continuity of derivatives of a convex solution to a
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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 Gigaโ and Shuntaro Tsubouchi โ
August 23, 2021
Abstract
We consider a one-Laplace equation perturbed by ๐-Laplacian with 1 < ๐ <โ. We prove that a weak solutionis continuously differentiable (๐ถ1) if it is convex. Note that similar result fails to hold for the unperturbedone-Laplace equation. The main difficulty is to show ๐ถ1-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 ๐ถ1-regularity, one-Laplace equation, strong maximum principle
1 IntroductionWe consider a one-Laplace equation perturbed by ๐-Laplacian of the form
๐ฟ๐,๐๐ข = ๐ in ฮฉ (1.1)
with๐ฟ๐,๐๐ข := โ๐ฮ1๐ขโฮ๐๐ข,
whereฮ1๐ข := div (โ๐ข/|โ๐ข |) , ฮ๐๐ข = div
(|โ๐ข |๐โ2โ๐ข
)in a domain ฮฉ in R๐, โ๐ข = (๐๐ฅ1๐ข, . . . , ๐๐ฅ๐๐ข) with ๐๐ฅ ๐๐ข = ๐๐ข/๐๐ฅ ๐ for a function ๐ข = ๐ข(๐ฅ1, . . . , ๐ฅ๐), and div๐ =๐โ๐=1๐๐ฅ๐๐๐ for a vector field ๐ = (๐1, . . . , ๐๐). The constants ๐ > 0 and ๐ โ (1,โ) are given and fixed. It has been a
long-standing open problem whether its weak solution is๐ถ1 up to a facet, the place where the gradient โ๐ข vanishes,even if ๐ is smooth. This is a non-trivial question since a weak solution to the unperturbed one-Laplace equation,i.e., โฮ1๐ข = ๐ may not be ๐ถ1. This is because the ellipticity degenerates in the direction of โ๐ข for ฮ1๐ข. 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 โ ๐ฟ๐loc (ฮฉ) (๐ < ๐ โค โ), i.e., | ๐ |๐ is locally integrable in ฮฉ. Our main result is
Theorem 1 (๐ถ1-regularity theorem). Let ๐ข be a convex weak solution to (1.1) with ๐ โ ๐ฟ๐loc (ฮฉ) (๐ < ๐ โค โ). Then๐ข is in ๐ถ1 (ฮฉ).
โGraduate 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]
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Difficulty on proving regularity on gradients of solutions to (1.1) can be explained from a viewpoint of ellipticityratio. We set a convex function ๐ธ : R๐ โ [0,โ) by
๐ธ (๐ง) B ๐ |๐ง | + |๐ง |๐๐
for ๐ง โ R๐.
We rewrite (1.1) byโdiv(โ๐ง๐ธ (โ๐ข)) = ๐ in ฮฉ. (1.2)
By differentiating (1.2) by ๐ฅ๐ (๐ โ {1, . . . , ๐ }), we get
โdiv(โ2๐ง๐ธ (โ๐ข)โ๐๐ฅ๐๐ข
)= ๐๐ฅ๐ ๐ . (1.3)
By elementary calculations, ellipticity ratio of the Hessian โ2๐ง๐ธ at ๐ง0 โ R๐ \ {0} is given by(
ellipticity ratio of โ2๐ง๐ธ (๐ง0)
)B
(the largest eigenvalue of โ2๐ง๐ธ (๐ง0))
(the lowest eigenvalue of โ2๐ง๐ธ (๐ง0))
=max(๐โ1, 1) + ๐ |๐ง0 |1โ๐
min(๐โ1, 1) .
Since the exponent 1โ ๐ is negative, the ellipticity ratio of โ2๐ง๐ธ (๐ง0) blows up as ๐ง0 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)-growth problem, since for (๐, ๐)-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(๐ข) Bโซ๐ธ๐ (โ๐ข) ๐๐ฅ +
โซ๐(๐ฅ)๐ธ๐ (โ๐ข) ๐๐ฅ with 1 < ๐ โค ๐ <โ, ๐(๐ฅ) โฅ 0
were discussed in scalar and even in vectorial cases by Colombo and Mingione [6, 7], their results do not recover our๐ถ1-regularity results. This is basically derived from the fact that, unlike โ2
๐ง๐ธ๐ with 1 < ๐ <โ, the Hessian matrixโ2๐ง๐ธ1 (๐ง0) (๐ง0 โ 0) always takes 0 as its eigenvalue. In other words, ellipticity of the operator ฮ1๐ข degenerates in
the direction of โ๐ข, which seems to be difficult to handle analytically.On the other hand, the ellipticity ratio of โ2
๐ง๐ธ (๐ง0) is uniformly bounded over |๐ง0 | > ๐ฟ for each fixed ๐ฟ > 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 GiorgiโNashโMoser 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 ๐ถ1-regularity criterion for a convex function, to show ๐ข is ๐ถ1 at ๐ฅ โ ฮฉ it suffices to prove that
the subdifferential ๐๐ข(๐ฅ) at ๐ฅ โ ฮฉ is a singleton; (1.4)
see [1, Appendix D], [30, ยง25] and Remark 1 for more detail. Here the subdifferential of ๐ข at ๐ฅ0 โ ฮฉ is defined by
๐๐ข(๐ฅ0) B {๐ง โ R๐ | ๐ข(๐ฅ) โฅ ๐ข(๐ฅ0) + โจ๐ง | ๐ฅโ ๐ฅ0โฉ for all ๐ฅ โ ฮฉ}.
Here โจ ยท | ยท โฉ stands for the standard inner product in R๐. For a convex function ๐ข : ฮฉโ R, we can simply expressthe facet of ๐ข as
๐น B {๐ฅ โ ฮฉ | ๐๐ข(๐ฅ) โ 0} = {๐ฅ โ ฮฉ | ๐ข(๐ฅ) โค ๐ข(๐ฆ) for all ๐ฆ โ ฮฉ}.By definition it is clear that the facet ๐น is non-empty if and only if a minimum of ๐ข in ฮฉ exists. By convexity of ๐ข,we can easily check that ๐น โ ฮฉ is a relatively closed convex set in ฮฉ. We also define an open set
๐ท B ฮฉ \๐น = {๐ฅ โ ฮฉ | ๐ข(๐ฆ) < ๐ข(๐ฅ) for some ๐ฆ โ ฮฉ}.
Our strategy to show (1.4) depends on whether ๐ฅ is inside ๐น or not.
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Remark 1. [Some properties on differentiability of convex functions] Let ๐ฃ a real-valued convex function in aconvex domain ฮฉ โ R๐, then following property holds.
1. ๐ฃ is locally Lipschitz continuous in ฮฉ, and therefore ๐ฃ is a.e. differentiable in ฮฉ by Rademacherโs theorem([1, Theorem 1.19], see also [9, Theorem 3.1 and 3.2] and [30, Theorem 25.5]).
2. For ๐ฅ โ ฮฉ, ๐ฃ is differentiable at ๐ฅ if and only if the subdifferential set ๐๐ฃ(๐ฅ) is a singleton. Moreover, if ๐ฅ โ ฮฉsatisfies either of these equivalent conditions, then we have ๐๐ฃ(๐ฅ) = {โ๐ฃ(๐ฅ)} ([1, Proposition D.5], see also[30, Theorem 25.1]). In particular, Rademacherโs theorem implies that ๐๐ฃ(๐ฅ) = {โ๐ฃ(๐ฅ)} for a.e. ๐ฅ โ ฮฉ.
3. ๐ฃ โ ๐ถ1 (ฮฉ) if and only if ๐๐ฃ 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 ๐ฅ โ ๐ท. Our goal is to show directly that ๐ข is ๐ถ1, ๐ผ near a neighborhood of ๐ฅ andtherefore ๐๐ข(๐ฅ) = {โ๐ข(๐ฅ)} โ {0} for all ๐ฅ โ ๐ท. This strategy roughly consists of three steps. Among them the firststep, a kind of separation of ๐ฅ โ ๐ท from the facet ๐น, plays an important role. Precisely speaking, we first find aneighborhood ๐ต๐ (๐ฅ) โ ๐ท, an open ball centered at ๐ฅ with its radius ๐ > 0, such that
๐๐๐ข โฅ ๐ > 0 a.e. in ๐ต๐ (๐ฅ) (1.5)
for some direction ๐ and some constant ๐ > 0. In order to justify (1.5), we fully make use of convexity of ๐ข (Lemma8 in Section A), not elliptic regularity theory. Then with the aid of local Lipschitz continuity of ๐ข, the inclusion๐ต๐ (๐ฅ) โ {0 < ๐ โค ๐๐๐ข โค |โ๐ข | โค ๐} holds for some finite positive constant ๐ . Secondly, this inclusion allows us tocheck that ๐ข admits local๐2, 2-regularity in ๐ต๐ (๐ฅ) 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 GiorgiโNashโMosertheory to obtain local ๐ถ1, ๐ผ-regularity at ๐ฅ โ ๐ท, since the equation (1.3) is uniformly elliptic in ๐ต๐ (๐ฅ). Here theconstant ๐ผ โ (0, 1) we have obtained may depend on the location of ๐ฅ โ ๐ท through ellipticity, so ๐ผ may tend to zeroas ๐ฅ tends to the facet.
It takes much efforts to prove that ๐๐ข(๐ฅ) = {0} for all ๐ฅ โ ๐น. 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 ๐ข : ฮฉโ R and a point ๐ฅ0 โ ฮฉ,we set a sequence of rescaled functions {๐ข๐}๐>0 defined by
๐ข๐ (๐ฅ) B๐ข(๐(๐ฅโ ๐ฅ0) + ๐ฅ0) โ๐ข(๐ฅ0)
๐.
We show that ๐ข๐ locally uniformly converges to some convex function ๐ข0 : R๐โR, which satisfies ๐๐ข(๐ฅ0) โ ๐๐ข0 (๐ฅ0)by construction. Moreover, we prove that ๐ข0 satisfies ๐ฟ๐, ๐๐ข0 = 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 ๐ฅ0 โ ๐น, then the convex weak solution ๐ข0 constructed as above satisfies ๐๐ข0 (๐ฅ0) = {0}.Moreover, we are going to prove that ๐ข0 is constant (a Liouville-type theorem). For this purpose we establish themaximum principle.
Theorem 2 (Strong maximum principle). Let ๐ข be a convex weak solution to ๐ฟ๐, ๐๐ข = 0 in a convex domain ฮฉ โ R๐and ๐น โ ฮฉ be the facet of ๐ข. Then ๐ข is affine in each connected component of the open set ๐ท B ฮฉ\๐น. In particular,if ๐น = โ , then ๐ข is affine in ฮฉ.
It should be noted that this result is a kind of strong maximum principle in the sense that
๐ข โฅ ๐ in ๐ท0 and ๐ข(๐ฅ0) = ๐(๐ฅ0) for ๐ฅ0 โ ๐ท0 imply that ๐ข โก ๐ in ๐ท0, (1.6)
where ๐(๐ฅ) B ๐ข(๐ฅ0) + โจโ๐ข(๐ฅ0) | ๐ฅ โ ๐ฅ0โฉ and ๐ท0 is a connected component of ๐ท. The affine function ๐ clearlysatisfies ๐ฟ๐, ๐๐ = 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 ๐ข, and the third is a construction of suitable barrier subsolutions, all of which are essentiallyneeded in the classical proof of E. Hopfโs 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 ๐ท โ ฮฉ, we are able to show the following Liouville-type theorem.
Theorem 3 (Liouville-type theorem). Let ๐ข be a convex weak solution to ๐ฟ๐, ๐๐ข = 0 in R๐. Then ๐น โ R๐, the facetof ๐ข, satisfies either ๐น = โ or ๐น = R๐. In particular, ๐ข satisfies either of the followings.
1. If ๐ข attains its minimum in R๐, then ๐ข is constant.
2. If ๐ข does not attains its minimum in R๐, then ๐ข 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 โ โ ๐น โ 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., ๐๐ข(๐ฅ) = {0} for all ๐ฅ โ ๐น, and we complete the proof of the ๐ถ1-regularity theorem.Note that the statements in Theorem 2 and 3 should not hold for unperturbed one-Laplace equation โฮ1๐ข = ๐ , sinceany absolutely continuous non-decreasing function of one variable ๐ข = ๐ข(๐ฅ1) satisfies โฮ1๐ข = 0.
Finally we mention that we are able to refine our strategy, and obtain ๐ถ1-regularity of convex solutions to moregeneral equations. We replace the one-Laplacian ฮ1 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. Hopfโs strong maximum principle is one of the well-known results on maximum principles.In Hopfโs proof of the strong maximum principle [20], he defined an auxiliary function
โ(๐ฅ) B ๐โ๐ผ |๐ฅโ๐ฅโ |2 โ ๐โ๐ผ๐ 2
for ๐ฅ โ R๐, (1.7)
which becomes a classical subsolution in a fixed open annulus ๐ธ๐ = ๐ธ๐ (๐ฅโ) B ๐ต๐ (๐ฅโ) \ ๐ต๐ /2 (๐ฅโ) for sufficientlylarge ๐ผ > 0. An alternative function
โ(๐ฅ) B |๐ฅโ ๐ฅโ |โ๐ผ โ๐ โ๐ผ for ๐ฅ โ R๐ \ {๐ฅโ} (1.8)
is given in [29, Chapter 2.8]. E. Hopfโs classical results on maximum principles are extensively contained in [17,Chapter 3], [28, Chapter 2] and [29, Chapter 2].
The materials [17, Chapter 8โ9] and [29, Chapter 3โ6] 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., โdiv(๐ด(๐ฅ, โ๐ข(๐ฅ))) = 0,
which covers the ๐-Laplace equation with 1 < ๐ <โ. Even in the distributional schemes, the proof of the maximumprinciple [29, Theorem 5.4.1] is partially similar to E. Hopfโs 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 ๐ถ1 class, which is given in [29, Chapter 4], is based on LerayโSchauderโsfixed 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 ๐ด(๐ฅ, ยท ) : R๐ โ R๐, whereas Hopfโsproof appeals to direct constructions of auxiliary functions.
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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 ๐ข and the construction of subsolutions, cannot be resolvedaffirmatively by just imitating arguments given in [29, Chapter 4โ5]. 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 ๐ถ1. We recall that ๐ถ1-regularity of convex weaksolutions can be guaranteed in ๐ท โ ฮฉ (the outside of the facet) by the classical De GiorgiโNashโMoser theory, andthis result enables us to overcome the problem whether ๐ข is differentiable at certain points. This is the reason whyTheorem 2 need to restrict on ๐ท. 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 ๐ฃ = ๐ฝโ+ ๐ in R๐ \ {๐ฅโ}, where๐ฝ > 0 is a constant and โ is defined as in (1.7) or (1.8). We will determine the constants ๐ผ, ๐ฝ > 0 so precisely that ๐ฃsatisfies ๐ฟ๐, ๐๐ฃ โค 0 in the classical sense over a fixed open annulus ๐ธ๐ = ๐ธ๐ (๐ฅโ). We also make |โ๐ฃ | very close to|โ๐ | โก |โ๐ข(๐ฅ0) | > 0 over ๐ธ๐ , so that โ๐ฃ no longer degenerates there. By direct calculation of ๐ฟ๐, ๐๐ฃ, we explicitlyconstruct classical subsolutions to ๐ฟ๐, ๐๐ข = 0 in ๐ธ๐ . 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. Krรผgelโs thesis in 2013 [25]. The significant difference is that Krรผgel did not regard the term โ๐ข/|โ๐ข | asa subgradient vector field. Since monotonicity of ๐ | ยท | is not used at all, it seems that Krรผgelโs 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 EulerโLagrange equation;
๐ =๐ฟ๐บ
๐ฟ๐ข, where ๐บ (๐ข) B ๐
โซฮฉ|โ๐ข | ๐๐ฅ + 1
๐
โซฮฉ|โ๐ข |๐ ๐๐ฅ.
The energy functional ๐บ 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
describing the height of the crystal for a two-dimensional domain ฮฉ is modeled as
โ๐ก +div ๐ = 0
with ๐ = โโ๐, where ๐ is a chemical potential. In [31], its evolution is given as
๐ =๐ฟฮฆ๐ฟโ
with ฮฆ(โ) =โซฮฉ|โโ| ๐๐ฅ + ๐
โซฮฉ|โโ|3 ๐๐ฅ
with ๐ > 0. This ฮฆ is essentially the same as ๐บ with ๐ = 3. Then, the resulting evolution equation for โ is of theform
๐โ๐ก = ฮ๐ฟ๐, 3โ with ๐ =13๐ .
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 must satisfies
ฮ๐ฟ๐, 3โ = 0.
If ๐ฟ๐, 3โ is a constant, our Theorem 1 implies that the height function โ is ๐ถ1 provided that โ is convex.For a second order problem,
i.e., ๐โ๐ก = ๐ฟ๐, ๐โ,
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].
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Another important mathematical model for the equation (1.1) is found in fluid mechanics. Especially for ๐ = 2and ๐ = 2, the energy functional ๐บ 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 ฮ1๐ข as well as that of viscosity corresponding ฮ2๐ข = ฮ๐ข in (1.1). Let us consider a parallelstationary flow with velocity ๐ = (0,0, ๐ข(๐ฅ1, ๐ฅ2)) in a cylinder ฮฉรR. Of course, this is incompressible flow, i.e.,div๐ = 0. If this flow is the classical Newtonian fluid, then the NavierโStokes equations become (1.1) inฮฉwith ๐ = 0and ๐ = โ๐๐ฅ3๐, where ๐ 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 ๐ฮ1๐ข, 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 ๐ถ1-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 ๐ถ1, ๐ผ-regularity at regular points of convex weak solutions (Lemma 1). In order to applyDe GiorgiโNashโMoser theory, we will need to justify local ๐2, 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 that๐ข0 : R๐ โ R, a limit of rescaled solutions, satisfies ๐ฟ๐, ๐๐ข0 = 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 ๐ฟ๐, ๐๐ข = 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 4.1โ4.2, 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 ฮ1 and ฮ๐ . 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 ๐ข is ๐ถ1 at any ๐ฅ โ ๐ท, and therefore (1.4) holds for all ๐ฅ โ ๐ท. This resultwill be used in the proof of the strong maximum principle (Theorem 2).
We first give a precise definition of weak solutions to ๐ฟ๐, ๐๐ข = ๐ in a convex domain ฮฉ โ R๐, which is notnecessarily bounded.
Definition 1. Let ฮฉ โ R๐ be a domain, which is not necessarily bounded, and ๐ โ ๐ฟ๐loc (ฮฉ) (๐ < ๐ โค โ). We saythat a function ๐ข โ๐1, ๐
loc (ฮฉ) is a weak solution to (1.1), when for any bounded Lipschitz domain ๐ โ ฮฉ, there existsa vector field ๐ โ ๐ฟโ (๐, R๐) such that the pair (๐ข, ๐) โ๐1, ๐ (๐) ร ๐ฟโ (๐, R๐) satisfies
๐
โซ๐โจ๐ | โ๐โฉ ๐๐ฅ +
โซ๐
โจ|โ๐ข |๐โ2โ๐ข
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ = โซ๐๐ ๐ ๐๐ฅ (2.1)
6
for all ๐ โ๐1, ๐0 (๐), and
๐ (๐ฅ) โ ๐ | ยท | (โ๐ข(๐ฅ)) (2.2)
for a.e. ๐ฅ โ ๐. For such pair (๐ข, ๐), we say that (๐ข, ๐) satisfies ๐ฟ๐, ๐๐ข = ๐ in ๐โ1, ๐โฒ (๐) or simply say that ๐ขsatisfies ๐ฟ๐, ๐๐ข = ๐ in๐โ1, ๐โฒ (๐). Here ๐โฒ โ (1,โ) denotes the Hรถlder conjugate exponent of ๐ โ (1,โ).
The aim of Section 2 is to show Lemma 1 below.
Lemma 1. Let ๐ข be a convex weak solution to (1.1) in a convex domain ฮฉ โ R๐, and ๐ โ ๐ฟ๐loc (ฮฉ) (๐ < ๐ โค โ). If๐ฅ0 โ ๐ท, then we can take a small radius ๐0 > 0, a unit vector ๐0 โ R๐, and a small number ๐ > 0 such that
๐ต๐0 (๐ฅ0) โ ๐ท and โจโ๐ข(๐ฅ) | ๐0โฉ โฅ ๐ for a.e. ๐ฅ โ ๐ต๐0 (๐ฅ0), (2.3)
and there exists a small number ๐ผ = ๐ผ(๐) โ (0, 1) such that ๐ข โ ๐ถ1, ๐ผ (๐ต๐0/2 (๐ฅ0)). In particular, ๐ข is ๐ถ1 in ๐ท, and๐๐ข(๐ฅ) = {โ๐ข(๐ฅ)} โ {0} for all ๐ฅ โ ๐ท.
Before proving Lemma 1, we introduce difference quotients. For given ๐ : ฮฉโR๐ (๐ โN), ๐ โ {1, . . . , ๐ }, โ โR \ {0}, we define
ฮ ๐ , โ๐(๐ฅ) B๐(๐ฅ + โ๐ ๐ ) โ๐(๐ฅ)
โโ R๐ for ๐ฅ โ ฮฉ with ๐ฅ + โ๐ ๐ โ ฮฉ,
where ๐ ๐ โ R๐ denotes the unit vector in the direction of the ๐ฅ ๐ -axis.In the proof of Lemma 1, we will use Lemma 7โ8 without proofs. For precise proofs, see Section A.
Proof. For each fixed ๐ฅ0 โ ๐ท, we may take and fix ๐ฅ1 โ ฮฉ such that ๐ข(๐ฅ0) > ๐ข(๐ฅ1). We set 3๐ฟ0 B ๐ข(๐ฅ0) โ๐ข(๐ฅ1) >0, ๐0 B |๐ฅ0 โ ๐ฅ1 | > 0 and ๐0 B ๐โ1
0 (๐ฅ0 โ ๐ฅ1). By ๐ข โ ๐ถ (ฮฉ), we may take a sufficiently small ๐0 > 0 such that
๐ข(๐ฆ0) โ๐ข(๐ฆ1) โฅ ๐ฟ0 > 0 for all ๐ฆ0 โ ๐ต๐0 (๐ฅ0), ๐ฆ1 โ ๐ต๐0 (๐ฅ1). (2.4)
From (2.4), the inclusion ๐ต๐0 (๐ฅ0) โ ๐ท clearly holds. (2.4) also allows us to check that for all ๐ฆ0 โ ๐ต๐0 (๐ฅ0), ๐ง0 โ๐๐ข(๐ฆ0),
โจ๐ง0 | ๐0โฉ โฅ๐ข(๐ฆ0) โ๐ข(๐ฆ0 โ ๐๐0)
๐0โฅ ๐ฟ0
๐0C ๐0 > 0. (2.5)
For the first inequality in (2.5), we have used Lemma 8, which is basically derived from convexity of ๐ข. Recall that๐๐ข(๐ฅ) = {โ๐ข(๐ฅ)} for a.e. ๐ฅ โ ฮฉ, and hence we are able to recover (2.3) from (2.5).
In order to obtain ๐ถ1-regularity in ๐ท, we will appeal to the classical De GiorgiโNashโMoser theory. Forpreliminaries, we check that the operator ๐ฟ๐, ๐๐ข assures uniform ellipticity in ๐ต๐0 (๐ฅ0). Local Lipschitz continuityof ๐ข implies that there exists a sufficiently large number ๐0 โ (0,โ) such that
ess sup๐ต๐0 (๐ฅ0)
|โ๐ข | โค ๐0 and |๐ข(๐ฅ) โ๐ข(๐ฆ) | โค ๐0 |๐ฅโ ๐ฆ | for all ๐ฅ, ๐ฆ โ ๐ต๐0 (๐ฅ0). (2.6)
For notational simplicity, we write subdomains by
๐1 B ๐ต๐0 (๐ฅ0) โ ๐2 B ๐ต15๐0/16 (๐ฅ0) โ ๐3 B ๐ต7๐0/8 (๐ฅ0) โ ๐4 B ๐ต3๐0/4 (๐ฅ0) โ ๐5 B ๐ต๐0/2 (๐ฅ0).
It should be noted that ๐ธ (๐ง) B ๐ |๐ง | + |๐ง |๐/๐ (๐ง โ R๐) satisfies ๐ธ โ ๐ถโ (R๐ \ {0}), and there exists two constants0 < ๐(๐, ๐0, ๐0) โค ฮ(๐, ๐, ๐0, ๐0) <โ such that
๐ |๐ |2 โคโจโ2๐ง๐ธ (๐ง0)๐
๏ฟฝ๏ฟฝ ๐โฉ (2.7)โจโ2๐ง๐ธ (๐ง0)๐
๏ฟฝ๏ฟฝ ๐โฉ โค ฮ|๐ | |๐ | (2.8)
for all ๐ง0, ๐ , ๐ โ R๐ with ๐0 โค |๐ง0 | โค ๐0. We can explicitly determine 0 < ๐ โค ฮ <โ by{๐(๐, ๐0, ๐0) B min๐0โค๐กโค๐0
(min{1, ๐โ1 }๐ก ๐โ2) ,
ฮ(๐, ๐, ๐0, ๐0) B max๐0โค๐กโค๐0
(๐๐กโ1 +max{1, ๐โ1 }๐ก ๐โ2)
Now we check that ๐ข โ๐2, 2 (๐4) 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{โซ๐4
|โ(ฮ ๐ , โ๐ข) |2 ๐๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ โ R, 0 < |โ| < ๐0
16
}<โ for each ๐ โ {1, . . . , ๐ }. (2.9)
7
Since ๐ข โ๐1, ๐ (๐1) satisfies ๐ฟ๐, ๐๐ข = ๐ in๐โ1, ๐โฒ (๐1), we obtainโซ๐1
โจโ๐ง๐ธ (โ๐ข) | โ๐โฉ ๐๐ฅ =โซ๐1
๐ ๐ ๐๐ฅ (2.10)
for all ๐ โ ๐1, ๐0 (๐1). Here we note that โ๐ข no longer degenerates in ๐1 by (2.3). We fix a cutoff function
๐ โ ๐ถ1๐ (๐3) such that
0 โค ๐ โค 1 in๐3, ๐ โก 1 in๐4, |โ๐ | โค๐
๐0(2.11)
for some constant ๐ > 0. For each fixed ๐ โ {1, . . . , ๐ }, โ โ R with 0 < |โ| < ๐0/16, we test ๐ B ฮ ๐ ,โโ (๐2ฮ ๐ , โ๐ข)into (2.10). We note that ๐ โ ๐1,โ (๐1) โ ๐1, ๐ (๐1) by (2.8), and this is compactly supported in ๐2. Hence๐ โ๐1, ๐
0 (๐2) is an admissible test function. By testing ๐, we have
0 =โซ๐2
โจฮ ๐ , โ (โ๐ง๐ธ (โ๐ข(๐ฅ)))
๏ฟฝ๏ฟฝ ๐2โ(ฮ ๐ , โ๐ข) +2๐ฮ ๐ , โ๐ขโ๐โฉโโซ๐2
๐ฮโ ๐ , โ (๐2ฮ ๐ , โ๐ข) ๐๐ฅ
=โซ๐2
๐2โจ๐ดโ (๐ฅ, โ๐ข(๐ฅ))โ(ฮ ๐ , โ๐ข) ๏ฟฝ๏ฟฝ โ(ฮ ๐ , โ๐ข)โฉ ๐๐ฅ+2
โซ๐2
๐ฮ ๐ , โ๐ขโจ๐ดโ (๐ฅ, โ๐ข(๐ฅ))โ(ฮ ๐ , โ๐ข)
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅโโซ๐2
๐ฮโ ๐ , โ (๐2ฮ ๐ , โ๐ข) ๐๐ฅ
C ๐ผ1 + ๐ผ2 โ ๐ผ3. (2.12)
Here ๐ดโ = ๐ดโ (๐ฅ, โ๐ข(๐ฅ)) denotes a matrix-valued function in๐2 given by
๐ดโ (๐ฅ, โ๐ข(๐ฅ)) Bโซ 1
0โ2๐ง๐ธ ((1โ ๐ก)โ๐ข(๐ฅ) + ๐กโ๐ข(๐ฅ + โ๐ ๐ )) ๐๐ก.
We note that with the aid of (2.3)โ(2.6), we obtain
๐0 โค |(1โ ๐ก)โ๐ข(๐ฅ) + ๐กโ๐ข(๐ฅ + โ๐ ๐ ) |โค ๐0
for a.e. ๐ฅ โ๐2 and for all 0 โค ๐ก โค 1. Combining this result with (2.7)โ(2.8), we conclude that ๐ดโ satisfies
๐ |๐ |2 โค โจ๐ดโ (๐ฅ, โ๐ข(๐ฅ))๐ | ๐โฉ (2.13)
โจ๐ดโ (๐ฅ, โ๐ข(๐ฅ))๐ | ๐โฉ โค ฮ|๐ | |๐ | (2.14)
for all ๐, ๐ โ R๐ and for a.e. ๐ฅ โ๐2. We set an integral
๐ฝ Bโซ๐2
๐2 |โ(ฮ ๐ , โ๐ข) |2 ๐๐ฅ.
By (2.13), it is clear that ๐ผ1 โฅ ๐๐ฝ. By Youngโs inequality and applying a Poincarรฉ-type inequality (Lemma 7) to๐2ฮ ๐ , โ๐ข โ๐1, 2
0 (๐2), we obtain for any ๐ > 0,
|๐ผ3 | โค14๐
โฅ ๐ โฅ2๐ฟ2 (๐2) + ๐
โซ๐2
|โ(๐2ฮ ๐ , โ๐ข) |2 ๐๐ฅ
โค 14๐
โฅ ๐ โฅ2๐ฟ2 (๐2) +4๐
โซ๐2
|ฮ ๐ , โ๐ข |2 |โ๐ |2 ๐๐ฅ +2๐โซ๐2
๐2 |โ(ฮ ๐ , โ๐ข) |2 ๐๐ฅ.
Here we have invoked the property 0 โค ๐ โค 1 in ๐2. We fix ๐ B ๐/6 > 0. By (2.14) and Youngโs inequality, wehave
|๐ผ2 | โค 2ฮโซ๐2
๐ |โ(ฮ ๐ , โ๐ข) | ยท |ฮ ๐ , โ๐ข | |โ๐ | ๐๐ฅ
โค ๐
3๐ฝ + 3ฮ2
๐
โซ๐2
|ฮ ๐ , โ๐ข |2 |โ๐ |2 ๐๐ฅ.
8
It follows from (2.6) that โฅฮ ๐ , โ๐ขโฅ๐ฟโ (๐2) โค ๐0. Therefore we obtain from (2.12),โซ๐4
|โ(ฮ ๐ , โ๐ข) |2 ๐๐ฅ โค ๐ฝ =โซ๐2
๐2 |โ(ฮ ๐ , โ๐ข) |2 ๐๐ฅ โค ๐ถ (๐, ฮ)(๐2
0 โฅโ๐โฅ2๐ฟ2 (๐2) + โฅ ๐ โฅ2
๐ฟ2 (๐2)
).
The estimate (2.9) follows from this, and therefore ๐ข โ๐2, 2 (๐4).For each ๐ โ ๐ถโ
๐ (๐4), we test ๐๐ฅ ๐๐ โ ๐ถโ๐ (๐4) into (2.10). Integrating by parts, we obtainโซ
๐4
โจโ2๐ง๐ธ (โ๐ข)โ๐๐ฅ ๐๐ข
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ = โโซ๐4
๐ ๐๐ฅ ๐๐ ๐๐ฅ (2.15)
for all ๐ โ ๐ถโ๐ (๐4). Noting that ๐ โ ๐ฟ๐ (๐4) โ ๐ฟ2 (๐4), ๐๐ฅ ๐๐ข โ ๐1, 2 (๐4), and (2.7)โ(2.8), we may extend ๐ โ
๐1, 20 (๐4) by a density argument. The conditions (2.7)โ(2.8) imply that โ2
๐ง๐ธ (โ๐ข) is uniformly elliptic over ๐1.Hence by [17, Theorem 8.22], there exists๐ผ =๐ผ(๐, ฮ, ๐, ๐) โ (0, 1) such that ๐๐ฅ ๐๐ข โ๐ถ๐ผ (๐5) for each ๐ โ {1, . . . , ๐ }.This regularity result implies ๐๐ข(๐ฅ) = {โ๐ข(๐ฅ)} โ {0} for all ๐ฅ โ ๐ท. โก
3 A blow-up argumentIn order to show that (1.4) holds true even for ๐ฅ โ ๐น, 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 โ R๐ be a convex domain, and ๐ โ ๐ฟ๐loc (ฮฉ) (๐ < ๐ โค โ). Assume that ๐ข is a convex weaksolution to (1.1), and ๐ฅ0 โ ฮฉ. Then there exists a convex function ๐ข0 : R๐ โ R such that
1. ๐ข0 is a weak solution to ๐ฟ๐, ๐๐ข0 = 0 in R๐.
2. The inclusion ๐๐ข(๐ฅ0) โ ๐๐ข0 (๐ฅ0) holds. That is, if ๐ โ ๐๐ข(๐ฅ0), then we have
๐ข0 (๐ฅ) โฅ ๐ข0 (๐ฅ0) + โจ๐ | ๐ฅโ ๐ฅ0โฉ for all ๐ฅ โ R๐.
In particular, if ๐ฅ0 โ ๐น, then the facet of ๐ข0 is non-empty.
Proof. Without loss of generality, we may assume that ๐ฅ0 = 0 and ๐ข(๐ฅ0) = 0. First we fix a closed ball ๐ต๐ (0) =๐ต๐ โ ฮฉ. We note that ๐ข โ Lip(๐ต๐ ) since ๐ข is convex. Hence there exists a sufficiently large number ๐ โ (0,โ)such that
ess sup๐ต๐
|โ๐ข | โค ๐ and |๐ข(๐ฅ) โ๐ข(๐ฆ) | โค ๐ |๐ฅโ ๐ฆ | for all ๐ฅ, ๐ฆ โ ๐ต๐ .
We take and fix a vector field ๐ โ ๐ฟโ (๐ต๐ , R๐) such that the pair (๐ข, ๐) โ ๐1, ๐ (๐ต๐ ) ร ๐ฟโ (๐ต๐ , R๐) satisfies๐ฟ๐, ๐๐ข = ๐ in๐โ1, ๐โฒ (๐ต๐ ). For each ๐ > 0, we define a rescaled convex function ๐ข๐ : ๐ต๐ /๐ โR and a dilated vectorfield ๐๐ โ ๐ฟโ (๐ต๐ /๐, R๐) by
๐ข๐ (๐ฅ) B๐ข(๐๐ฅ)๐
, ๐๐ (๐ฅ) B ๐ (๐๐ฅ) for ๐ฅ โ ๐ต๐ /๐ .
We also set ๐๐ โ ๐ฟ๐ (๐ต๐ /๐) by๐๐ (๐ฅ) B ๐ ๐ (๐๐ฅ) for ๐ฅ โ ๐ต๐ /๐ .
Then it is easy to check that the pair (๐ข๐, ๐๐) โ๐1,โ (๐ต๐ /๐)ร๐ฟโ (๐ต๐ /๐, R๐) satisfies ๐ฟ๐, ๐๐ข๐ = ๐๐ in๐โ1, ๐โฒ (๐ต๐ /๐).By definition of ๐ข๐, we clearly have
sup๐ต๐ /๐
|๐ข๐ | โค ๐ <โ, โฅโ๐ข๐โฅ๐ฟโ (๐ต๐ /๐) โค ๐ <โ for all ๐ > 0. (3.1)
Hence by the Arzelร โAscoli theorem and a diagonal argument, we can take a decreasing sequence {๐๐ }โ๐=1 โ(0,โ), such that ๐๐ โ 0 as ๐ โโ, and
๐ข๐๐ โ ๐ข0 locally uniformly in R๐. (3.2)
9
for some function ๐ข0 : R๐ โ R. Clearly ๐ข0 is convex in R๐, and the inclusion ๐๐ข(๐ฅ0) โ ๐๐ข0 (๐ฅ0) holds true by theconstruction of rescaled functions ๐ข๐. If ๐ฅ0 โ ๐น, then we have {0} โ ๐๐ข(๐ฅ0) โ ๐๐ข0 (๐ฅ0) and therefore ๐ฅ0 lies in thefacet of ๐ข0. We are left to show that ๐ข0 is a weak solution to ๐ฟ๐, ๐๐ข0 = 0 in R๐. Before proving this, we note thatfrom (3.1)โ(3.2) and Lemma 9, it follows that
โ๐ข๐๐ (๐ฅ) โ โ๐ข0 (๐ฅ) and |โ๐ข0 (๐ฅ) | โค ๐ for a.e. ๐ฅ โ R๐ (3.3)
as ๐ โโ. We arbitrarily fix an open ball ๐ต๐ โ R๐. Note that the inclusion ๐ต๐ โ ๐ต๐ /๐ holds for all 0 < ๐ < ๐ /๐ .Hence we easily realize that a family of pairs {(๐ข๐, ๐๐)}0<๐<๐ /๐ โ๐1,โ (๐ต๐ ) ร ๐ฟโ (๐ต๐ , R๐) satisfies
๐๐ (๐ฅ) โ ๐ | ยท | (โ๐ข๐ (๐ฅ)) for a.e. ๐ฅ โ ๐ต๐ , (3.4)
๐
โซ๐ต๐
โจ๐๐ | โ๐โฉ ๐๐ฅ +โซ๐ต๐
โจ|โ๐ข๐ |๐โ2โ๐ข๐
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ = โซ๐ต๐
๐๐๐๐๐ฅ for all ๐ โ๐1, ๐0 (๐ต๐ ). (3.5)
By definition of ๐๐, we get โฅ ๐๐โฅ๐ฟ๐ (๐ต๐ ) = ๐1โ๐/๐ โฅ ๐ โฅ๐ฟ๐ (๐ต๐๐ ) โค ๐1โ๐/๐ โฅ ๐ โฅ๐ฟ๐ (๐ต๐ ) for all 0 < ๐ < ๐ /๐ . Hence by the
continuous embedding ๐ฟ๐ (๐ต๐ ) โฉโ๐โ1, ๐โฒ (๐ต๐ ), we obtain
๐๐๐ โ 0 in๐โ1, ๐โฒ (๐ต๐ ) as ๐ โโ. (3.6)
By (3.1) and (3.3), we can apply Lebesgueโs dominated convergence theorem and get
|โ๐ข๐๐ |๐โ2โ๐ข๐๐ โ |โ๐ข0 |๐โ2โ๐ข0 in ๐ฟ ๐โฒ (๐ต๐ , R๐) as ๐ โโ. (3.7)
It is clear that โฅ๐๐โฅ๐ฟโ (๐ต๐ ,R๐) โค 1 for all 0 < ๐ < ๐ /๐ . Hence by [5, Corollary 3.30], up to a subsequence, we mayassume that
๐๐๐โโ ๐0, ๐ in ๐ฟโ (๐ต๐ , R๐) as ๐ โโ (3.8)
for some ๐0, ๐ โ ๐ฟโ (๐ต๐ , R๐). By lower-semicontinuity of the norm with respect to the weakโ topology and(3.3)โ(3.4), we get
โฅ๐0, ๐ โฅ๐ฟโ (๐ต๐ ,R๐) โค 1, ๐0, ๐ (๐ฅ) =โ๐ข0 (๐ฅ)|โ๐ข0 (๐ฅ) |
for a.e. ๐ฅ โ ๐ต๐ with โ๐ข0 (๐ฅ) โ 0,
which implies that๐0, ๐ (๐ฅ) โ ๐ | ยท | (โ๐ข0 (๐ฅ)) for a.e. ๐ฅ โ ๐ต๐ . (3.9)
Letting ๐ = ๐๐ in (3.5) and ๐ โโ, we obtain
๐
โซ๐ต๐
โจ๐0, ๐ | โ๐โฉ ๐๐ฅ +โซ๐ต๐
โจ|โ๐ข0 |๐โ2โ๐ข0
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ = 0 for all ๐ โ๐1, ๐0 (๐ต๐ ) (3.10)
by (3.5)โ(3.8). Since ๐ต๐ โ R๐ is arbitrary, (3.9)โ(3.10) means that ๐ข0 is a weak solution to ๐ฟ๐, ๐๐ข0 = 0 in R๐, inthe sense of Definition 1. โก
4 Maximum principlesIn Section 4, we justify maximum principles for the equation ๐ฟ๐, ๐๐ข = 0.
We first define subsolutions and supersolutions in the weak sense.
Definition 2. Letฮฉ โ R๐ be a bounded domain. A pair (๐ข, ๐) โ๐1, ๐ (ฮฉ) ร๐ฟโ (ฮฉ, R๐) is called a weak subsolutionto ๐ฟ๐, ๐๐ข = 0 in ฮฉ, if it satisfies
๐
โซฮฉโจ๐ | โ๐โฉ ๐๐ฅ +
โซฮฉ
โจ|โ๐ข |๐โ2โ๐ข
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ โค 0 (4.1)
for all 0 โค ๐ โ ๐ถโ๐ (ฮฉ), and
๐ (๐ฅ) โ ๐ | ยท | (โ๐ข(๐ฅ)) for a.e. ๐ฅ โ ฮฉ. (4.2)
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Similarly we call a pair (๐ข, ๐) โ๐1, ๐ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) a weak supersolution ๐ฟ๐, ๐๐ข = 0 in ฮฉ, if it satisfies (4.2)and
๐
โซฮฉโจ๐ | โ๐โฉ ๐๐ฅ +
โซฮฉ
โจ|โ๐ข |๐โ2โ๐ข
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ โฅ 0 (4.3)
for all 0 โค ๐ โ ๐ถโ๐ (ฮฉ). For ๐ข โ๐1, ๐ (ฮฉ), we simply say that ๐ข is respectively a subsolution and a supersolution to
๐ฟ๐, ๐๐ข = 0 in the weak sense if there is ๐ โ ๐ฟโ (ฮฉ, R๐) such that the pair (๐ข, ๐) is a weak subsolution and a weaksupersolution to ๐ฟ๐, ๐๐ข = 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
๐ท+ (ฮฉ) B {๐ โ๐1, ๐ (ฮฉ) | ๐ โฅ 0 a.e. in ฮฉ, supp๐ โ ฮฉ}.
Indeed, for ๐ โ ๐ท+ (ฮฉ) and 0 < ๐ < dist(supp๐, ๐ฮฉ), the function,
๐๐ (๐ฅ) =โซฮฉ๐(๐ฅโ ๐ฆ)๐๐ (๐ฆ) ๐๐ฆ for ๐ฅ โ ฮฉ
satisfies 0 โค ๐๐ โ ๐ถโ๐ (ฮฉ). Here for 0 < ๐ <โ, 0 โค ๐๐ โ ๐ถโ
๐ (๐ต๐ (0)) denotes a standard mollifier so that
0 โค ๐ โ ๐ถโ๐ (๐ต1), โฅ๐โฅ๐ฟ1 (R๐) = 1, ๐๐ (๐ฅ) B ๐โ๐๐(๐ฅ/๐) for ๐ฅ โ R๐.
By testing ๐๐ into (4.1) for sufficiently small ๐ > 0 and letting ๐โ 0, we conclude that if the pair (๐ข, ๐)satisfies (4.1) for all 0 โค ๐ โ ๐ถโ
๐ (ฮฉ), then (4.1) holds for all ๐ โ ๐ท+ (ฮฉ). A similar result is also valid for(4.3).
2. By Definition 1โ2, if a pair (๐ข, ๐) โ ๐1, ๐ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) satisfies ๐ฟ๐, ๐๐ข = 0 in ๐โ1, ๐โฒ (ฮฉ), then ๐ข isclearly both a subsolution and a supersolution to ๐ฟ๐, ๐๐ข = 0 in ฮฉ in the weak sense. Conversely, if a pair(๐ข, ๐) โ๐1, ๐ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) is both a weak subsolution and a weak supersolution to ๐ฟ๐, ๐๐ข = 0 in ฮฉ, thenthe pair (๐ข, ๐) satisfies ๐ฟ๐, ๐๐ข = 0 in๐โ1, ๐โฒ (ฮฉ). Indeed, by the previous remark we have already known thatthe pair (๐ข, ๐) satisfies (4.1) and (4.3) for all ๐ โ ๐ท+ (ฮฉ), which clearly yields
๐
โซฮฉโจ๐ | โ๐โฉ ๐๐ฅ +
โซฮฉ
โจ|โ๐ข |๐โ2โ๐ข
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ = 0 (4.4)
for all ๐ โ ๐ท+ (ฮฉ). We decompose arbitrary ๐ โ ๐ถโ๐ (ฮฉ) by ๐ = ๐+ โ ๐โ, where ๐+ B max{๐, 0 }, ๐โ B
max{โ๐, 0 } โ ๐ท+ (ฮฉ). By testing ๐+, ๐โ โ ๐ท+ (ฮฉ) into (4.4), we conclude that (4.4) holds for all ๐ โ๐ถโ๐ (ฮฉ).
By density of ๐ถโ๐ (ฮฉ) โ๐1, ๐
0 (ฮฉ), it is clear that (4.4) is valid for all ๐ โ๐1, ๐0 (ฮฉ).
3. For a bounded domain ฮฉ โ R๐, let ๐ข โ ๐ถ2 (ฮฉ) satisfy the following two conditions (4.5)โ(4.6);
โ๐ข(๐ฅ) โ 0 for all ๐ฅ โ ฮฉ, (4.5)
(๐ฟ๐, ๐๐ข)(๐ฅ) = โ(๐ฮ1๐ข +ฮ๐๐ข)(๐ฅ) โค 0 for all ๐ฅ โ ฮฉ. (4.6)
Then for any fixed 0 โค ๐ โ ๐ถโ๐ (ฮฉ), we have
0 โฅโซฮฉ(๐ฟ๐, ๐๐ข)๐๐๐ฅ = ๐
โซฮฉ
โจโ๐ข|โ๐ข |
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ +โซฮฉ
โจ|โ๐ข |๐โ2โ๐ข
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ,with the aid of integration by parts and (4.6). We also note that
๐ | ยท | (โ๐ข(๐ฅ)) ={โ๐ข(๐ฅ)|โ๐ข(๐ฅ) |
}for all ๐ฅ โ ฮฉ
by (4.5). Therefore the pair (๐ข, โ๐ข/|โ๐ข |) โ ๐1, ๐ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) satisfies (4.1)โ(4.2). For such ๐ข, wesimply say that ๐ข satisfies ๐ฟ๐, ๐๐ข โค 0 in ฮฉ in the classical sense.
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4.1 Comparison principleWe justify the comparison principle, i.e., for any subsolution ๐ขโ and supersolution ๐ข+,
๐ขโ โค ๐ข+ on ๐ฮฉ implies that ๐ขโ โค ๐ข+ in ฮฉ,
under the condition that ๐ข+ and ๐ขโ admits continuity properties in ฮฉ.
Proposition 2. Let ฮฉ โ R๐ be a bounded domain. Assume that ๐ข+, ๐ขโ โ ๐ถ (ฮฉ) โฉ๐1, ๐ (ฮฉ) is a subsolution and asupersolution to ๐ฟ๐, ๐๐ข = 0 in the weak sense respectively. If ๐ข+, ๐ขโ satisfies
๐ขโ (๐ฅ) โค ๐ข+ (๐ฅ) for all ๐ฅ โ ๐ฮฉ, (4.7)
then ๐ขโ โค ๐ข+ in ฮฉ.
Before proving Proposition 2, we recall that the mapping ๐ด : R๐ โ ๐ง โฆโ |๐ง |๐โ2๐ง โ R๐ satisfies strict monotonicity,
i.e., โจ๐ด(๐ง2) โ ๐ด(๐ง1) | ๐ง2 โ ๐ง1โฉ > 0 for all ๐ง1, ๐ง2 โ R๐ with ๐ง1 โ ๐ง2. (4.8)
Proof. We take arbitrary ๐ฟ > 0. By ๐ข+, ๐ขโ โ ๐ถ (ฮฉ) and (4.7), we can take a subdomain ฮฉโฒ โ ฮฉ such that ๐ขโ โค๐ข+ + ๐ฟ in ฮฉ \ฮฉโฒ. This implies that the support of the truncated non-negative function ๐ค ๐ฟ B (๐ข+โ๐ขโ + ๐ฟ)โ โ๐1, ๐ (ฮฉ) is contained in ฮฉโฒ โ ฮฉ and therefore ๐ค ๐ฟ โ ๐ท+ (ฮฉ). Let ๐+, ๐โ โ ๐ฟโ (ฮฉ, R๐) be vector fields such that(๐ข+, ๐+), (๐ขโ, ๐โ) satisfies (4.1)โ(4.2), (4.2)โ(4.3) respectively. As in Remark 2, we may test ๐ค ๐ฟ in (4.1) and (4.3).Note that โ๐ค ๐ฟ = โ๐๐ฟโ(๐ข+โ๐ขโ), where ๐๐ฟ denotes the characteristic function of ๐ด๐ฟ B {๐ฅ โ ฮฉ | ๐ข+ + ๐ฟ โค ๐ขโ}.Hence, we have
0 โค โ๐โซ๐ด๐ฟ
โจ๐+โ ๐โ
๏ฟฝ๏ฟฝ โ๐ข+โโ๐ขโโฉ๐๐ฅโ
โซ๐ด๐ฟ
โจ|โ๐ข+ |๐โ2โ๐ข+โ |โ๐ขโ |๐โ2โ๐ขโ
๏ฟฝ๏ฟฝ โ๐ข+โโ๐ขโโฉ๐๐ฅ
โค โโซ๐ด๐ฟ
โจ|โ๐ข+ |๐โ2โ๐ข+โ |โ๐ขโ |๐โ2โ๐ขโ
๏ฟฝ๏ฟฝ โ๐ข+โโ๐ขโโฉ๐๐ฅ.
Here we have invoked (4.2) and monotonicity of the subdifferential operator ๐ | ยท |. From (4.8) we can easily checkthat โ๐ข+ = โ๐ขโ in ๐ด๐ฟ , and therefore ๐ค ๐ฟ = 0 in ๐1, ๐
0 (ฮฉ). This means that ๐ขโ โค ๐ข+ + ๐ฟ a.e. in ฮฉ. By regularityassumptions ๐ข+, ๐ขโ โ๐ถ (ฮฉ), we conclude that ๐ขโ โค ๐ข++๐ฟ inฮฉ. Since ๐ฟ > 0 is arbitrary, this completes the proof. โก
Remark 3. In 2013, Krรผgel gave another type of definitions of weak subsolutions and weak supersolutions to๐ฟ๐, ๐ = ๐, where ๐ โ R is a constant. In Krรผgelโs definition [25, Definition 4.6], a function ๐ขโ โ๐1, ๐ (ฮฉ) is calleda subsolution to ๐ฟ๐, ๐ = ๐ if ๐ขโ satisfiesโซ
๐ทโ
โจโ๐ขโ|โ๐ขโ |
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ +โซ๐นโ
|โ๐| ๐๐ฅ +โจ|โ๐ขโ |๐โ2โ๐ขโ
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ โค โซฮฉ๐๐ ๐๐ฅ (4.9)
for all ๐ โ ๐ท+ (ฮฉ). Here ๐นโ B {๐ฅ โ ฮฉ | โ๐ขโ (๐ฅ) = 0}, ๐ทโ B ฮฉ\๐นโ. Similarly a function ๐ข+ โ๐1, ๐ (ฮฉ) is called asupersolution to ๐ฟ๐, ๐ = ๐ if ๐ข+ satisfiesโซ
๐ท+
โจโ๐ข+|โ๐ข+ |
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ +โซ๐น+
|โ๐| ๐๐ฅ +โจ|โ๐ข+ |๐โ2โ๐ข+
๏ฟฝ๏ฟฝ โ๐โฉ ๐๐ฅ โฅ โซฮฉ๐๐ ๐๐ฅ (4.10)
for all ๐ โ ๐ท+ (ฮฉ). Here ๐น+ B {๐ฅ โ ฮฉ | โ๐ข+ (๐ฅ) = 0}, ๐ท+ B ฮฉ \๐น+.The comparison principle discussed by Krรผgel [25, Theorem 4.8] states that
(๐ขโโ๐ข+)+ โ ๐ท+ (ฮฉ) implies ๐ขโ โค ๐ข+ a.e. in ฮฉ. (4.11)
By testing (๐ขโโ๐ข+)+ โ ๐ท+ (ฮฉ) into (4.9)(4.10) and substracting the two inequalities, Krรผgel claims that โ๐ขโ = โ๐ข+over ฮฉโฒ B {๐ฅ โ ฮฉ | ๐ขโ (๐ฅ) โฅ ๐ข+ (๐ฅ)} and hence ๐ขโ = ๐ข+ a.e. in ฮฉโฒ. Despite Krรผgelโs comment that integrals over ๐นโ
and ๐น+ cancel out, however, it seems unclear whetherโซ๐นโ
|โ(๐ขโโ๐ข+)+ | ๐๐ฅ =โซ๐น+
|โ(๐ขโโ๐ข+)+ | ๐๐ฅ (4.12)
is valid. This problem is essentially due to the fact that Krรผgel did not appeal to monotonicity of the subdifferentialoperator ๐ | ยท | and did not regard the term โ๐ข/|โ๐ข | as an ๐ฟโ-vector field satisfying the property (4.2). In our proofof the comparison principle (Proposition 2), we make use of monotonicity of the operator ๐ | 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 ๐ฟ๐, ๐๐ข = 0 in an open annulus.
Lemma 2. Let ๐ โ R๐ \ {0}, ๐ > 0. Then for each fixed open ball ๐ต๐ (๐ฅโ) โ R๐, there exists a function โ โ๐ถโ (R๐ \ {๐ฅโ}) such that
โ = 0 on ๐๐ต๐ (๐ฅโ), 0 โค โ โค ๐ on ๐ธ๐ (๐ฅโ), (4.13)
๐๐โ < 0 on ๐๐ต๐ (๐ฅโ), (4.14)
|โโ| โค |๐ |2
in ๐ธ๐ (๐ฅโ), (4.15)
๐ฃ(๐ฅ) B โ(๐ฅ) + โจ๐ | ๐ฅโฉ satisfies ๐ฟ๐, ๐๐ฃ โค 0 in ๐ธ๐ (๐ฅโ), in the classical sense. (4.16)
Here ๐ธ๐ (๐ฅโ) B ๐ต๐ (๐ฅโ) \ ๐ต๐ /2 (๐ฅโ) is an open annulus, and ๐ in (4.14) denotes the exterior unit vector normal to๐ต๐ (๐ฅโ).
Before proving Lemma 2, we fix some notations on matrices. For a given ๐ร๐ matrix ๐ด, we write tr(๐ด) as thetrace of ๐ด. We denote 1๐ by the ๐ร๐ unit matrix. For column vectors ๐ฅ = (๐ฅ๐)๐ , ๐ฆ = (๐ฆ๐)๐ โ R๐, we define a tensor๐ฅ โ ๐ฆ, which is regarded as a real-valued ๐ร๐ matrix
๐ฅ โ ๐ฆ B (๐ฅ๐๐ฆ ๐ )๐, ๐ =ยฉยญยญยซ๐ฅ1๐ฆ1 ยท ยท ยท ๐ฅ1๐ฆ๐...
. . ....
๐ฅ๐๐ฆ1 ยท ยท ยท ๐ฅ๐๐ฆ๐
ยชยฎยฎยฌ .Assume that โ satisfies (4.15). Then the triangle inequality implies that
0 <12|๐ | โค |โ๐ฃ | โค 3
2|๐ | in ๐ธ๐ (๐ฅโ). (4.17)
The estimate (4.17) allows us to calculate ๐ฟ๐, ๐๐ฃ in the classical sense over ๐ธ๐ (๐ฅโ). By direct calculations we have
โ๐ฟ๐, ๐๐ฃ = +div(โ๐ง๐ธ (โ๐ฃ)) =๐โ
๐, ๐=1๐๐ง๐ ๐ง ๐๐ธ (โ๐ฃ)๐๐ฅ๐ ๐ฅ ๐ ๐ฃ = tr
(โ2๐ง๐ธ (โ๐ฃ)โ2โ
)in ๐ธ๐ (๐ฅโ).
We note that โ2๐ฃ = โ2โ by definition. Here we recall a well-known result on Pucciโs extremal operators. For givenconstants 0 < ๐ โค ฮ <โ and a fixed ๐ร๐ symmetric matrix ๐ , we define
Mโ (๐, ๐, ฮ) B ๐โ๐๐>0
๐๐ +ฮโ๐๐<0
๐๐ ,
where ๐๐ โ R are the eigenvalues of ๐ . The following formula is a well-known result [1, Remark 5.36] ;
Mโ (๐, ๐, ฮ) = inf{tr(๐ด๐)
๏ฟฝ๏ฟฝ ๐ด โ A๐,ฮ},
where A๐,ฮ denotes the set of all symmetric matrices whose eigenvalues all belong to the closed interval [๐, ฮ].By (4.17) ๐ฟ๐, ๐๐ฃ is an uniformly elliptic operator in ๐ธ๐ (๐ฅโ). This enables us to find constants 0 < ๐ โค ฮ < โ,depending on 0 < ๐ < โ, 1 < ๐ < โ, |๐ | > 0, such that โ2
๐ง๐ธ (โ๐ฃ) โ [๐, ฮ] in ๐ธ๐ (๐ฅโ). Combining these results, itsuffices to show that
Mโ(โ2โ(๐ฅ), ๐, ฮ
)= ๐
โ๐๐>0
๐๐ (๐ฅ) +ฮโ๐๐<0
๐๐ (๐ฅ) > 0 for all ๐ฅ โ ๐ธ๐ (๐ฅโ), (4.18)
where ๐๐ (๐ฅ) โ R denotes the eigenvalues of โ2โ(๐ฅ).Now we construct classical subsolutions. Our first construction is a modification of that by E. Hopf [20].
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Proof. Without loss of generality we may assume ๐ฅโ = 0. We define
โ(๐ฅ) B ๐โ๐ผ |๐ฅ |2 โ ๐โ๐ผ๐ 2
for ๐ฅ โ R๐. (4.19)
Here ๐ผ = ๐ผ(๐, ๐, ๐, |๐ |, ๐ ) > 0 is a sufficiently large constant to be chosen later. It is clear that 0 โค โ(๐ฅ) โค๐โ๐ผ๐
2/4 โ ๐โ๐ผ๐ 2 in ๐ธ๐ (0). We first let ๐ผ > 0 be so large that
๐๐๐ผ๐ 2 โฅ ๐3๐ผ๐ 2/4 โ1. (4.20)
From (4.20), we can easily check (4.13). By direct calculation we get
โโ(๐ฅ) = โ2๐ผ๐โ๐ผ |๐ฅ |2๐ฅ, and โ2โ(๐ฅ) = โ2๐ผ๐โ๐ผ |๐ฅ |
21๐ +4๐ผ2๐โ๐ผ |๐ฅ |
2๐ฅ โ ๐ฅ for each ๐ฅ โ R๐.
From this result, (4.14) is clear. Also, we have
|โโ(๐ฅ) | โค 2๐ผ๐ ๐โ๐ผ๐ 2/4 for all ๐ฅ โ ๐ธ๐ (0).
Let ๐ผ > 0 be so large that
๐ผ๐โ๐ผ๐ 2/4 โค |๐ |
4๐ , (4.21)
then we can check that โ satisfies (4.15). Now we prove (4.16) to complete the proof. For ๐ฅ โ 0, the eigenvalues ofโ2โ(๐ฅ) are given by{
๐ โฅ (๐ฅ) B 4๐ผ2 |๐ฅ |2๐โ๐ผ |๐ฅ |2 โ2๐ผ๐โ๐ผ |๐ฅ |2 ,๐โฅ (๐ฅ) B โ2๐ผ๐โ๐ผ |๐ฅ |2 ,
and the geometric multiplicities are{
1,๐โ1.
Assume that ๐ผ satisfies๐ผ >
2๐ 2 , (4.22)
so that ๐ โฅ > 0 > ๐โฅ in ๐ธ๐ (0). Therefore we get
Mโ(โ2โ(๐ฅ), ๐, ฮ
)= ๐๐ โฅ (๐ฅ) + (๐โ1)ฮ๐โฅ (๐ฅ) = 2๐ผ๐โ๐ผ |๐ฅ |
2 [๐(2๐ผ |๐ฅ |2 โ1) โ (๐โ1)ฮ
]โฅ 2๐ผ๐โ๐ผ |๐ฅ |
2[๐
(๐ 2
2๐ผโ1
)โ (๐โ1)ฮ
].
We can take sufficiently large ๐ผ = ๐ผ( |๐ |, ๐, ๐, ๐ , ๐, ฮ) > 0 so that ๐ผ satisfies (4.18) and (4.20)โ(4.22). For suchconstant ๐ผ > 0, the function ๐ฃ defined as in (4.19) satisfies (4.13)โ(4.16). โก
It is possible to construct an alternative function โ โ ๐ถโ (R๐ \ {๐ฅ0}) 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 ๐ฅโ = 0. We define
โ(๐ฅ) B ๐ฝ[|๐ฅ |โ๐ผ โ๐ โ๐ผ] for ๐ฅ โ R๐ \ {0}. (4.23)
We will later determine positive constants ๐ผ, ๐ฝ > 0, depending on ๐, ๐, ๐, ๐, |๐ |, ๐ . It is clear that 0 โค โ(๐ฅ) โค๐ฝ๐ โ๐ผ (2๐ผ โ1) in ๐ธ๐ (0). We first let ๐ผ, ๐ฝ > 0 satisfy
๐ฝ โค ๐๐ ๐ผ
2๐ผ โ1. (4.24)
Then โ satisfies (4.13). By direct calculation we get
โโ(๐ฅ) = โ ๐ผ๐ฝ๐ฅ
|๐ฅ |๐ผ+2 , and โ2โ(๐ฅ) = ๐ผ๐ฝ
|๐ฅ |๐ผ+2
[(๐ผ+2) ๐ฅ โ ๐ฅ
|๐ฅ |2โ1๐
]
14
for each ๐ฅ โ R๐ \ {0}. The estimate (4.14) is clear by this result. Also, we have
|โโ(๐ฅ) | โค ๐ผ๐ฝ
(๐ /2)๐ผ+1 for all ๐ฅ โ ๐ธ๐ (0).
Let ๐ผ, ๐ฝ > 0 satisfy
๐ฝ โค |๐ | (๐ /2)๐ผ+1
2๐ผ, (4.25)
then we can check that โ satisfies (4.15). Now we prove (4.16) to complete the proof. For ๐ฅ โ 0, the eigenvalues ofโ2โ(๐ฅ) are given by{
๐ โฅ (๐ฅ) B (๐ผ+1)๐ผ๐ฝ |๐ฅ |โ๐ผโ2,๐โฅ (๐ฅ) B โ๐ผ๐ฝ |๐ฅ |โ๐ผโ2,
and the geometric multiplicities are{
1,๐โ1.
It is clear that ๐ โฅ > 0 > ๐โฅ in R๐ \ {0}, and therefore
Mโ(โ2โ(๐ฅ), ๐, ฮ
)= ๐ผ๐ฝ |๐ฅ |โ๐ผโ2 [(๐ผ+1)๐โ (๐โ1)ฮ]
for all ๐ฅ โ ๐ธ๐ (0). We take and fix sufficiently large ๐ผ = ๐ผ(๐, ๐, ฮ) > 0 so that ๐ผ satisfies (4.18). For such ๐ผ > 0,we choose sufficiently small ๐ฝ = ๐ฝ( |๐ |, ๐ , ๐ผ) > 0 so that ๐ฝ satisfies (4.24)โ(4.25). Then the function โ defined asin (4.23) satisfies (4.13)โ(4.16). โก
4.3 Strong maximum principleWe prove the strong maximum principle (Theorem 2).
Proof. Let ๐ท0 โ ๐ท be a connected component of the open set ๐ท, and ๐ฅ0 โ ๐ท0. Without loss of generality wemay assume that ๐ฅ0 = 0 and ๐ข(0) = 0. By Lemma 1, it is clear that ๐๐ข(0) = {โ๐ข(0)} โ {0}. We set a vector๐ B โ๐ข(0) โ R๐ \ {0} and a relatively closed set
ฮฃ B {๐ฅ โ ๐ท0 | ๐ข(๐ฅ) = โจ๐ | ๐ฅโฉ}.
and we will prove that ฮฃ = ๐ท0. It is also clear that 0 โ ฮฃ and hence ฮฃ โ โ . Suppose for contradiction that ฮฃ โ ๐ท0.Then it follows that ๐ฮฃ โฉ๐ท0 โ โ , since ๐ท0 is connected. We may take and fix a point ๐ฅโ โ ๐ท0 \ฮฃ such thatdist(๐ฅโ, ฮฃ) < dist(๐ฅโ, ๐๐ท0). By extending a closed ball centered at ๐ฅโ until it hits ฮฃ, we can take a point ๐ฆโ โ ๐ท0and a closed ball ๐ต๐ (๐ฅโ) โ ๐ท0 such that ๐ฆโ โ ๐๐ต๐ (๐ฅโ) โฉฮฃ and ๐ข(๐ฅ) > โจ๐ | ๐ฅโฉ for all ๐ฅ โ ๐ต๐ (๐ฅโ). We note that
0 = min๐ฅโ๐๐ต๐ (๐ฅโ)
(๐ข(๐ฅ) โ โจ๐ | ๐ฅโฉ), achieved at ๐ฆโ โ ๐๐ต๐ (๐ฅโ),๐ B min
๐ฅโ๐๐ต๐ /2 (๐ฅโ)(๐ข(๐ฅ) โ โจ๐ | ๐ฅโฉ) > 0, (4.26)
by construction of ๐ต๐ (๐ฅโ). Let โ โ ๐ถโ (R๐ \ {๐ฅโ}) be an auxiliary function as in Lemma 2. Then from (4.26) itis easy to check that ๐ฃ B โ + โจ๐ | ๐ฅโฉ satisfies ๐ฃ โค ๐ข on ๐๐ธ๐ (๐ฅโ), in the sense of (4.7). By Proposition 2, we have๐ฃ โค ๐ข on ๐ธ๐ (๐ฅโ). Hence 0 โค ๐ขโ โจ๐ | ๐ฅโฉ โ โ in ๐ธ๐ (๐ฅโ). This inequality becomes equality at ๐ฆโ โ ๐๐ธ๐ (๐ฅโ) by (4.13)and (4.26). Therefore the function ๐ข(๐ฅ) โ โจ๐ | ๐ฅโฉ โ โ(๐ฅ) (๐ฅ โ ๐ธ๐ (๐ฅโ)) takes its minimum at ๐ฆโ โ ๐๐ต๐ (๐ฅโ). Also by๐ฆโ โ ฮฃ and the subgradient inequality
๐ข(๐ฅ) โฅ โจ๐ | ๐ฅโฉ for all ๐ฅ โ ฮฉ,
it is clear that the function ๐ค(๐ฅ) B ๐ข(๐ฅ) โ โจ๐ | ๐ฅโฉ (๐ฅ โ ๐ท0) takes its minimum 0 at ๐ฆโ โ ๐ท0. We note that๐ค, ๐คโ โ โ ๐ถ1 (๐ท0) by Lemma 1. By calculating classical partial derivatives at ๐ฆโ in the direction ๐0 B (๐ฆโโ๐ฅโ)/๐ ,we obtain
0 โฅ ๐๐0 (๐คโ โ) (๐ฆโ) = โ๐๐0โ(๐ฆโ) > 0.
This is a contradiction, and therefore ฮฃ = ๐ท0. โก
5 Proofs of main theoremsIn Section 5, we give proofs of the Liouville-type theorem (Theorem 3) and the ๐ถ1-regularity theorem (Thorem 1).
15
5.1 Liouville-type theoremFor a preparation, we prove Lemma 3 below.
Lemma 3. Let ๐ข be a real-valued convex function in R๐. Assume that ๐ข satisfies the following,
1. The facet of ๐ข, ๐น โ R๐, satisfies โ โ ๐น โ R๐.
2. ๐ข attains its minimum 0.
3. ๐ข is affine in each connected component of ๐ท B R๐ \๐น.
Then up to a rotation and a shift translation, ๐ข can be expressed as either of the following three types ofpiecewise-linear functions.
๐ข(๐ฅ) = max{ ๐ก1๐ฅ1, 0 } for all ๐ฅ โ R๐, (5.1)
๐ข(๐ฅ) = max{ ๐ก1๐ฅ1, โ๐ก2๐ฅ1 } for all ๐ฅ โ R๐, (5.2)
๐ข(๐ฅ) = max{ ๐ก1๐ฅ1, 0, โ๐ก2 (๐ฅ1 + ๐0) } for all ๐ฅ โ R๐. (5.3)
Here ๐ก1, ๐ก2, ๐ > 0 are constants.
Before starting the proof of Lemma 3, we introduce notations on affine hyperplanes. For ๐ โ R๐ \ {0} and๐ฅ0 โ R๐, we define
๐ป๐, ๐ฅ0 B {๐ฅ โ R๐ | โจ๐ | ๐ฅโ ๐ฅ0โฉ = 0},๐ปโ๐, ๐ฅ0 B {๐ฅ โ R๐ | โจ๐ | ๐ฅโ ๐ฅ0โฉ < 0},
๐ป+๐, ๐ฅ0 B {๐ฅ โ R๐ | โจ๐ | ๐ฅโ ๐ฅ0โฉ > 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 ๐ถ โ R๐ and ๐ฅ0 โ ๐๐ถ, there exists ๐ โ R๐ \ {0} such that
sup๐ฅโ๐ถ
โจ๐ | ๐ฅโฉ โค โจ๐ | ๐ฅ0โฉ, and in particular ๐ป+๐, ๐ฅ0 โ R
๐ \๐ถ.
For such ๐ โ R๐ \ {0}, a hyperplane ๐ป๐, ๐ฅ0 is often called a supporting hyperplane for ๐ถ at the boundary point ๐ฅ0.For the proof of the supporting hyperplane theorem, see [3, Proposition 1.5.1].
Proof. Since R๐ is connected and ๐น โ R๐ is a closed convex set, it follows that ๐๐น โ โ . Without loss of generalitywe may assume that 0 โ ๐๐น and ๐ข(0) = 0.
By the supporting hyperplane theorem, we can take and fix a supporting hyperplane for ๐น at the boundary point0, which we write ๐ป๐, 0 โ R๐. By rotation, we may assume that ๐ = ๐1. Let ๐ท1 be the connected component of๐ท which contains ๐ป+
๐1 , 0 โ R๐ \๐น = ๐ท. By the assumption 3 and ๐ข(0) = 0, it follows that there exists ๐ โ R๐ \ {0}such that ๐ข(๐ฅ) = โจ๐ | ๐ฅโฉ for all ๐ฅ โ ๐ท1. We should note that ๐ป๐, 0 = ๐ป๐1 , 0 and hence ๐ = ๐ก1๐1 for some ๐ก1 โ (0,โ),since otherwise it follows that ๐ป+
๐1 , 0 โฉ๐ปโ๐, 0 โ โ and 0 โค ๐ข(๐ฅ0) = โจ๐ | ๐ฅ0โฉ < 0 for any ๐ฅ0 โ ๐ป+
๐1 , 0 โฉ๐ปโ๐, 0. The result
๐ป๐, 0 = ๐ป๐1 , 0 also implies that ๐ป๐1 , 0 โ ๐๐น โ ๐น โ {๐ฅ โ R๐ | ๐ฅ1 โค 0} = ๐ปโ๐1 , 0 โช๐ป๐1 , 0. Now we will deduce three
possible representations of ๐ข.If ๐๐น = ๐ป๐1 , 0, then we have either ๐น = ๐ปโ
๐1 , 0 โช๐ป๐1 , 0 or ๐น = ๐ป๐1 , 0, since the open set ๐ปโ๐1 , 0 = {๐ฅ โ R๐ | ๐ฅ1 < 0}
is connected. For the first case, ๐ข is clearly expressed by (5.1). For the second case, it is clear that ๐ท consists of twoconnected components ๐ท1 = ๐ป+
๐1 , 0 and ๐ท2 = ๐ปโ๐1 , 0. Again by the condition 3 and similar arguments to the above,
we can determine ๐ข |๐ท2 as ๐ข(๐ฅ) = โจโ๐ก2๐1 | ๐ฅโฉ for all ๐ฅ โ ๐ท2. Here ๐ก2 โ (0,โ) is a constant. Hence we obtain (5.2).For the case ๐ป๐1 , 0 โ ๐๐น, we take and fix ๐ง0 โ ๐๐น \๐ป๐1 , 0 and a supporting hyperplane for ๐น at ๐ง0, which we write by๐ป๐โฒ, ๐ง0 . Let ๐ท2 be the connected component of ๐ท which contains ๐ป+
๐โฒ, ๐ง0โ ๐ท. By the assumption 3 and ๐ข(๐ง0) = 0,
it follows that there exists ๐โฒโฒ โ R๐ \ {0} such that ๐ข(๐ฅ) = โจ๐โฒโฒ | ๐ฅโ ๐ง0โฉ for all ๐ฅ โ ๐ท2. Completely similarly to thearguments above for showing that ๐ป๐, 0 = ๐ป๐1 , 0, we can easily notice that ๐ป๐โฒโฒ, ๐ง0 = ๐ป๐โฒ, ๐ง0 and hence ๐โฒโฒ = ๐ก โฒ1๐
โฒ forsome constant ๐ก โฒ1 โ (0,โ). Moreover, we also realize that ๐โฒ = ๐กโ๐1 for some ๐กโ โ R\ {0}. Otherwise it follows thatthe two hyperplanes ๐ป๐1 , 0 and ๐ป๐โฒ, ๐ง0 cross, and hence we get ๐ท1 = ๐ท2 and ๐ป+
๐1 , 0 โฉ๐ปโ๐โฒ, ๐ง0
โ โ , which implies thatthere exists a point ๐ฅ0 โ ๐ท such that ๐ข(๐ฅ0) < 0. This is clearly a contradiction. This result and convexity of ๐ข implythat ๐ท consists of two connected components ๐ท1 = ๐ป+
๐1 , 0 and ๐ท2 = ๐ป+โ๐1 , ๐ง0 , and that ๐น = {๐ฅ โ R๐ | โ๐0 โค ๐ฅ1 โค 0}.
Here ๐0 B dist(๐ป๐1 , 0, ๐ปโ๐1 , ๐ง0 ) > 0. Finally we obtain the last possible expression (5.3). ๐ข can be expressed byeither of (5.1)โ(5.3). โก
16
Now we give the proof of Theorem 3.
Proof. Assume by contradiction that ๐น, the facet of ๐ข, would satisfy โ โ ๐น โ R๐. Without loss of generality, wemay assume that ๐ข attains its minimum 0. By the strong maximum principle (Theorem 2), the convex weak solution๐ข is affine in each connected component of ๐ท B R๐ \๐น. Therefore we are able to apply Lemma 3. By rotation andtranslation, ๐ข can be expressed as (5.1)โ(5.3). Now we prove that ๐ข is no longer a weak solution to ๐ฟ๐, ๐๐ข = 0 inR๐. We set open cubes ๐ โฒ B (โ1, 1)๐โ1 โ R๐โ1 and ๐ B (โ๐, ๐) ร๐ โฒ โ R๐, where ๐ > 0 is to be chosen later.We claim that ๐ข does not satisfy ๐ฟ๐, ๐๐ข = 0 in๐โ1, ๐โฒ (๐). Assume by contradiction that there exists a vector field๐ โ ๐ฟโ (๐, R๐) such that the pair (๐ข, ๐) โ๐1, ๐ (๐) ร ๐ฟโ (๐, R๐) satisfies ๐ฟ๐, ๐๐ข = 0 in๐โ1, ๐โฒ (๐).
For the first case (5.1), we have
|๐ (๐ฅ) | โค 1 for a.e. ๐ฅ โ ๐, and ๐ (๐ฅ) = ๐1 for a.e. ๐ฅ โ ๐๐ B (0, ๐) ร๐ โฒ โ R๐. (5.4)
by definition of ๐ . We also set another open cube ๐๐ B (โ๐, 0) ร๐ โฒ โ R๐. We take and fix non-negative functions๐1 โ ๐ถ1
๐ ((โ๐, ๐)), ๐2 โ ๐ถ1๐ (๐ โฒ) such that
๐โฒ1 โฅ 0 in (โ๐, 0), max(โ๐, ๐)
๐1 = ๐1 (0) > 0, and ๐2 . 0. (5.5)
We define an admissible test function ๐ โ ๐ถ1๐ (๐) by ๐(๐ฅ1, ๐ฅ
โฒ) B ๐1 (๐ฅ1)๐2 (๐ฅ โฒ) for (๐ฅ1, ๐ฅโฒ) โ (โ๐, ๐) ร๐ โฒ =๐. Test
๐ โ ๐ถ1๐ (๐) into ๐ฟ๐, ๐๐ข = 0 in ๐โ1, ๐โฒ (๐), and divide the integration over ๐ into that over ๐๐ and ๐๐ . Then (5.4)
implies that
0 = ๐โซ๐๐
โจ๐ + |0|๐โ20 | โ(๐1๐2)โฉ ๐๐ฅ +โซ๐๐
โจ(๐ + ๐ก ๐โ1
1 )๐1
๏ฟฝ๏ฟฝ๏ฟฝ โ(๐1๐2)โฉ๐๐ฅ
โค ๐โซ๐๐
๐โฒ1๐2 ๐๐ฅ + ๐โซ๐๐
๐1 |โ๐2 | ๐๐ฅ
+ ๐๐1 (0)โซ๐โฒ๐2 (๐ฅ โฒ)โจ๐1 | โ๐1โฉ ๐๐ฅ โฒ+ ๐ก ๐โ1
1 ๐1 (0)โซ๐โฒ๐2 (๐ฅ โฒ)โจ๐1 | โ๐1โฉ ๐๐ฅ โฒ
C ๐ผ1 + ๐ผ2 + ๐ผ3 + ๐ผ4.Here we have applied the GaussโGreen theorem to the integration over ๐๐ , and the CauchyโSchwarz inequality tothe integration over ๐๐ . For the integrations ๐ผ1 and ๐ผ2, we make use of Fubiniโs theorem and (5.5). Then we have
๐ผ1 =โซ๐โฒ
(โซ 0
โ๐๐โฒ1 (๐ฅ1)๐๐ฅ1
)๐2 (๐ฅ โฒ)๐๐ฅ โฒ = ๐๐1 (0)
โซ๐โฒ๐2 (๐ฅ โฒ)๐๐ฅ โฒ = ๐๐1 (0)โฅ๐2โฅ๐ฟ1 (๐โฒ) = โ๐ผ3,
๐ผ2 โค ๐๐1 (0)โซ 0
โ๐๐๐ฅ1
โซ๐โฒ|โ๐2 (๐ฅ โฒ) | ๐๐ฅ โฒ = ๐๐๐1 (0)โฅโ๐2โฅ๐ฟ1 (๐โฒ) .
Finally we obtain
0 โค ๐ผ1 + ๐ผ2 + ๐ผ3 + ๐ผ4 โค ๐ผ2 + ๐ผ4 โค ๐1 (0)(๐๐โฅโ๐2โฅ๐ฟ1 (๐โฒ) โ ๐ก
๐โ11 โฅ๐2โฅ๐ฟ1 (๐โฒ)
). (5.6)
From (5.6), we can easily deduce a contradiction by choosing sufficiently small ๐ = ๐ (๐, ๐, ๐ก1, ๐2) > 0. Similarlywe can prove that ๐ข defined as in (5.3) does not satisfy ๐ฟ๐, ๐๐ข = 0 in๐โ1, ๐โฒ (๐), since it suffices to restrict ๐ < ๐0.We consider the remaining case (5.2). We have
๐ (๐ฅ) ={๐1 for a.e. ๐ฅ โ ๐๐ ,โ๐1 for a.e. ๐ฅ โ ๐๐ .
by definition of ๐ . We test the same function ๐ โ ๐ถ1๐ (๐) in ๐ฟ๐, ๐๐ข = 0, then it follows that
0 =โซ๐๐
โจโ(๐ + ๐ก ๐โ1
2 )๐1
๏ฟฝ๏ฟฝ๏ฟฝ โ(๐1๐2)โฉ๐๐ฅ +
โซ๐๐
โจ(๐ + ๐ก ๐โ1
1 )๐1
๏ฟฝ๏ฟฝ๏ฟฝ โ(๐1๐2)โฉ๐๐ฅ
= โ(๐ + ๐ก ๐โ12 )
โซ๐โฒ๐1 (0)๐2 (๐ฅ โฒ)โจ๐1 | ๐1โฉ ๐๐ฅ โฒ+ (๐ + ๐ก ๐โ1
1 )โซ๐โฒ๐1 (0)๐2 (๐ฅ โฒ)โจ๐1 | โ๐1โฉ ๐๐ฅ โฒ
= โ๐1 (0)(2๐ + ๐ก ๐โ1
1 + |๐ก2 |๐โ1) โซ
๐โฒ๐2 (๐ฅ โฒ) ๐๐ฅ โฒ < 0,
which is a contradiction. This completes the proof. โก
17
Remark 4. The estimate (5.6) breaks for ๐ = 1, since the equation |0|๐โ20 = 0 is no longer valid for ๐ = 1. Thismeans that we have implicitly used differentiability of the function |๐ง |๐/๐ at 0 โ R๐. Also it should be noted thatfor the one-variable case, functions as in (5.1), which are in general not in ๐ถ1, are one-harmonic in R.
5.2 ๐ถ1-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 that๐๐ข(๐ฅ0) = {0} for all ๐ฅ0 โ ๐น. Let ๐ฅ0 โ ๐น. We get a convex function ๐ข0 : R๐ โ R as a blow-up limit as in Proposition1. We note that the facet of ๐ข0 is non-empty by Proposition 1. Hence by the Liouville-type theorem (Theorem 3),๐ข0 is constant and we obtain ๐๐ข0 (๐ฅ0) = {0}. Combining these results, we have {0} โ ๐๐ข(๐ฅ0) โ ๐๐ข0 (๐ฅ0) = {0} andtherefore ๐๐ข(๐ฅ0) = {0}. This completes the proof. โก
6 GeneralizationIn Section 6, we would like to discuss ๐ถ1-regularity of convex weak solutions to
๐ฟ๐ข B โdiv(โ๐งฮจ(โ๐ข)) โdiv(โ๐ง๐ (โ๐ข)) = ๐ in ฮฉ โ R๐, (6.1)
which covers (1.1). Precisely speaking, throughout Section 6, we make these following assumptions for ฮจ and ๐on regularity and ellipticity. For regularity, we only require
ฮจ โ ๐ถ (R๐) โฉ๐ถ2 (R๐ \ {0}), ๐ โ ๐ถ1 (R๐) โฉ๐ถ2 (R๐ \ {0}). (6.2)
For๐ , we assume that for each fixed 0 < ๐ โค ๐ <โ, there exist constants 0 < ๐พ < ฮ <โ such that๐ satisfies
๐พ |๐ |2 โคโจโ2๐ง๐ (๐ง0)๐
๏ฟฝ๏ฟฝ ๐โฉ, (6.3)๏ฟฝ๏ฟฝโจโ2๐ง๐ (๐ง0)๐
๏ฟฝ๏ฟฝ ๐โฉ๏ฟฝ๏ฟฝ โค ฮ |๐ | |๐ | (6.4)
for all ๐ง0, ๐ , ๐ โ R๐ with ๐ โค |๐ง0 | โค ๐ . Also, there is no loss of generality in assuming that
โ๐ง๐ (0) = 0. (6.5)
Finally, we assume that ฮจ is positively homogeneous of degree 1. In other words, ฮจ satisfies
ฮจ(๐๐ง0) = ๐ฮจ(๐ง0) (6.6)
holds for all ๐ง0 โ R๐ and ๐ > 0. This clearly yields ฮจ(0) = 0.By modifying some of our arguments, we are able to show that
Theorem 4 (๐ถ1-regularity theorem for general equations). Let ฮฉ โ R๐ be a domain. Assume that ๐ โ ๐ฟ๐loc (ฮฉ) (๐ <๐ โค โ) and the functionals ฮจ and๐ satisfy (6.2)โ(6.5). If ๐ข is a convex weak solution to (6.1), then ๐ข is in ๐ถ1 (ฮฉ).
If we setฮจ(๐ง) B ๐ |๐ง |, ๐ (๐ง) B |๐ง |๐
๐, where 1 < ๐ <โ,
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 , which are derived from the assumptions (6.2)โ(6.5).
For ๐ , by (6.2)โ(6.3) and (6.5) it is easy to check that the continuous mapping ๐ด : R๐ โ ๐ง โฆโ โ๐ (๐ง) โ R๐satisfies strict monotonicity (4.8). In particular, by (6.5) we have
โจ๐ด(๐ง) | ๐งโฉ > 0 for all ๐ง โ 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(๐ง1 + ๐ง2) โค ฮจ(๐ง1) +ฮจ(๐ง2) for all ๐ง1, ๐ง2 โ R๐. (6.8)
We define a function ฮจฬ : R๐ โ [0,โ] by
ฮจฬ(๐) B sup{โจ๐ | ๐งโฉ | ๐ง โ R๐, ฮจ(๐ง) โค 1}.
ฮจฬ is the support function for the closed convex set๐ถฮจ B {๐ง โ R๐ | ฮจ(๐ง) โค 1}. By definition it is easy to check that ฮจฬis convex and lower semicontinuous. Also, if ๐ โ R๐ satisfies ฮจฬ(๐) <โ, then the following CauchyโSchwarz-typeinequality holds;
โจ๐ง | ๐โฉ โค ฮจ(๐ง)ฮจฬ(๐) for all ๐ง โ R๐. (6.9)
If a convex function ฮจ is positively homogeneous of degree 1, then the subdifferential operator ๐ฮจ is explicitlygiven by
๐ฮจ(๐ง) ={๐ โ R๐
๏ฟฝ๏ฟฝ ฮจฬ(๐) โค 1, ฮจ(๐ง) = โจ๐ง | ๐โฉ}
(6.10)
for all ๐ง โ R๐. In particular, we have the following formula
โจโ๐งฮจ(๐ง0) | ๐ง0โฉ = ฮจ(๐ง0) for all ๐ง0 โ R๐ \ {0}, (6.11)
which is often called Eulerโs identity. Also, assumptions (6.2) and (6.6) imply that
โฮจ(๐๐ง0) = โฮจ(๐ง0), โ2ฮจ(๐๐ง0) = ๐โ1โ2ฮจ(๐ง0) (6.12)
for all ๐ > 0 and ๐ง0 โ R๐ \ {0}. Proofs of (6.8)โ(6.10) are given in Lemma 11 of the appendices for the readerโsconvenience.
Remark 5. The results (6.11)โ(6.12) give us the following basic property for ฮจ.
1. We set a constant๐พ B sup{|โ๐งฮจ(๐ง0) | | ๐ง0 โ R๐, |๐ง0 | = 1},
which is finite. Then we have ๐ฮจ(๐ง0) โ ๐ต๐พ (0) for all ๐ง0 โ R๐. For the case ๐ง0 โ 0, this inclusion is clearby (6.12) and ๐ฮจ(๐ง0) = {โ๐งฮจ(๐ง0)}. For ๐ง0 = 0, we take arbitrary ๐ค โ ๐ฮจ(0) \ {0}. Then by the subgradientinequality, Eulerโs identity (6.11) and the CauchyโSchwarz inequality, we have
|๐ค |2 = โจ๐ค | ๐คโ0โฉ +ฮจ(0)โค ฮจ(๐ค) = โจโ๐งฮจ(๐ค) | ๐คโฉ โค ๐พ |๐ค |.
This estimate yields the inclusion ๐ฮจ(0) โ ๐ต๐พ (0).
2. For ๐ง0 โ R๐ \ {0}, the Hessian matrix โ2๐งฮจ(๐ง0) satisfies
0 โคโจโ2๐งฮจ(๐ง0)๐
๏ฟฝ๏ฟฝ ๐โฉ, (6.13)๏ฟฝ๏ฟฝโจโ2๐งฮจ(๐ง0)๐
๏ฟฝ๏ฟฝ ๐โฉ๏ฟฝ๏ฟฝ โค ๐ถ
|๐ง0 ||๐ | |๐ | (6.14)
for all ๐, ๐ โ R๐. Here the finite constant ๐ถ is explicitly given by
๐ถ B sup{๏ฟฝ๏ฟฝโจโ2
๐งฮจ(๐ค)๐๏ฟฝ๏ฟฝ ๐โฉ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ ๐ง, ๐ , ๐ โ R๐, |๐ค | = |๐ | = |๐ | = 1
}.
Lemma 4 states lower semicontinuity of a functional in the weakโ topology of an ๐ฟโ-space. This result is usedin the justification of a blow-up argument for the equation (6.1).
Lemma 4. Let ฮฉ โ R๐ be a Lebesgue measurable set, and let ฮจ : R๐ โ [0,โ) be a convex function whichsatisfies (6.6). Assume that a vector field ๐ โ ๐ฟโ (ฮฉ, R๐) and a sequence {๐๐ }๐ โ ๐ฟโ (ฮฉ, R๐) satisfy ๐๐
โโ ๐ in
๐ฟโ (ฮฉ, R๐). Then we haveess sup๐ฅโฮฉ
ฮจฬ(๐ (๐ฅ)) โค liminf๐โโ
ess sup๐ฅโฮฉ
ฮจฬ(๐๐ (๐ฅ)), (6.15)
where ฮจฬ denotes the support function of the closed convex set ๐ถฮจ B {๐ง โ R๐ | ฮจ(๐ง) โค 1}.
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We give an elementary proof of Lemma 4, which is based on a definition of ฮจฬ.
Proof. We consider the case ๐ถโ B liminf๐โโ
ฮจฬ(๐๐ ) ๐ฟโ (ฮฉ) <โ, since otherwise (6.15) is clear. Fix arbitrary ๐ > 0.
Then we may take a subsequence {๐๐ ๐ }โ๐=1 such that
ess sup๐ฅโฮฉ
ฮจฬ(๐๐ ๐ (๐ฅ)) โค ๐ถโ + ๐ <โ. (6.16)
Take arbitrary 0 โค ๐ โ ๐ฟ1 (ฮฉ) and ๐ค โ ๐ถฮจ. Then with the aid of (6.9), we have
โจ๐๐ ๐ (๐ฅ) | ๐คโฉ โค ๐ถโ + ๐
for all ๐ โ N and for a.e. ๐ฅ โ ฮฉ, which yieldsโซฮฉ
[๐ถโ + ๐โ โจ๐๐ ๐ (๐ฅ) | ๐คโฉ
]๐(๐ฅ) ๐๐ฅ โฅ 0 (6.17)
for all ๐ โ N. Letting ๐ โโ, we have โซฮฉ[๐ถโ + ๐โ โจ๐ (๐ฅ) | ๐คโฉ]๐(๐ฅ) ๐๐ฅ โฅ 0
by ๐๐ ๐
โโ ๐ in ๐ฟโ (ฮฉ, R๐). Since 0 โค ๐ โ ๐ฟ1 (ฮฉ) is arbitrary, for each ๐ค โ ๐ถฮจ, there exists an L๐-measurable set
๐๐ค โ ฮฉ, such that L๐ (๐๐ค) = 0 and
โจ๐ (๐ฅ) | ๐คโฉ โค ๐ถโ + ๐ for all ๐ฅ โ ฮฉ \๐๐ค .
Here we denote L๐ by the ๐-dimensional Lebesgue measure. Since ๐ถฮจ โ R๐ is separable, we may take a countableand dense set ๐ท โ ๐ถ๐ . We set an L๐-measurable set
๐ Bโ๐คโ๐ท
๐๐ค โ ฮฉ,
which clearly satisfies L๐ (๐) = 0. Then we conclude that
โจ๐ (๐ฅ) | ๐คโฉ โค ๐ถโ + ๐ for all ๐ฅ โ ฮฉ \๐, ๐ค โ ๐ถฮจ
from density of ๐ท โ ๐ถฮจ. Hence by definition of ฮจฬ, it is clear that
ฮจฬ(๐ (๐ฅ)) โค ๐ถโ + ๐ for a.e. ๐ฅ โ ฮฉ.
Since ๐ > 0 is arbitrary, this completes the proof of (6.15). โก
6.2 Sketches of the proofsWe first give definitions of weak solutions to (6.1). We also define weak subsolutions, and supersolutions to anequation ๐ฟ๐ข = 0 in a bounded domain.
Definition 3. Let ฮฉ โ R๐ be a domain.
1. Let ๐ โ ๐ฟ๐loc (ฮฉ) (๐ < ๐ โค โ). We say that a function ๐ข โ ๐1,โloc (ฮฉ) is a weak solution to (6.1), when
for any bounded Lipschitz domain ๐ โ ฮฉ, there exists a vector field ๐ โ ๐ฟโ (๐, R๐) such that the pair(๐ข, ๐) โ๐1,โ (๐) ร ๐ฟโ (๐, R๐) satisfiesโซ
๐โจ๐ | โ๐โฉ ๐๐ฅ +
โซ๐โจ๐ด(โ๐ข) | โ๐โฉ ๐๐ฅ =
โซ๐๐ ๐ ๐๐ฅ (6.18)
for all ๐ โ๐1, 10 (๐), and
๐ (๐ฅ) โ ๐ฮจ(โ๐ข(๐ฅ)) (6.19)
for a.e. ๐ฅ โ ๐. Here ๐ด denotes the continuous mapping ๐ด : R๐ โ ๐ฅ โฆโ โ๐ง๐ (๐ฅ) โ R๐. For such pair (๐ข, ๐),we say that (๐ข, ๐) satisfies ๐ฟ๐ข = ๐ in๐โ1,โ (๐) or simply say that ๐ข satisfies ๐ฟ๐ข = ๐ in๐โ1,โ (๐).
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2. Assume that ฮฉ is bounded. A pair (๐ข, ๐) โ๐1,โ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) is called a weak subsolution to ๐ฟ๐ข = 0 inฮฉ, if it satisfies โซ
ฮฉโจ๐ | โ๐โฉ ๐๐ฅ +
โซฮฉโจ๐ด(โ๐ข) | โ๐โฉ ๐๐ฅ โค 0 (6.20)
for all 0 โค ๐ โ ๐ถโ๐ (ฮฉ), and
๐ (๐ฅ) โ ๐ฮจ(โ๐ข(๐ฅ)) for a.e. ๐ฅ โ ฮฉ. (6.21)
Similarly we call a pair (๐ข, ๐) โ๐1, ๐ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) a weak supersolution ๐ฟ๐, ๐๐ข = 0 in ฮฉ, if it satisfies(6.21) and โซ
ฮฉโจ๐ | โ๐โฉ ๐๐ฅ +
โซฮฉโจ๐ด(โ๐ข) | โ๐โฉ ๐๐ฅ โฅ 0
for all 0 โค ๐ โ๐ถโ๐ (ฮฉ). For ๐ข โ๐1, ๐ (ฮฉ), we simply say that ๐ข is respectively a subsolution and a supersolution
to ๐ฟ๐ข = 0 in the weak sense if there is ๐ โ ๐ฟโ (ฮฉ, R๐) such that the pair (๐ข, ๐) is a weak subsolution and aweak supersolution to ๐ฟ๐ข = 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 ๐1,โ-regularity for solutions in Definition 3. Also it should be notedthat if a vector field ๐ satisfies (6.19), then ๐ is in ๐ฟโ by Remark 5. Hence our regularity assumptions of thepair (๐ข, ๐) involve no loss of generality.
2. Integrals in (6.18) make sense by ๐, โ๐ข โ ๐ฟโ (๐, R๐), ๐ด โ ๐ถ (R๐, R๐), and the continuous embedding๐1, 1
0 (๐) โฉโ ๐ฟ๐โฒ (๐).
3. For a bounded domain ฮฉ โ R๐, let ๐ข โ ๐ถ2 (ฮฉ) satisfy
โ๐ข(๐ฅ) โ 0 for all ๐ฅ โ ฮฉ, and
๐ฟ๐ข(๐ฅ) โค 0 for all ๐ฅ โ ฮฉ.
Then the pair (๐ข, โ๐งฮจ(โ๐ข)) โ๐1, ๐ (ฮฉ) ร ๐ฟโ (ฮฉ, R๐) satisfies (6.20)โ(6.21). For such ๐ข, we simply say that๐ข satisfies ๐ฟ๐ข โค 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 ๐ข to (6.1) satisfies (1.4) for all ๐ฅ โ ฮฉ.
For the case ๐ฅ โ ๐ท, we can show (1.4) by De GiorgiโNashโMoser theory. This is basically due to the fact thatthe functional
๐ธ (๐ง) B ฮจ(๐ง) +๐ (๐ง) for ๐ง โ R๐
satisfy the following property. For each fixed constants 0 < ๐ โค ๐ <โ, there exists constants 0 < ๐ โค ฮ <โ suchthat the estimates (2.7)โ(2.8) hold for all ๐ง0, ๐ , ๐ โ R๐ with ๐ โค |๐ง0 | โค ๐ . In other words, the operator ๐ฟ is locallyuniformly elliptic outside a facet, in the sense that for a function ๐ฃ the operator ๐ฟ๐ฃ becomes uniformly elliptic in aplace where 0 < ๐ โค |โ๐ฃ | โค ๐ <โ 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 ๐ฟ outside the facet and De GiorgiโNashโMoser theory, weare able to show that a convex solution to ๐ฟ๐ข = ๐ is ๐ถ1, ๐ผ near a neighborhood of each fixed point ๐ฅ โ ๐ท, similarlyto the proof of Lemma 1.
For the case ๐ฅ โ ๐น, we first make a blow-argument to construct a convex function ๐ข0 : R๐ โ R satisfying๐๐ข(๐ฅ) โ ๐๐ข0 (๐ฅ), and ๐ฟ๐ข0 = 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 ๐ท. 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 ๐ข0, which is constructed by the previous blow-argument, should be constant. Finally the inclusions{0} โ ๐๐ข(๐ฅ) โ ๐๐ข0 (๐ฅ) โ {0} hold, and this completes the proof of (1.4), i.e., ๐๐ข(๐ฅ) = {0}.
For maximum principles on the equation ๐ฟ๐ข = 0, the proofs are almost similar to those in Section 4. Indeed,we first recall that the operator ๐ด : R๐ โ ๐ง0 โฆโ โ๐ง๐ (๐ง0) โ R๐ satisfies strict monotonicity (4.8). Combining withmonotonicity of the subdifferential operator ๐ฮจ, we can easily prove a comparison principle as in Proposition 2.Also, similarly to Lemma 2, we can construct classical barrier subsolutions to ๐ฟ๐ข = 0 in an open annulus, since
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the operator ๐ฟ 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 โAscoli theorem and a diagonal argument. By proving Lemma 5below, we are able to demonstrate that ๐ข0, a limit of rescaled solutions, is a weak solution to ๐ฟ๐ข = 0 in R๐, and thisfinishes our blow-up argument.
Lemma 5. Let ๐ โ R๐ be a bounded domain. Assume that sequences of functions {๐ข๐ }โ๐=1 โ ๐1,โ (๐) and{ ๐๐ }โ๐=1 โ ๐ฟ๐ (๐) (๐ < ๐ โค โ) satisfy all of the following.
1. For each ๐ โ N, ๐ข๐ satisfies ๐ฟ๐ข๐ = ๐๐ in๐โ1,โ (๐).
2. There exists a constant ๐ > 0, independent of ๐ โ N, such that
|โ๐ข๐ (๐ฅ) | โค ๐ for a.e. ๐ฅ โ๐. (6.22)
3. There exists a function ๐ข โ๐1,โ (๐) such that
โ๐ข๐ (๐ฅ) โ โ๐ข(๐ฅ) for a.e. ๐ฅ โ๐. (6.23)
4. ๐๐ strongly converges to 0 in ๐ฟ๐ (๐).
Then ๐ข satisfies ๐ฟ๐ข = 0 in๐โ1,โ (๐).
Proof. For each ๐ โ N, there exists a vector field ๐๐ โ ๐ฟโ (๐, R๐) such that
๐๐ (๐ฅ) โ ๐ฮจ(โ๐ข(๐ฅ)) for a.e. ๐ฅ โ๐, (6.24)โซ๐โจ๐๐ | โ๐โฉ ๐๐ฅ +
โซ๐โจ๐ด(โ๐ข๐ ) | โ๐โฉ ๐๐ฅ =
โซ๐๐๐๐๐๐ฅ for all ๐ โ๐1, 1
0 (๐). (6.25)
Combining the assumption ๐๐ โ ๐ in ๐ฟ๐ (๐) with the continuous embedding ๐ฟ๐ (๐) โฉโ๐โ1,โ (๐), we get
๐๐ โ 0 in๐โ1,โ (๐). (6.26)
By ๐ด โ ๐ถ (R๐, R๐) and (6.22), the vector fields {๐ด(โ๐ข๐ )}โ๐=1 satisfy
๐ด(โ๐ข๐ (๐ฅ)) โ ๐ด(โ๐ข(๐ฅ)) for a.e. ๐ฅ โ๐,
|๐ด๐ (โ๐ข๐ (๐ฅ)) โ ๐ด(โ๐ข(๐ฅ)) | โค ๐ถ for a.e. ๐ฅ โ๐,
where ๐ถ is independent of ๐ โ N. From these and Lebesgueโs dominated convergence theorem, it follows that
๐ด(โ๐ข๐ )โโ ๐ด(โ๐ข) in ๐ฟโ (๐, R๐). (6.27)
As mentioned in Remark 5โ6, the {๐๐ }โ๐=1 โ ๐ฟโ (๐, R๐) is bounded. Hence by [5, Corollary 3.30], we may takea subsequence {๐๐ ๐ }โ๐=1 so that
๐๐ ๐
โโ ๐ in ๐ฟโ (๐, R๐) (6.28)
for some ๐ โ ๐ฟโ (๐, R๐). By (6.25)โ(6.28) we obtainโซ๐โจ๐ | ๐โฉ ๐๐ฅ +
โซ๐โจ๐ด(โ๐ข) | โ๐โฉ ๐๐ฅ = 0 for all ๐ โ๐1, 1
0 (๐).
Now we are left to prove that๐ (๐ฅ) โ ๐ฮจ(โ๐ข(๐ฅ)) for a.e. ๐ฅ โ๐.
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By (6.10), it suffices to show that ๐ satisfiesฮจฬ(๐ (๐ฅ)) โค 1, (6.29)
ฮจ(โ๐ข(๐ฅ)) = โจ๐ | โ๐ข(๐ฅ)โฉ (6.30)
for a.e. ๐ฅ โ๐. Similarly, it follows that for each ๐ โ N, the vector field ๐๐ satisfies{ฮจฬ(๐๐ (๐ฅ)) โค 1,
ฮจ(โ๐ข๐ (๐ฅ)) = โจ๐๐ | โ๐ข(๐ฅ)โฉ, for a.e. ๐ฅ โ๐.
Hence (6.29) is an easy consequence of Lemma 4. We recall (6.2), and thus ๐ฮจ(๐ง0) = {โ๐งฮจ(๐ง0)} holds for all๐ง0 โ R๐ \ {0}. Combining (6.23), we can check that ๐๐ (๐ฅ) โ ๐ (๐ฅ) for a.e. ๐ฅ โ ๐ท B {๐ฅ โ๐ | โ๐ข(๐ฅ) โ 0}. Hence(6.30) holds for a.e. ๐ฅ โ ๐ท. Note that (6.30) is clear for ๐ฅ โ๐ \๐ท, and this completes the proof. โก
We prove a Liouville-type theorem as in Theorem 3. In other words, for a convex solution to ๐ฟ๐ข = 0 in R๐,we show that ๐น, the facet of ๐ข, would satisfy either ๐น = โ or ๐น = R๐. Assume by contradiction that ๐น satisfiesโ โ ๐น โ R๐. Then by Lemma 3, we may write a convex solution ๐ข by either of (5.1)โ(5.3). However, Lemma 6below states that ๐ข is no longer a weak solution, and this completes our proof.
Lemma 6. Let ๐ข be a piecewise-linear function defined as in either of (5.1)โ(5.3). Then ๐ข is not a weak solutionto ๐ฟ๐ข = 0 in R๐.
Proof. As in the proof of Theorem 3, we introduce a constant ๐ > 0, and set open cubes๐ โฒ โ R๐โ1 and๐, ๐๐ , ๐๐ โR๐. By choosing sufficiently small ๐ > 0, we show that ๐ข does not satisfy ๐ฟ๐ข = 0 in ๐โ1,โ (๐). Assume bycontradiction that there exists a vector field ๐ โ ๐ฟโ (๐, R๐) such that the pair (๐ข, ๐) satisfies ๐ฟ๐ข = 0 in๐โ1,โ (๐).
We first show that a function ๐ข defined as in (5.1) is not a weak solution. For this case, (6.12) implies that ๐satisfies ๐ (๐ฅ) = โ๐งฮจ(๐1) for a.e. ๐ฅ โ ๐๐ . We take and fix non-negative functions ๐1 โ ๐ถ1
๐ ((โ๐, ๐)), ๐2 โ ๐ถ1๐ (๐ โฒ)
such that (5.5) holds, and define ๐ โ ๐ถ1๐ (๐) by ๐(๐ฅ1, ๐ฅ
โฒ) B ๐1 (๐ฅ1)๐2 (๐ฅ โฒ) for (๐ฅ1, ๐ฅโฒ) โ (โ๐, ๐) ร๐ โฒ = ๐. Testing
๐ into ๐ฟ๐ข = 0 in๐โ1,โ (๐), we have
0 =โซ๐๐
โจ๐ + ๐ด(0) | โ(๐1๐2)โฉ ๐๐ฅ +โซ๐๐
โจโ๐งฮจ(๐1) + ๐ด(๐ก1๐1) | โ(๐1๐2)โฉ ๐๐ฅ
โคโซ๐๐
ฮจ(โ(๐1๐2))ฮจฬ(๐ (๐ฅ)) ๐๐ฅ
+๐1 (0)โซ๐โฒ๐2 (๐ฅ โฒ)โจโ๐งฮจ(๐1) | โ๐1โฉ ๐๐ฅ โฒ+๐1 (0)
โซ๐โฒ๐2 (๐ฅ โฒ)โจ๐ด(๐ก1๐1) | โ๐1โฉ ๐๐ฅ โฒ
C ๐ผ1 + ๐ผ2 + ๐ผ3.
Here we have used the CauchyโSchwarz-type inequality (6.5) for the integral over๐๐ , and applied the GaussโGreentheorem to the integration over ๐๐ . For ๐ผ1, we make use of (6.9)โ(6.8), Fubiniโs theorem and (5.5). Then we have
๐ผ1 โคโซ๐๐
๐1 (๐ฅ1)ฮจ(0, โ๐ฅโฒ๐2 (๐ฅ โฒ)) ๐๐ฅ +โซ๐๐
๐โฒ1 (๐ฅ1)๐2 (๐ฅ โฒ)ฮจ(๐1) ๐๐ฅ
โค ๐1 (0)(๐ ยท โฅฮจ(0, โ๐ฅโฒ๐2)โฅ๐ฟ1 (๐โฒ) +ฮจ(๐1)โฅ๐2โฅ๐ฟ1 (๐โฒ)
),
where โ๐ฅโฒ๐2 B (๐๐ฅ2๐2, . . . , ๐๐ฅ๐๐2). For ๐ผ2, recalling Eulerโs identity (6.11), we get ๐ผ2 = โ๐1 (0)ฮจ(๐1)โฅ๐2โฅ๐ฟ1 (๐โฒ) .We set a constant ๐ B โจ๐ด(๐ก1๐1) | ๐1โฉ, which is positive by (6.7). Then we obtain
๐ผ1 + ๐ผ2 + ๐ผ3 โค ๐1 (0)(๐ ยท โฅฮจ(0, โ๐ฅโฒ๐2)โฅ๐ฟ1 (๐โฒ) โ ๐โฅ๐2โฅ๐ฟ1 (๐โฒ)
).
Choosing ๐ = ๐ (๐, ฮจ, ๐2) > 0 sufficiently small, we have 0 โค ๐ผ1 + ๐ผ2 + ๐ผ3 < 0, which is a contradiction. Similarlywe can deduce that ๐ข defined as in (5.3) does not satisfy ๐ฟ๐ข = 0 in ๐โ1,โ (๐), since it suffices to restrict ๐ < ๐0.For the remaining case (5.2), we have already known that
๐ (๐ฅ) ={
โ๐งฮจ(๐1) for a.e. ๐ฅ โ ๐๐ ,โ๐งฮจ(โ๐1) for a.e. ๐ฅ โ ๐๐
23
by definition of ๐ and (6.12). We set two constants ๐1 B โจ๐ด(๐ก1๐1) | ๐1โฉ, ๐2 B โจ๐ด(โ๐ก2๐1) | โ๐1โฉ, both of which arepositive by (6.7). Testing the same function ๐ โ ๐ถ1
๐ (๐) into ๐ฟ๐ข = 0 in๐โ1,โ (๐), we obtain
0 =โซ๐๐
โจโ๐งฮจ(โ๐1) + ๐ด(โ๐ก2๐1) | โ(๐1๐2)โฉ ๐๐ฅ +โซ๐๐
โจโ๐งฮจ(๐1) + ๐ด(๐ก1๐1) | โ(๐1๐2)โฉ ๐๐ฅ
=โซ๐โฒ๐1 (0)๐2 (๐ฅ โฒ)โจโ๐งฮจ(โ๐1) + ๐ด(โ๐ก2๐1) | ๐1โฉ ๐๐ฅ โฒ+
โซ๐โฒ๐1 (0)๐2 (๐ฅ โฒ)โจโ๐งฮจ(๐1) + ๐ด(๐ก1๐1) | โ๐1โฉ ๐๐ฅ โฒ
= โ๐1 (0) (ฮจ(๐1) +ฮจ(โ๐1) + ๐1 + ๐2)โซ๐โฒ๐2 (๐ฅ โฒ) ๐๐ฅ โฒ < 0,
which is a contradiction. Here we have used the GaussโGreen theorem and Eulerโs identity (6.11). This completesthe proof. โก
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 in๐1, ๐
0 (1 โค๐ <โ). 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 space๐1, ๐
0 [10, Proposition 3.10].
Lemma 7. Let ฮฉ โ R๐ be a bounded open set and 1 โค ๐ <โ. For all ๐ข โ๐1, ๐0 (ฮฉ), ๐ โ {1, . . . , ๐ }, โ โ R \ {0},
we haveโฅฮ ๐ , โ๐ขโฅ๐ฟ๐ (ฮฉ) โค โฅโ๐ขโฅ๐ฟ๐ (ฮฉ) . (A.1)
Here ฮ ๐ , โ๐ข is defined by
ฮ ๐ , โ๐ข(๐ฅ) B๐ข(๐ฅ + โ๐ ๐ ) โ๐ข(๐ฅ)
โfor ๐ฅ โ ฮฉ.
Before the proof of Lemma 7, we note that ฮ ๐ , โ๐ข(๐ฅ) makes sense for a.e. ๐ฅ โ ฮฉ by the zero extension of๐ข โ๐1, ๐
0 (๐). That is, for a given ๐ข โ๐1, ๐0 (๐), we set ๐ข โ๐1, ๐ (R๐) by
๐ข(๐ฅ) B{๐ข(๐ฅ) ๐ฅ โ๐,
0 ๐ฅ โ R๐ \๐. (A.2)
Proof. We fix ๐ โ {1, . . . , ๐ }, โ โ R \ {0}. We first note that the operator ฮ ๐ , โ : ๐1, ๐0 (๐) โ ๐ฟ ๐ (๐) is bounded,
since for all ๐ข โ๐1, ๐0 (๐) we have
โฅฮ ๐ , โ๐ขโฅ๐ฟ๐ (๐) โค1|โ|
[(โซ๐|๐ข(๐ฅ + โ) |๐ ๐๐ฅ
)1/๐+(โซ๐|๐ข(๐ฅ) |๐ ๐๐ฅ
)1/๐]โค 2
|โ| โฅ๐ขโฅ๐ฟ๐ (๐) โค
๐ถ (๐, ๐)|โ| โฅโ๐ขโฅ๐ฟ๐ (๐)
by the Minkowski inequality and the Poincarรฉ inequality. Here ๐ข โ๐1, ๐ (R๐) is defined as in (A.2). Hence by adensity argument, it suffices to check that (A.1) holds true for all ๐ข โ ๐ถโ
๐ (๐). Let ๐ข โ ๐ถโ๐ (๐). Then for all ๐ฅ โ๐,
24
we have
|๐ข(๐ฅ + โ๐ ๐ ) โ๐ข(๐ฅ) | =๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโซ 1
0
โจโ๐ข(๐ฅ + ๐กโ๐ ๐ )
๏ฟฝ๏ฟฝ โ๐ ๐ โฉ ๐๐ก๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโค |โ|
โซ 1
0|โ๐ข(๐ฅ + ๐กโ๐ ๐ ) | ๐๐ก โค |โ|
(โซ 1
0|โ๐ข(๐ฅ + ๐กโ๐ ๐ ) |๐ ๐๐ก
)1/๐
by the Cauchy-Schwarz inequality and Hรถlderโs inequality. From this estimate we get
โฅฮ ๐ , โ๐ขโฅ ๐๐ฟ๐ (๐) โคโซฮฉ
โซ 1
0|โ๐ข(๐ฅ + ๐กโ๐ ๐ ) |๐ ๐๐ก ๐๐ฅ
=โซ 1
0
โซ๐|โ๐ข(๐ฅ + ๐กโ๐ ๐ ) |๐ ๐๐ฅ๏ธธ ๏ธท๏ธท ๏ธธ
โคโฅโ๐ข โฅ๐๐ฟ๐ (๐)
๐๐ก (by Fubiniโs theorem)
โค โฅโ๐ขโฅ ๐๐ฟ๐ (๐) .
Hence we obtain (A.1) for all ๐ข โ ๐ถโ๐ (๐), and this completes the proof. โก
A.2 Convex analysisLemma 8 is used in the proof of Lemma 1 for a justification of local ๐2, 2-regularity of a convex weak solutionoutside of the facet.
Lemma 8. Let ๐ข be a real-valued convex function in a convex domain ฮฉ โ R๐. Assume that ๐ฅ1, ๐ฅ2 โ ฮฉ satisfy๐ฅ1 โ ๐ฅ2, and set ๐ B |๐ฅ2 โ ๐ฅ1 | > 0, ๐ B ๐โ1 (๐ฅ2 โ ๐ฅ1). Then for all ๐ง2 โ ๐๐ข(๐ฅ2), we have
โจ๐ง2 | ๐โฉ โฅ ๐ข(๐ฅ2) โ๐ข(๐ฅ1)๐
. (A.3)
Proof. By ๐ง2 โ ๐๐ข(๐ฅ2), we have a subgradient inequality
๐ข(๐ฅ) โฅ ๐ข(๐ฅ2) + โจ๐ง2 | ๐ฅโ ๐ฅ2โฉ
for all ๐ฅ โ ฮฉ. Substituting ๐ฅ B ๐ฅ1 = ๐ฅ2 โ ๐๐ โ ฮฉ, we obtain
๐ข(๐ฅ1) โฅ ๐ข(๐ฅ2) โ ๐โจ๐ง2 | ๐โฉ,
which yields (A.3). โก
Remark 7. Instead of subgradient inequalities, we are able to show (A.3) by monotonicity of ๐๐ข. For each fixed๐ฅ1, ๐ฅ2 โ ฮฉ with ๐ฅ1 โ ๐ฅ2, we may take and fix ๐ฅ3 B ๐ฅ1 + ๐ก (๐ฅ2 โ ๐ฅ1) for some 0 < ๐ก < 1 and ๐ง3 โ ๐๐ข(๐ฅ3) such that
๐ข(๐ฅ2) โ๐ข(๐ฅ1) = โจ๐ง3 | ๐ฅ2 โ ๐ฅ1โฉ, (A.4)
with the aid of the mean value theorem for non-smooth convex functions [1, Theorem D.6]. ๐ฅ2 โ ๐ฅ1 = ๐๐ is clearby definitions of ๐, ๐. Noting ๐ฅ2 โ ๐ฅ3 = (1โ ๐ก)๐๐, we can check that
โจ๐ง2 โ ๐ง3 | ๐โฉ = 1(1โ ๐ก)๐ โจ๐ง2 โ ๐ง3 | ๐ฅ2 โ ๐ฅ3โฉ โฅ 0
by monotonicity of ๐๐ข. Combining these results with (A.4), we obtain
๐ข(๐ฅ2) โ๐ข(๐ฅ1) = ๐โจ๐ง3 | ๐โฉ โค ๐โจ๐ง2 | ๐โฉ,
which yields (A.3).
The following lemma is used in the proof of Proposition 1.
25
Lemma 9. Let๐ โ R๐ be a convex open set, and let {๐ข๐ }โ๐=1 be a sequence of real-valued convex functions in๐.Assume that this sequence is uniformly Lipschitz. In other words, there is a constant ๐ฟ > 0 independent of ๐ โ Nsuch that
|๐ข๐ (๐ฅ) โ๐ข๐ (๐ฆ) | โค ๐ฟ |๐ฅโ ๐ฆ | for all ๐ฅ, ๐ฆ โ๐. (A.5)
If there exists a function ๐ขโ : ๐โ R such that
๐ข๐ (๐ฅ) โ ๐ขโ (๐ฅ) for all ๐ฅ โ๐, (A.6)
then we have โ๐ข๐ (๐ฅ) โ โ๐ขโ (๐ฅ) for a.e. ๐ฅ โ๐.
Remark 8. From (A.5)โ(A.6), it is easy to show that ๐ขโ is also convex, ๐ข๐ โ ๐ขโ uniformly in๐, and
|๐ขโ (๐ฅ) โ๐ขโ (๐ฆ) | โค ๐ฟ |๐ฅโ ๐ฆ | for all ๐ฅ, ๐ฆ โ๐.
Our proof of Lemma 9 is inspired by [11, Lemma A.3].
Proof. We define L๐-measurable sets
๐๐ B {๐ฅ โ๐ | ๐ข๐ is not differentiable at ๐ฅ} for ๐ โ Nโช {โ}.
Clearly ๐๐ (๐ โ Nโช {โ}) satisfies L๐ (๐๐ ) = 0 by Lipschitz continuity of ๐ข๐ , and therefore the L๐-measurableset
๐ Bโ
๐ โNโช{โ}๐๐ โ ๐
also satisfies L๐ (๐) = 0. We claim that
โ๐ข๐ (๐ฅ0) โ โ๐ขโ (๐ฅ0) for all ๐ฅ0 โ๐ \๐. (A.7)
We take and fix arbitrary ๐ฅ0 โ๐ \๐. We note that โ๐ข๐ (๐ฅ0) exists for each ๐ โ N since ๐ฅ0 โ ๐๐ , and we obtain
sup๐ โN
|โ๐ข๐ (๐ฅ0) | โค ๐ฟ
with the aid of (A.5). Hence it suffices to check that, if a subsequence {๐ข๐๐ }๐ โ {๐ข๐ }๐ satisfies
โ๐ข๐๐ (๐ฅ0) โ ๐ฃ (๐ โโ) for some ๐ฃ โ R๐ , (A.8)
then ๐ฃ = โ๐ขโ (๐ฅ0). Since ๐ฅ0 โ ๐๐๐ and therefore ๐๐ข๐๐ (๐ฅ0) = {โ๐ข๐๐ (๐ฅ0)} for each ๐ โ N, we easily get
๐ข๐๐ (๐ฅ) โฅ ๐ข๐๐ (๐ฅ0) + โจโ๐ข๐๐ (๐ฅ0) | ๐ฅโ ๐ฅ0โฉ for all ๐ฅ โ๐, ๐ โ N.
Letting ๐ โโ, we have๐ขโ (๐ฅ) โฅ ๐ขโ (๐ฅ0) + โจ๐ฃ | ๐ฅโ ๐ฅ0โฉ for all ๐ฅ โ๐
by (A.6) and (A.8). This means that ๐ฃ โ ๐๐ขโ (๐ฅ0). Note again that ๐ฅ0 โ ๐โ and therefore ๐๐ขโ (๐ฅ0) = {โ๐ขโ (๐ฅ0)},which yields ๐ฃ = โ๐ขโ (๐ฅ0). This completes the proof of (A.7). โก
A.3 Convex functionalsWe prove some basic property of convex functionals ฮจ and๐ in Section 6.
Lemma 10. Let ๐ be a convex function which satisfies (6.2)-(6.3) and (6.5). Then the mapping ๐ด : R๐ โ ๐ง โฆโโ๐ (๐ง) โ R๐ satisfies strict monotonicity (4.8).
Proof. We take arbitrary ๐ง1, ๐ง2 โ R๐ with ๐ง1 โ ๐ง2 and define a line segment ๐ฟ B {๐ง1 + ๐ก (๐ง2 โ ๐ง1) โ R๐ | 0 โค ๐ก โค 1}.We first consider the case 0 โ ๐ฟ. Then there exist constants 0 < ๐ โค ๐ <โ such that ๐ โค |๐ง0 | โค ๐ holds for all
๐ง0 โ ๐ฟ. Here we can take a constant ๐พ > 0 such that (6.3) holds for all ๐ง0 โ ๐ฟ. Then by๐ โ ๐ถ2 (R๐ \ {0}), we have
โจ๐ด(๐ง1) โ ๐ด(๐ง2) | ๐ง2 โ ๐ง1โฉ =โซ 1
0
โจโ2๐ง๐ (๐ง1 + ๐ก (๐ง2 โ ๐ง1)) (๐ง2 โ ๐ง1)
๏ฟฝ๏ฟฝ ๐ง2 โ ๐ง1โฉ ๐๐ก โฅ ๐พ |๐ง2 โ ๐ง1 |2 > 0.
26
To consider the remaining case 0 โ ๐ฟ, it suffices to show (6.7). Indeed, the assumption 0 โ ๐ฟ allows us to write๐ง1 = โ๐1๐, ๐ง2 = ๐2๐ for some unit vector ๐ and some constants ๐1, ๐2 โฅ 0. Under this notation, we obtain
โจ๐ด(๐ง2) โ ๐ด(๐ง1) | ๐ง2 โ ๐ง1โฉ = โจ๐ด(๐2๐) | (๐1 + ๐2)๐โฉ + โจ๐ด(โ๐1๐) | โ(๐1 + ๐2)๐โฉ > 0
by (6.7). Here we note that at least one of ๐1, ๐2 is positive since ๐1 + ๐2 = |๐ง2 โ ๐ง1 | > 0.We prove (6.7) to complete the proof. Let ๐ง โ R๐ \ {0}. Then we obtain
๐๐ Bโจ๐ด(๐ง/2๐โ1) โ ๐ด(๐ง/2๐ )
๏ฟฝ๏ฟฝ ๐งโฉ > 0
for each ๐ โ N, since we have already shown (4.8) for the case 0 โ ๐ฟ. By definition of ๐ ๐ ( ๐ โ N), it is clear thatโจ๐ด(๐ง) โ ๐ด(๐ง/2๐ )
๏ฟฝ๏ฟฝ ๐งโฉ = ๐1 + ยท ยท ยท + ๐๐ โฅ ๐1.
Letting ๐ โโ, we obtain โจ๐ด(๐ง) | ๐งโฉ โฅ ๐1 > 0 by ๐ด โ ๐ถ (R๐, R๐). โก
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,โ) be a convex function which is positively homogeneous of degree 1.
1. ฮจ satisfies the triangle inequality (6.8).
2. Assume that ๐ โ R๐ satisfies ฮจฬ(๐) <โ. Then the CauchyโSchwarz-type inequality (6.9) holds.
3. The subdifferential operator ๐ฮจ is given by (6.10).
Proof. By convexity of ฮจ and (6.6), ฮจ satisfies
ฮจ(๐ง1 + ๐ง2)2
= ฮจ( ๐ง1 + ๐ง2
2
)โค ฮจ(๐ง1) +ฮจ(๐ง2)
2for all ๐ง1, ๐ง2 โ R๐,
which yields (6.8).We next show the CauchyโSchwarz inequality (6.9). Let ๐ง โ R๐. If ฮจ(๐ง) > 0, then we have
โจ๐ง | ๐โฉ = ฮจ(๐ง)โจ
๐ง
ฮจ(๐ง)
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โฉ โค ฮจ(๐ง)ฮจฬ(๐)
by ๐ง/ฮจ(๐ง) โ ๐ถฮจ. For the case ฮจ(๐ง) = 0, we note that ๐๐ง โ ๐ถฮจ for all ๐ > 0. Hence it follows that
โจ๐ง | ๐โฉ = โจ๐๐ง | ๐โฉ๐
โค ฮจฬ(๐ค)๐
for all ๐ > 0. By ฮจฬ(๐) <โ, we obtain โจ๐ง | ๐โฉ โค 0 = ฮจ(๐ง)ฮจฬ(๐). This completes the proof of (6.9).Finally we prove (6.10). Let ๐ง0 โ R๐ be arbitrarily fixed. Assume that ๐ โ R๐ satisfies ฮจฬ(๐) โค 1 and ฮจ(๐ง0) =
โจ๐ง0 | ๐โฉ. Then by combining these assumptions with (6.9), we have
ฮจ(๐ง) โฅ ฮจ(๐ง)ฮจฬ(๐)โฅ โจ๐ง | ๐โฉ = โจ๐ง0 | ๐โฉ + โจ๐งโ ๐ง0 | ๐โฉ= ฮจ(๐ง0) + โจ๐ | ๐งโ ๐ง0โฉ
for all ๐ง โ R๐. Hence ๐ โ ๐ฮจ(๐ง0). Conversely, if ๐ โ ๐ฮจ(๐ง0), then we have the subgradient inequality
ฮจ(๐ง) โฅ ฮจ(๐ง0) + โจ๐ | ๐งโ ๐ง0โฉ for all ๐ง โ R๐. (A.9)
By testing ๐๐ง0 into (A.9), where ๐ โ [0,โ) is arbitrary, we have
(๐ โ1)ฮจ(๐ง0) = ฮจ(๐๐ง0) โฮจ(๐ง0) โฅ โจ๐ | (๐ โ1)๐ง0โฉ = (๐ โ1)โจ๐ | ๐ง0โฉ. (A.10)
If we let 0 โค ๐ < 1 so that ๐ โ 1 < 0, then we have ฮจ(๐ง0) โค โจ๐ | ๐ง0โฉ. Similarly, letting 1 < ๐ < โ, we haveฮจ(๐ง0) โฅ โจ๐ | ๐ง0โฉ. Hence we obtain ฮจ(๐ง0) = โจ๐ | ๐ง0โฉ. Combining with (A.9), we have
โจ๐ง | ๐โฉ โค ฮจ(๐ง) for all ๐ง โ R๐,
which yields ฮจฬ(๐) โค 1 by definition of ฮจฬ. This completes the proof of (6.10). โก
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References[1] L. Ambrosio, A. Carlotto, and A. Massaccesi. Lectures on elliptic partial differential equations, volume 18
of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore diPisa (New Series)]. Edizioni della Normale, Pisa, 2018.
[2] F. Andreu-Vaillo, V. Caselles, and J. M. Mazรณn. Parabolic quasilinear equations minimizing linear growthfunctionals, volume 223 of Progress in Mathematics. Birkhรคuser Verlag, Basel, 2004.
[3] D. P. Bertsekas. Convex optimization theory. Athena Scientific, Nashua, NH, 2009.
[4] H. Brรฉzis. Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differentialequations. In Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ.Wisconsin, Madison, Wis., 1971), pages 101โ156, 1971.
[5] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer,New York, 2011.
[6] M. Colombo and G. Mingione. Bounded minimisers of double phase variational integrals. Arch. Ration.Mech. Anal., 218(1):219โ273, 2015.
[7] M. Colombo and G. Mingione. Regularity for double phase variational problems. Arch. Ration. Mech. Anal.,215(2):443โ496, 2015.
[8] G. Duvaut and J.-L. Lions. Inequalities in mechanics and physics, volume 219 of Grundlehren derMathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1976. Translated from the Frenchby C. W. John.
[9] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Textbooks in Mathematics.CRC Press, Boca Raton, FL, revised edition, 2015.
[10] M. Giaquinta and L. Martinazzi. An introduction to the regularity theory for elliptic systems, harmonic mapsand minimal graphs, volume 11 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes.Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, second edition, 2012.
[11] M.-H. Giga and Y. Giga. Stability for evolving graphs by nonlocal weighted curvature. Comm. PartialDifferential Equations, 24(1-2):109โ184, 1999.
[12] M.-H. Giga and Y. Giga. Very singular diffusion equations: second and fourth order problems. Jpn. J. Ind.Appl. Math., 27(3):323โ345, 2010.
[13] Y. Giga and R. V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations.Discrete Contin. Dyn. Syst., 30(2):509โ535, 2011.
[14] Y. Giga, H. Kuroda, and H. Matsuoka. Fourth-order total variation flow with Dirichlet condition:characterization of evolution and extinction time estimates. Adv. Math. Sci. Appl., 24(2):499โ534, 2014.
[15] Y. Giga, M. Muszkieta, and P. Rybka. A duality based approach to the minimizing total variation flow in thespace ๐ปโ๐ . Jpn. J. Ind. Appl. Math., 36(1):261โ286, 2019.
[16] Y. Giga and Y. Ueda. Numerical computations of split Bregman method for fourth order total variation flow.J. Comput. Phys., 405:109114, 24, 2020.
[17] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics.Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
[18] E. Giusti. Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ,2003.
[19] R. Glowinski, J.-L. Lions, and R. Trรฉmoliรจres. Numerical analysis of variational inequalities, volume 8 ofStudies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York, 1981.Translated from the French.
28
[20] E. Hopf. A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc., 3:791โ793,1952.
[21] Y. Kashima. A subdifferential formulation of fourth order singular diffusion equations. Adv. Math. Sci. Appl.,14(1):49โ74, 2004.
[22] Y. Kashima. Characterization of subdifferentials of a singular convex functional in Sobolev spaces of orderminus one. J. Funct. Anal., 262(6):2833โ2860, 2012.
[23] R. V. Kohn. Surface relaxation below the roughening temperature: some recent progress and open questions.In Nonlinear partial differential equations, volume 7 of Abel Symp., pages 207โ221. Springer, Heidelberg,2012.
[24] R. V. Kohn and H. M. Versieux. Numerical analysis of a steepest-descent PDE model for surface relaxationbelow the roughening temperature. SIAM J. Numer. Anal., 48(5):1781โ1800, 2010.
[25] F. Krรผgel. A variational problem leading to a singular elliptic equation involving the 1-Laplacian. Berlin:Mensch und Buch Verlag, 2013.
[26] G. Mingione. Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math.,51(4):355โ426, 2006.
[27] I. V. Odisharia. Simulation and analysis of the relaxation of a crystalline surface. New York University, 2006.
[28] M. H. Protter and H. F. Weinberger. Maximum principles in differential equations. Prentice-Hall, Inc.,Englewood Cliffs, N.J., 1967.
[29] P. Pucci and J. Serrin. The maximum principle, volume 73 of Progress in Nonlinear Differential Equationsand their Applications. Birkhรคuser Verlag, Basel, 2007.
[30] R. T. Rockafellar. Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press,Princeton, NJ, 1997. Reprint of the 1970 original, Princeton Paperbacks.
[31] H. Spohn. Surface dynamics below the roughening transition. Journal de Physique I, 3(1):69โ81, 1993.
[32] S. Tsubouchi. Local Lipschitz bounds for solutions to certain singular elliptic equations involving theone-Laplacian. Calc. Var. Partial Differential Equations, 60(1):Paper No. 33, 35, 2021.
[33] X. Xu. Mathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in2 space dimensions. Calc. Var. Partial Differential Equations, 59(5):158, 2020.
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