Workshop

Abstracts

Abdellaoui, Boumediene

Fractional regularity of the solution of the fractional Heat Equation and Applications

Barrios, Begoña

Higher regularity in the nonlocal Bernoulli free boundary problem

Abstract

In this talk we present a very recent result obtained in collaboration with M. Weidner (UB) and X. Ros Oton (UB) about the higher regularity of nonlocal free boundary problems. More precisely we will focus on the one phase (or Bernoulli) problem for a general 2s-stable integro differential operator. By proving a Holder bootstrap's regularity up to the boundary result for a particular equation with weights that explode at the free boundary, we obtain that if the free boundary is C^{2\alpha}, 0<\alpha<1 then has to be $C^{\infty}$. This approach requires a dedicated study of a nonlocal problem with a kind of local Neumann type boundary condition as is different from those known to study the higher regularity of other nonlocal free boundary problems like the obstacle problem.

Bonforte, Matteo

Sharp regularity estimates for nonlocal 0-order p-Laplacian evolution problems

Dipierro, Serena

Boundary behavior of solutions to fractional elliptic problems

Abstract

Solutions of nonlocal equations typically depend rather significantly on their values outside of a given region of interest and, in this sense, it is often convenient to assume “global” conditions to deduce “local” results.

In this talk, we present instead a Hopf Lemma for solutions to some integro-differential equations that does not assume any global condition on the sign of the solutions.

This is a joint work with Nicola Soave (University of Turin) and Enrico Valdinoci (University of Western Australia).

García-Ferrero, María Ángeles

Reconstruction for the Calderón problem with Lipschitz conductivities

Abstract

The Calderón problem seeks to determine the conductivity of the interior of a body from electrical measurements on its boundary. In the eighties, a reconstruction procedure was pro- vided for twice continuously differentiable conductivities. In this talk, we will address the reconstruction in the case that the conductivities are only Lipschitz continuous. For that, we will introduce Sobolev spaces with norms depending on auxiliary parameters, in which we can construct suitable CGO solutions.

This is a joint work with P. Caro and K. Rogers.

Manfredi, Juan José

On the Theorem of Sums

Abstract

We present an extension of the Theorem of Sums of Crandall-Ishii-Jensen-Lions and some of its applications to uniqueness for a general class of elliptic equations. We will also present some considerations about regularity obtained by using the Theorem of Sums.

González, María del Mar

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

Abstract

In this paper we produce families of complete, noncompact Riemannian metrics with positive constant σ_2–curvature on the sphere with a prescribed singular set given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than (n − √n − 2)/2. The σ_2–curvature in conformal geometry is defined as the second elementary symmetric polynomial of the eigenvalues of the Schouten tensor, which yields a fully non-linear PDE for the conformal factor. We show that the classical gluing method of Mazzeo-Pacard (JDG 1996) for the scalar curvature still works in the fully non-linear setting. This is a consequence of the conformal properties of the σ_2 equation, which imply that the linearized operator has good mapping properties in weighted spaces.

de Pablo, Arturo

Blow-up for nonlocal equations with memory

del Teso, Félix

Mean Value Properties and Finite Difference Schemes for Local and Nonlocal p-Laplace Problems

Abstract

The aim of this talk is to present recent advances on asymptotic expansions, mean value properties, and finite difference schemes for parabolic and elliptic equations involving the p-Laplacian and fractional p-Laplacian operators.

Valdinoci, Enrico

Sheet happens (but only as the root of 1-s)

Abstract

We discuss the regularity properties of two-dimensional stable s-minimal surfaces, presenting a robust regularity estimate and an optimal sheet separation bound, according to which the distance between different connected components of the surface must be at least the square root of 1-s.

Vondraček, Zoran

Markov processes with jump kernels decaying at the boundary

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