Luis A. Caffarelli

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.

The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian

On etudie le probleme de Dirichlet dans un domaine borne Ω de R n a frontiere lisse ∂Ω:F(D 2 u)=ψ dans Ω, u=φ sur ∂Ω

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.

On W1,p estimates for elliptic equations in divergence form

First, we will apply the method to study W 1,p regularity for a nonlinear elliptic operator in divergence form. We would like to point out that in the particular case of a linear elliptic equation, this method gives an alternative proof to the classical one, which uses the general theory of singular integrals. The utility of the method that we describe below is that hypotheses (A) and (B) are obtained directly by studying the deviation of the “coefficients” of A from the “coefficients” of A0, and this is usually not a difficult task. A more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give W 1,p estimates with a weak hypothesis of regularity in the coefficients. For instance, we are able to study the following cases:

The regularity of mappings with a convex potential

In this work, we apply the techniques developed in [Cl] to the problem of mappings with a convex potential between domains. That is, given two bounded domains Q, Q2 of Rn and two nonnegative real functions fi defined in Qi that are bounded away from zero and infinity, we want to study the map vf = V W for a Lipschitz convex ,v, such that V ,/ maps Ql onto Q?2 in the a.e. sense and in some (weak) sense. (1) f2(VyV) det Dij V = f1 (X) . In recent work Y. Brenier showed existence and uniqueness of such a map (provided that JaQil = 0) under the obvious compatibility condition

Interior W2 P estimates for solutions of the Monge-Ampere equation

In this work we adapt the techniques developed in [C1] to prove interior estimates for solutions of perturbations of the Monge-Ampere equation

A geometric approach to free boundary problems

Elliptic problems: An introductory problem Viscosity solutions and their asymptotic developments The regularity of the free boundary Lipschitz free boundaries are

Vortex Condensation in the Chern-Simons Higgs Model: An Existence Theorem

C^{1,\gamma}

On viscosity solutions of fully nonlinear equations with measurable ingredients

Flat free boundaries are Lipschitz Existence theory Evolution problems: Parabolic free boundary problems Lipschitz free boundaries: Weak results Lipschitz free boundaries: Strong results Flat free boundaries are smooth Complementary chapters: Main tools: Boundary behavior of harmonic functions Monotonicity formulas and applications Boundary behavior of caloric functions Bibliography Index.