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
Structure of the thesis PART I : Integro-differential equations � � � Lu ( x ) = PV u ( x ) − u ( x + y ) K ( y ) dy R n PART II : Regularity of stable solutions to elliptic equations in Ω ⊂ R n − ∆ u = λ f ( u ) PART III : Isoperimetric inequalities with densities | ∂ Ω | ≥ | ∂ B 1 | n − 1 n − 1 | Ω | | B 1 | n n 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 u ( x ) − u ( x + y ) K ( y ) dy , R n with K ≥ 0, K ( y ) = K ( − y ), and � � 1 , | y | 2 � R n min 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
Expected payoff Brownian motion ∆ u = 0 in Ω u = φ on ∂ Ω � � u ( x ) = E φ ( X τ ) (expected payoff) = Random process, X 0 = x X t τ = first time X t exits Ω Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 8 / 43
Expected payoff Brownian motion L´ evy processes ∆ u = 0 in Ω Lu = 0 in Ω in R n \ Ω u = φ on ∂ Ω u = φ � � u ( x ) = E φ ( X τ ) (expected payoff) = Random process, X 0 = x X t τ = first time X t exits Ω Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 8 / 43
More equations from Probability Distribution of the process X t Fractional heat equation ∂ t u + 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 X t Fractional heat equation ∂ t u + Lu = 0 Expected hitting time / running cost Dirichlet problem Lu = f ( x ) in Ω in R n \ Ω u = 0 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 X t Fractional heat equation ∂ t u + Lu = 0 Expected hitting time / running cost Dirichlet problem Lu = f ( x ) in Ω in R n \ Ω u = 0 Controlled diffusion Fully nonlinear equations sup L α u = 0 α ∈A Optimal stopping time Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 9 / 43
More equations from Probability Distribution of the process X t Fractional heat equation ∂ t u + Lu = 0 Expected hitting time / running cost Dirichlet problem Lu = f ( x ) in Ω in R n \ Ω u = 0 Controlled diffusion Fully nonlinear equations sup L α u = 0 α ∈A 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: � u ( x ) − u ( x + y ) ( − ∆) s u ( x ) = c n , s PV dy , s ∈ (0 , 1) . | y | n +2 s R n Notation justified by ( − ∆) s ◦ ( − ∆) t = ( − ∆) s + t . ( − ∆) s u ( ξ ) = | ξ | 2 s � � u ( ξ ) , → It corresponds to stable and radially symmetric L´ evy process. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 10 / 43
Stable L´ evy processes Special class of L´ evy processes: stable processes � � � a ( y / | y | ) Lu ( x ) = PV u ( x ) − u ( x + y ) | y | n +2 s dy R n Very important and well studied in Probability These are processes with self-similarity properties ( X t ≈ t − 1 /α X 1 ) Central Limit Theorems ← → stable L´ evy processes a ( θ ) is called the spectral measure (defined on S n − 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): u t + 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 u t + 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) � � � � � ∂ u � 2 n F ( u ) − n − 2 = 1 u f ( u ) ( x · ν ) d σ 2 2 ∂ν Ω ∂ Ω Follows from: For any function u with u = 0 on ∂ Ω, � � � � ∂ u � 2 ( x · ∇ u ) ∆ u = 2 − n u ∆ u + 1 ( x · ν ) d σ 2 2 ∂ν Ω Ω ∂ Ω 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: n +2 Nonexistence of solutions: critical exponent − ∆ u = u n − 2 Unique continuation “from the boundary” Monotonicity formulas Concentration-compactness phenomena Radial symmetry Stable solutions: uniqueness results, H 1 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 � � � ( − ∆) s u � ≤ C in Ω in R n \ Ω , = 0 u (+ some interior regularity on u ) Theorem (R-Serra’12; ARMA) If Ω is C 1 , 1 , � � � � u � 2 u ( − ∆) s u − Γ(1 + s ) 2 ( x · ∇ u ) ( − ∆) s u = 2 s − n d s ( x ) ( x · ν ) 2 2 Ω Ω ∂ Ω Here, Γ is the gamma function. Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 17 / 43
Remark � � ∂ u � 2 � � u � 2 Γ(1 + s ) 2 1 ( x · ν ) ( x · ν ) � 2 ∂ν 2 d s ∂ Ω ∂ Ω � u ∂ u � � plays the role that plays in 2nd order PDEs d s ∂ν ∂ Ω Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 18 / 43
Pohozaev identities for ( − ∆) s Changing the origin in our identity, we find � � � u � 2 u x i ( − ∆) s u = Γ(1 + s ) 2 ν i d s 2 Ω ∂ Ω Thus, Corollary Under the same hypotheses as before � � � u v u x i ( − ∆) s v + Γ(1 + s ) 2 ( − ∆) s u v x i = − d s ν i d s Ω Ω ∂ Ω � � � Ω ( − ∆) s w = Note the contrast with the nonlocal flux in the formula Ω ... R n \ Ω Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 19 / 43
Ideas of the proof u λ ( x ) = u ( λ x ) , λ > 1, ⇒ 1 � � � � d � ( x · ∇ u )( − ∆) s u = u λ ( − ∆) s u � d λ Ω Ω λ =1 + Ω star-shaped ⇒ 2 � � � � � ( x · ∇ u )( − ∆) s u = 2 s − n u ( − ∆) s u + 1 d � R n w λ w 1 /λ , � 2 2 d λ Ω Ω λ =1 + s 2 u w = ( − ∆) s 2 u along ∂ Ω, and compute. Analyze very precisely the singularity of ( − ∆) 3 Deduce the result for general C 1 , 1 domains. 4 Xavier Ros Oton (UPC, Barcelona) PhD Thesis Barcelona, June 2014 20 / 43
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