integro-di erential equations: regularity theory and pohozaev...
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Integro-differential equations:
Regularity theory and Pohozaev identities
Xavier Ros Oton
Departament Matematica Aplicada I, Universitat Politecnica de Catalunya
PhD Thesis
Advisor: Xavier Cabre
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 1 / 43
Structure of the thesis
PART I: Integro-differential equations
Lu(x) = PV
∫Rn
(u(x)− u(x + y)
)K (y)dy
PART II: Regularity of stable solutions to elliptic equations
−∆u = λ f (u) in Ω ⊂ Rn
PART III: Isoperimetric inequalities with densities
|∂Ω||Ω| n−1
n
≥ |∂B1||B1|
n−1n
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 2 / 43
PART I
1. The Dirichlet problem for the fractional Laplacian: regularity up to the
boundary, [J. Math. Pures Appl. ’14]
2. The Pohozaev identity for the fractional Laplacian, [ARMA ’14]
3. Nonexistence results for nonlocal equations with critical and supercritical
nonlinearities, [Comm. PDE ’14]
4. Boundary regularity for fully nonlinear integro-differential equations, Preprint.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 3 / 43
PART II
5. Regularity of stable solutions up to dimension 7 in domains of double
revolution, [Comm. PDE ’13]
6. The extremal solution for the fractional Laplacian, [Calc. Var. PDE ’14]
7. Regularity for the fractional Gelfand problem up to dimension 7,
[J. Math. Anal. Appl. ’14]
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 4 / 43
PART III
8. Sobolev and isoperimetric inequalities with monomial weights,
[J. Differential Equations ’13]
9. Sharp isoperimetric inequalities via the ABP method, Preprint.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 5 / 43
PART I:
Integro-differential equations
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 6 / 43
Nonlocal equations
Linear elliptic integro-differential operators:
Lu(x) = PV
∫Rn
(u(x)− u(x + y)
)K (y)dy ,
with K ≥ 0, K (y) = K (−y), and∫Rn
min(1, |y |2
)K (y)dy <∞.
Brownian motion −→ 2nd order PDEs
Levy processes −→ Integro-Differential Equations
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 7 / 43
Expected payoff
Brownian motion ∆u = 0 in Ω
u = φ on ∂Ω
u(x) = E(φ(Xτ )
)(expected payoff)
Xt = Random process, X0 = x
τ = first time Xt exits Ω
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 8 / 43
Expected payoff
Brownian motion Levy processes ∆u = 0 in Ω
u = φ on ∂Ω
Lu = 0 in Ω
u = φ in Rn \ Ω
u(x) = E(φ(Xτ )
)(expected payoff)
Xt = Random process, X0 = x
τ = first time Xt exits Ω
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 8 / 43
More equations from Probability
Distribution of the process Xt Fractional heat equation
∂tu + Lu = 0
Expected hitting time / running cost
Controlled diffusion
Optimal stopping time
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability
Distribution of the process Xt Fractional heat equation
∂tu + Lu = 0
Expected hitting time / running cost Dirichlet problem Lu = f (x) in Ω
u = 0 in Rn \ Ω
Controlled diffusion
Optimal stopping time
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability
Distribution of the process Xt Fractional heat equation
∂tu + Lu = 0
Expected hitting time / running cost Dirichlet problem Lu = f (x) in Ω
u = 0 in Rn \ Ω
Controlled diffusion Fully nonlinear equations
supα∈A
Lαu = 0
Optimal stopping time
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability
Distribution of the process Xt Fractional heat equation
∂tu + Lu = 0
Expected hitting time / running cost Dirichlet problem Lu = f (x) in Ω
u = 0 in Rn \ Ω
Controlled diffusion Fully nonlinear equations
supα∈A
Lαu = 0
Optimal stopping time Obstacle problem
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
The fractional Laplacian
Most canonical example of elliptic integro-differential operator:
(−∆)su(x) = cn,sPV
∫Rn
u(x)− u(x + y)
|y |n+2sdy , s ∈ (0, 1).
Notation justified by
(−∆)su(ξ) = |ξ|2s u(ξ), → (−∆)s (−∆)t = (−∆)s+t .
It corresponds to stable and radially symmetric Levy process.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 10 / 43
Stable Levy processes
Special class of Levy processes: stable processes
Lu(x) = PV
∫Rn
(u(x)− u(x + y)
)a(y/|y |)|y |n+2s
dy
Very important and well studied in Probability
These are processes with self-similarity properties (Xt ≈ t−1/αX1)
Central Limit Theorems ←→ stable Levy processes
a(θ) is called the spectral measure (defined on Sn−1).
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 11 / 43
Why studying nonlocal equations?
Nonlocal equations are used to model (among others):
Prices in Finance (since the 1990’s)
Anomalous diffusions (Physics, Ecology, Biology): ut + Lu = f (x , u)
Also, they arise naturally when long-range interactions occur:
Image Processing
Relativistic Quantum Mechanics√−∆ + m
Boltzmann equation
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 12 / 43
Why studying nonlocal equations?
Still, these operators appear in:
Fluid Mechanics (surface quasi-geostrophic equation)
Conformal Geometry
Finally, all PDEs are limits of nonlocal equations (as s ↑ 1).
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 13 / 43
Important works
Works in Probability 1950-2014 (Kac, Getoor, Bogdan, Bass, Chen,...)
Fully nonlinear equations: Caffarelli-Silvestre ’07-10 [CPAM, Annals, ARMA]
Reaction-diffusion equations ut + Lu = f (x , u)
Obstacle problem, free boundaries
Nonlocal minimal surfaces, fractional perimeters
Math. Physics: (Lieb, Frank,...) [JAMS’08], [Acta Math.’13]
Fluid Mech.: Caffarelli-Vasseur [Annals’10], [JAMS’11]
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 14 / 43
The classical Pohozaev identity
−∆u = f (u) in Ω
u = 0 on ∂Ω,
Theorem (Pohozaev, 1965)∫Ω
n F (u)− n − 2
2u f (u)
=
1
2
∫∂Ω
(∂u
∂ν
)2
(x · ν)dσ
Follows from: For any function u with u = 0 on ∂Ω,
∫Ω
(x · ∇u) ∆u =2− n
2
∫Ω
u ∆u +1
2
∫∂Ω
(∂u
∂ν
)2
(x · ν)dσ
And this follows from the divergence theorem.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 15 / 43
The classical Pohozaev identity
Applications of the identity:
Nonexistence of solutions: critical exponent −∆u = un+2n−2
Unique continuation “from the boundary”
Monotonicity formulas
Concentration-compactness phenomena
Radial symmetry
Stable solutions: uniqueness results, H1 interior regularity
Other: Geometry, control theory, wave equation, harmonic maps, etc.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 16 / 43
Pohozaev identities for (−∆)s
Assume ∣∣ (−∆)su∣∣ ≤ C in Ω
u = 0 in Rn \ Ω,
(+ some interior regularity on u)
Theorem (R-Serra’12; ARMA)
If Ω is C 1,1,∫Ω
(x · ∇u) (−∆)su =2s − n
2
∫Ω
u (−∆)su − Γ(1 + s)2
2
∫∂Ω
( u
d s(x))2
(x · ν)
Here, Γ is the gamma function.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 17 / 43
Remark
1
2
∫∂Ω
(∂u
∂ν
)2
(x · ν) Γ(1 + s)2
2
∫∂Ω
( u
d s
)2
(x · ν)
u
d s
∣∣∣∂Ω
plays the role that∂u
∂νplays in 2nd order PDEs
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 18 / 43
Pohozaev identities for (−∆)s
Changing the origin in our identity, we find∫Ω
uxi (−∆)su =Γ(1 + s)2
2
∫∂Ω
( u
d s
)2
νi
Thus,
Corollary
Under the same hypotheses as before∫Ω
(−∆)su vxi = −∫
Ω
uxi (−∆)sv + Γ(1 + s)2
∫∂Ω
u
d s
v
d sνi
Note the contrast with the nonlocal flux in the formula∫
Ω(−∆)sw =
∫Rn\Ω
∫Ω...
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 19 / 43
Ideas of the proof
1 uλ(x) = u(λx) , λ > 1, ⇒∫Ω
(x · ∇u)(−∆)su =d
dλ
∣∣∣∣λ=1+
∫Ω
uλ(−∆)su
2 Ω star-shaped ⇒∫Ω
(x · ∇u)(−∆)su =2s − n
2
∫Ω
u(−∆)su +1
2
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ,
w = (−∆)s2 u
3 Analyze very precisely the singularity of (−∆)s2 u along ∂Ω, and compute.
4 Deduce the result for general C 1,1 domains.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 20 / 43
Fractional Laplacian: Two explicit solutions
1. u(x) = (x+)s satisfies (−∆)su = 0 in (0,+∞).
2. Explicit solution by [Getoor, 1961]:
(−∆)su = 1 in B1
u = 0 in Rn \ B1
=⇒ u(x) = c(1− |x |2
)s
They are C∞ inside Ω, but C s(Ω) and not better!
In both cases, they are comparable to d s , where d(x) = dist(x ,Rn \ Ω).
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 21 / 43
Boundary regularity: First results
(−∆)su = g in Ω
u = 0 in Rn\Ω,
Then, ‖u‖C s (Ω) ≤ C‖g‖L∞ . Moreover,
Theorem (R-Serra’12; J. Math. Pures Appl.)
Ω bounded and C 1,1 domain. Then,
‖u/d s‖Cγ(Ω) ≤ C‖g‖L∞ for some small γ > 0,
where d is the distance to ∂Ω.
Proof: Can not do odd reflection! (boundary behavior different from interior!)
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 22 / 43
Boundary for integro-differential operators?
(−∆)su = g(x) in Ω
u = 0 in Rn\Ω,=⇒ u/d s ∈ Cγ(Ω).
We answer an open question: What about boundary regularity for more general
operators of “order” 2s?
Lu(x) = PV
∫Rn
(u(x)− u(x + y)
)K (y)dy
Is it true that u/d s is Holder continuous? At least bounded?
We answer this for linear and also for fully nonlinear equations
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 23 / 43
Fully nonlinear integro-differential equations
Let us consider solutions to Iu = f in Ω
u = 0 in Rn\Ω,
where I is a fully nonlinear operator like
Iu(x) = supα
Lαu(x) (controlled diffusion)
Here, all Lα ∈ L for some class of linear operators L.
The class L is called the ellipticity class.
(When Lα are 2nd order operators, we have F (D2u) = f (x) in Ω)
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 24 / 43
Interior regularity:
Was developed by Caffarelli and Silvestre in 2007-2010 (CPAM, Annals,
ARMA)
They established: Krylov-Safonov, Evans-Krylov, perturbative theory, etc.
The reference ellipticity class of Caffarelli-Silvestre is L0 , with kernels
λ
|y |n+2s≤ K (y) ≤ Λ
|y |n+2s
Boundary regularity:
We establish boundary regularity for the class L∗ , with kernels
K (y) =a (y/|y |)|y |n+2s
, λ ≤ a(θ) ≤ Λ
L∗ = Subclass of L0, corresponding to stable Levy processes
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 25 / 43
Boundary regularity for fully nonlinear equations
Let I (u, x) be a fully nonlinear operator elliptic w.r.t. L∗, and I (u, x) = f (x) in Ω
u = 0 in Rn\Ω,
Theorem (R-Serra; preprint’14)
If Ω is C 1,1, then any viscosity solution satisfies
‖u/d s‖C s−ε(Ω) ≤ C‖f ‖L∞(Ω) for all ε > 0
Even for (−∆)s we improve our previous results!
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 26 / 43
Novelty: We obtain higher regularity for u/d s !
The exponent s − ε is optimal for f ∈ L∞
Also, it cannot be improved if a ∈ L∞(Sn−1)
Very important: L∗ is the good class for boundary regularity!
The class L0 is too large for fine boundary regularity
There exist positive numbers 0 < β1 < s < β2 such that
I1(x+)β1 ≡ 0, I2(x+)β2 ≡ 0 in x > 0
Solutions are not even comparable near the boundary!
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 27 / 43
Main steps of the proof of u/d s ∈ C s−ε(Ω):
1 Bounded measurable coefficients =⇒ u/d s ∈ Cγ(Ω)
2 Blow up the equation at x ∈ ∂Ω + compactness argument.
3 Liouville theorem in half-space
Advantages of the method:
It allows us to obtain higher regularity of u/d s , also in the normal direction!
After blow up, you do not see the geometry of the domain
Also non translation invariant equations
Discontinuous kernels a ∈ L∞(Sn−1)
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 28 / 43
PART II:
Regularity of stable solutions
to elliptic equations
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 29 / 43
Regularity of minimizers
Classical problem in the Calculus of Variations: Regularity of minimizers
Example in Geometry: Regularity of hypersurfaces in Rn which minimize the area
functional.
These hypersurfaces are smooth if n ≤ 7
In R8 the Simons cone minimizes area and has a singularity at x = 0
As we will see, the same happens for other nonlinear PDE in bounded domains.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 30 / 43
Regularity of minimizers
−∆u = f (u) in Ω ⊂ Rn
u = 0 on ∂Ω,
Open problem:
u local minimizer (or stable solution) & n ≤ 9 =⇒ u ∈ L∞?
In R10, u(x) = log 1|x|2 is a stable solution in B1
f (u) = λeu or f (u) = λ(1 + u)p & n ≤ 9 [Crandall-Rabinowitz ’75]
Ω = B1 & n ≤ 9 [Cabre-Capella ’06]
n ≤ 4 [n ≤ 3 Nedev ’00; n ≤ 4 Cabre ’10]
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 31 / 43
The extremal solution
If f (u) λf (u) , then there is λ∗ ∈ (0,+∞) s.t.
For 0 < λ < λ∗, there is a bounded solution uλ.
For λ > λ∗, there is no solution.
For λ = λ∗,
u∗(x) = limλ↑λ∗
uλ(x)
is a weak solution, called the extremal solution. Moreover, it is stable.
Question: Is the extremal solution bounded? [Brezis-Vazquez ’97]
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 32 / 43
Our work
We have studied the regularity of stable solutions to −∆u = f (u) in Ω
u = 0 on ∂Ω,[Comm. PDE ’13]
Thm: L∞ for n ≤ 7 & Ω of double revolution
(−∆)su = f (u) in Ω
u = 0 in Rn \ Ω,
[Calc. Var. PDE ’13]
[J. Math. Anal. Appl. ’13]
Thm: L∞ and Hs bounds in general domains;
Thm: optimal regularity for f (u) = λeu in xi -symmetric domains
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 33 / 43
Sobolev inequalities with weights
When studying −∆u = f (u), we needed∫Ω
s−α|us |2 + t−β |ut |2
ds dt ≤ C =⇒ u ∈ Lq(Ω) ? q(α, β) =?
After a change of variables, we want(∫Ω
|u|q xa1 xb
2 dx1 dx2
)1/q
≤ C
(∫Ω
|∇u|2 xa1 xb
2 dx1 dx2
)1/2
, q = q(a, b)
Thus, we want Sobolev inequalities with weights(∫Rn
|u|q w(x)dx
)1/q
≤ C
(∫Rn
|∇u|p w(x)dx
)1/p
, w(x) = xA11 · · · x
Ann
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 34 / 43
PART III:
Isoperimetric inequalities with densities
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 35 / 43
Sobolev inequalities with weights
Theorem (Cabre-R; J. Differential Equations’13)
Let Ai ≥ 0,
w(x) = xA11 · · · x
Ann , D = n + A1 + · · ·+ An.
Let 1 ≤ p < D. Then,(∫Rn
|u|q w(x)dx
)1/q
≤ Cp,A
(∫Rn
|∇u|p w(x)dx
)1/p
,
with q = pD/(D − p).
To prove the result, we establish a new weighted isoperimetric inequality.
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 36 / 43
Isoperimetric inequalities with monomial weights
Theorem (Cabre-R; J. Differential Equations’13)
Let Ai > 0, w(x) and D as before, and
Σ = x ∈ Rn : x1, ..., xn > 0.
Then, for any E ⊂ Σ,Pw (E )
w(E )D−1D
≥ Pw (B1 ∩ Σ)
w(B1 ∩ Σ)D−1D
.
We denoted the weighted volume and perimeter
w(E ) =
∫E
w(x)dx Pw (E ) =
∫Σ∩∂E
w(x)dS .
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 37 / 43
Isoperimetric inequalities with weights
This type of isoperimetric inequalities have been widely studied:
w(x) = e−|x|2
[Borell; Invent. Math.’75]
Existence and regularity of minimizers (Pratelli, Morgan,...)
log-convex radial densities w(|x |) [Figalli-Maggi ’13], [Chambers ’14]
with w(x) = e|x|2
, w(x) = |x |α, or other particular weights
...
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 38 / 43
Isoperimetric inequalities in cones
In general cones Σ, a well known result is the following:
Theorem (Lions-Pacella ’90)
Let Σ be any open convex cone in Rn. Then, for any E ⊂ Σ,
|Σ ∩ ∂E ||E |
nn−1
≥ |Σ ∩ ∂B1||B1 ∩ Σ|
nn−1
.
Important: Only the perimeter inside Σ is counted
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 39 / 43
New isoperimetric inequalities weights
Theorem (Cabre-R-Serra; preprint ’13)
Let Σ be any convex cone in Rn. Assume
w(x) homogeneous of degree α ≥ 0, & w 1/α concave in Σ.
Then, for any E ⊂ Σ,Pw (E )
w(E )D−1D
≥ Pw (B1 ∩ Σ)
w(B1 ∩ Σ)D−1D
.
Recall
w(E ) =
∫E
w(x)dx Pw (E ) =
∫Σ∩∂E
w(x)dS .
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 40 / 43
Comments
Minimizers are radial, while w(x) is not!
When w ≡ 1 we recover the result of Lions-Pacella (with new proof!).
We can also treat anisotropic perimeters
Pw ,H(E ) =
∫Σ∩∂E
H(ν) w(x)dS .
w ≡ 1 =⇒ new proof of the Wulff theorem.
Some examples of weights are
w(x) = dist(x , ∂Σ)α, w(x) =√
x +√
y , w(x) =xyz
x + y + z, ...
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 41 / 43
The proof
The proof uses the ABP technique applied to an appropriate PDE
When w ≡ 1, the idea goes back to the work [Cabre ’00] (for the classical
isoperimetric inequality)
Here, we need to consider a linear Neumann problem in E ⊂ Σ involving the
operator w−1div(w∇u)
Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 42 / 43