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Integro-differential equations: Regularity theory and Pohozaev - - PowerPoint PPT Presentation

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Integro-differential equations: Regularity theory and Pohozaev identities Xavier Ros Oton Departament Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya PhD Thesis Advisor: Xavier Cabr e Xavier Ros Oton (UPC, Barcelona) PhD

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Integro-differential equations: Regularity theory and Pohozaev identities

Xavier Ros Oton

Departament Matem` atica Aplicada I, Universitat Polit` ecnica de Catalunya

PhD Thesis Advisor: Xavier Cabr´ e

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 1 / 43

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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−1 n Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 2 / 43

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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

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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

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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

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PART I: Integro-differential equations

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 6 / 43

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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 L´ evy processes − → Integro-Differential Equations

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 7 / 43

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Expected payoff

Brownian motion    ∆u = 0 in Ω u = φ

  • n ∂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

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Expected payoff

Brownian motion L´ evy processes    ∆u = 0 in Ω u = φ

  • n ∂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

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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

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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

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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 sup

α∈A

Lαu = 0 Optimal stopping time

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43

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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 sup

α∈A

Lαu = 0 Optimal stopping time Obstacle problem

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43

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The fractional Laplacian

Most canonical example of elliptic integro-differential operator: (−∆)su(x) = cn,sPV

  • Rn

u(x) − u(x + y) |y|n+2s dy, s ∈ (0, 1). Notation justified by

  • (−∆)su(ξ) = |ξ|2s

u(ξ), → (−∆)s ◦ (−∆)t = (−∆)s+t. It corresponds to stable and radially symmetric L´ evy process.

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 10 / 43

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Stable L´ evy processes

Special class of L´ evy 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 L´ evy processes a(θ) is called the spectral measure (defined on Sn−1).

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 11 / 43

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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

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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

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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

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The classical Pohozaev identity

   −∆u = f (u) in Ω u =

  • n ∂Ω,

Theorem (Pohozaev, 1965)

  • Ω
  • n F(u) − n − 2

2 u 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

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The classical Pohozaev identity

Applications of the identity: Nonexistence of solutions: critical exponent −∆u = u

n+2 n−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

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Pohozaev identities for (−∆)s

Assume

  • (−∆)su
  • ≤

C in Ω u = 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 ds (x) 2 (x · ν) Here, Γ is the gamma function.

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 17 / 43

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Remark

1 2

  • ∂Ω

∂u ∂ν 2 (x · ν)

  • Γ(1 + s)2

2

  • ∂Ω

u ds 2 (x · ν) u ds

  • ∂Ω

plays the role that ∂u ∂ν plays in 2nd order PDEs

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 18 / 43

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Pohozaev identities for (−∆)s

Changing the origin in our identity, we find

  • Ω

uxi(−∆)su = Γ(1 + s)2 2

  • ∂Ω

u ds 2 νi Thus,

Corollary

Under the same hypotheses as before

  • Ω

(−∆)su vxi = −

  • Ω

uxi(−∆)sv + Γ(1 + s)2

  • ∂Ω

u ds v ds νi 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

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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 = (−∆)

s 2 u 3

Analyze very precisely the singularity of (−∆)

s 2 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

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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 = in Rn \ B1    = ⇒ u(x) = c

  • 1 − |x|2s

They are C ∞ inside Ω, but C s(Ω) and not better! In both cases, they are comparable to ds, where d(x) = dist(x, Rn \ Ω).

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 21 / 43

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Boundary regularity: First results

   (−∆)su = g in Ω u = in Rn\Ω, Then, uC s(Ω) ≤ CgL∞. Moreover,

Theorem (R-Serra’12; J. Math. Pures Appl.)

Ω bounded and C 1,1 domain. Then, u/dsC γ(Ω) ≤ CgL∞ 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

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Boundary for integro-differential operators?

   (−∆)su = g(x) in Ω u = in Rn\Ω, = ⇒ u/ds ∈ C γ(Ω). We answer an open question: What about boundary regularity for more general

  • perators of “order” 2s?

Lu(x) = PV

  • Rn
  • u(x) − u(x + y)
  • K(y)dy

Is it true that u/ds is H¨

  • lder 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

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Fully nonlinear integro-differential equations

Let us consider solutions to    Iu = f in Ω u = in Rn\Ω, where I is a fully nonlinear operator like Iu(x) = sup

α 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

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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 L´ evy processes

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 25 / 43

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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 = in Rn\Ω,

Theorem (R-Serra; preprint’14)

If Ω is C 1,1, then any viscosity solution satisfies u/dsC s−ǫ(Ω) ≤ Cf L∞(Ω) for all ǫ > 0 Even for (−∆)s we improve our previous results!

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 26 / 43

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Novelty: We obtain higher regularity for u/ds! The exponent s − ǫ 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

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Main steps of the proof of u/ds ∈ C s−ǫ(Ω):

1

Bounded measurable coefficients = ⇒ u/ds ∈ C γ(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/ds, 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

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PART II: Regularity of stable solutions to elliptic equations

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 29 / 43

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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

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Regularity of minimizers

   −∆u = f (u) in Ω ⊂ Rn u =

  • n ∂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 [Cabr´ e-Capella ’06] n ≤ 4 [n ≤ 3 Nedev ’00; n ≤ 4 Cabr´ e ’10]

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 31 / 43

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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) = lim

λ↑λ∗ uλ(x)

is a weak solution, called the extremal solution. Moreover, it is stable. Question: Is the extremal solution bounded? [Brezis-V´ azquez ’97]

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 32 / 43

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Our work

We have studied the regularity of stable solutions to    −∆u = f (u) in Ω u =

  • n ∂Ω,

[Comm. PDE ’13] Thm: L∞ for n ≤ 7 & Ω of double revolution    (−∆)su = f (u) in Ω u = 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

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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 xa

1xb 2 dx1 dx2

1/q ≤ C

  • ˜

Ω

|∇u|2 xa

1xb 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) = xA1

1 · · · xAn n

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 34 / 43

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PART III: Isoperimetric inequalities with densities

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 35 / 43

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Sobolev inequalities with weights

Theorem (Cabr´ e-R; J. Differential Equations’13)

Let Ai ≥ 0, w(x) = xA1

1 · · · xAn n ,

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

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Isoperimetric inequalities with monomial weights

Theorem (Cabr´ e-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−1 D

≥ Pw(B1 ∩ Σ) w(B1 ∩ Σ)

D−1 D .

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

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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

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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|

n n−1

≥ |Σ ∩ ∂B1| |B1 ∩ Σ|

n n−1 .

Important: Only the perimeter inside Σ is counted

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 39 / 43

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New isoperimetric inequalities weights

Theorem (Cabr´ e-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−1 D

≥ Pw(B1 ∩ Σ) w(B1 ∩ Σ)

D−1 D .

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

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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

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The proof

The proof uses the ABP technique applied to an appropriate PDE When w ≡ 1, the idea goes back to the work [Cabr´ e ’00] (for the classical isoperimetric inequality) Here, we need to consider a linear Neumann problem in E ⊂ Σ involving the

  • perator w −1div(w∇u)

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 42 / 43

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The end

Thank you!

Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 43 / 43