Joaquim Serra

BOUNDARY REGULARITY FOR FULLY NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0, which consists of all the infinitesimal generators of stable Levy processes belonging to the class L0 of Caffarelli-Silvestre. For fully nonlinear operators I elliptic with respec tt oL∗ ,w e prove that solutions to Iu = f in Ω, u = 0 in R n \ Ω, satisfy u/d s ∈ C s−ϵ (Ω) for all ϵ> 0, where d is the distance to ∂ Ωa ndf ∈ L ∞ . We expect the Holder exponent s − ϵ to be optimal (or almost optimal) for general right hand sides f ∈ L ∞ . Moreover, we also expect the class L∗ to be the largest scale invariant subclass of L0 for which this result is true. In this direction, we show that the class L0 is too large for all solutions to behave like d s . The constants in all the estimates in this paper remain bounded as th eo rder of the equation approaches 2.

Nonexistence Results for Nonlocal Equations with Critical and Supercritical Nonlinearities

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| n+σ is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case a ij ≡ 0 and for σ = 2 in case that (a ij ) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for L in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to n and σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ) s (here s > 1) or the fractional p-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.

Sharp isoperimetric inequalities via the ABP method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of

Pohozaev identities for anisotropic integrodifferential operators

\mathbb{R}^n

On the fine structure of the free boundary for the classical obstacle problem

. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

ABSTRACT We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, with s∈(0,1). These identities involve local boundary terms, in which the quantity plays the role that ∂u∕∂ν plays in the second-order case. Here, u is any solution to Lu = f(x,u) in Ω, with u = 0 in ℝn∖Ω, and d is the distance to ∂Ω.

The Pohozaev Identity for the Fractional Laplacian

AbstractIn the classical obstacle problem, the free boundary can be decomposed into “regular” and “singular” points. As shown by Caffarelli in his seminal papers (Caffarelli in Acta Math 139:155–184, 1977; J Fourier Anal Appl 4:383–402, 1998), regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of

Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

C^1