‘ง โˆˆ ๐ถฮจ for all ๐œ† > 0. Hence it follows that

โŸจ๐‘ง | ๐œโŸฉ = โŸจ๐œ†๐‘ง | ๐œโŸฉ๐œ†

โ‰ค ฮจฬƒ(๐‘ค)๐œ†

for all ๐œ† > 0. By ฮจฬƒ(๐œ) <โˆž, we obtain โŸจ๐‘ง | ๐œโŸฉ โ‰ค 0 = ฮจ(๐‘ง)ฮจฬƒ(๐œ). This completes the proof of (6.9).Finally we prove (6.10). Let ๐‘ง0 โˆˆ R๐‘› be arbitrarily fixed. Assume that ๐œ โˆˆ R๐‘› satisfies ฮจฬƒ(๐œ) โ‰ค 1 and ฮจ(๐‘ง0) =

โŸจ๐‘ง0 | ๐œโŸฉ. Then by combining these assumptions with (6.9), we have

ฮจ(๐‘ง) โ‰ฅ ฮจ(๐‘ง)ฮจฬƒ(๐œ)โ‰ฅ โŸจ๐‘ง | ๐œโŸฉ = โŸจ๐‘ง0 | ๐œโŸฉ + โŸจ๐‘งโˆ’ ๐‘ง0 | ๐œโŸฉ= ฮจ(๐‘ง0) + โŸจ๐œ | ๐‘งโˆ’ ๐‘ง0โŸฉ

for all ๐‘ง โˆˆ R๐‘›. Hence ๐œ โˆˆ ๐œ•ฮจ(๐‘ง0). Conversely, if ๐œ โˆˆ ๐œ•ฮจ(๐‘ง0), then we have the subgradient inequality

ฮจ(๐‘ง) โ‰ฅ ฮจ(๐‘ง0) + โŸจ๐œ | ๐‘งโˆ’ ๐‘ง0โŸฉ for all ๐‘ง โˆˆ R๐‘›. (A.9)

By testing ๐‘˜๐‘ง0 into (A.9), where ๐‘˜ โˆˆ [0,โˆž) is arbitrary, we have

(๐‘˜ โˆ’1)ฮจ(๐‘ง0) = ฮจ(๐‘˜๐‘ง0) โˆ’ฮจ(๐‘ง0) โ‰ฅ โŸจ๐œ | (๐‘˜ โˆ’1)๐‘ง0โŸฉ = (๐‘˜ โˆ’1)โŸจ๐œ | ๐‘ง0โŸฉ. (A.10)

If we let 0 โ‰ค ๐‘˜ < 1 so that ๐‘˜ โˆ’ 1 < 0, then we have ฮจ(๐‘ง0) โ‰ค โŸจ๐œ | ๐‘ง0โŸฉ. Similarly, letting 1 < ๐‘˜ < โˆž, we haveฮจ(๐‘ง0) โ‰ฅ โŸจ๐œ | ๐‘ง0โŸฉ. Hence we obtain ฮจ(๐‘ง0) = โŸจ๐œ | ๐‘ง0โŸฉ. Combining with (A.9), we have

โŸจ๐‘ง | ๐œโŸฉ โ‰ค ฮจ(๐‘ง) for all ๐‘ง โˆˆ R๐‘›,

which yields ฮจฬƒ(๐œ) โ‰ค 1 by definition of ฮจฬƒ. This completes the proof of (6.10). โ–ก

27

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