Luis Silvestre

An Extension Problem Related to the Fractional Laplacian

The operator square root of the Laplacian (− ▵)1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this article, we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

Regularity Theory for Fully Nonlinear Integro-Differential Equations

We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Levy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1,� regularity for general fully nonlinear integro- differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

Eventual regularization for the slightly supercritical quasi-geostrophic equation

Abstract We prove that weak solutions of the slightly supercritical quasi-geostrophic equation become smooth for large time. The proof uses ideas from a recent article of Caffarelli and Vasseur and is based on an argument in the style of De Giorgi.

Regularity for Fully Nonlinear Elliptic Equations with Neumann Boundary Data

We obtain local C α, C 1,α, and C 2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.

Global well-posedness of slightly supercritical active scalar equations

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasigeostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.

The Dirichlet problem for the convex envelope

The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability (C1,α regularity) of the boundary data implies the corresponding result for the solution in the interior, despite the fact that the solution need not be continuous up to the boundary. Secondary results are the characterization of the convex envelope as: (i) the value function of a stochastic control problem, and (ii) the optimal underestimator for a class of nonlinear elliptic PDEs.

ON THE LOSS OF CONTINUITY FOR SUPER-CRITICAL DRIFT-DIFFUSION EQUATIONS

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts, we prove that solutions satisfy a modulus of continuity depending only on the local L1 norm of the drift, which is a super-critical quantity.

Estimates on elliptic equations that hold only where the gradient is large

We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Holder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.

C1,a regularity of solutions of some degenerate fully non-linear elliptic equations

Abstract In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. Holder estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii–Lions method in order to get C 1 , α estimates for solutions of these equations.