Estimates for Dirichlet-to-Neumann maps as integro-differential operators

TL;DR

This paper extends the classical theory of Dirichlet-to-Neumann maps to nonlinear elliptic equations, providing detailed insights into the associated integro-differential operators and Lévy measures for both linear and nonlinear cases.

Contribution

It introduces a framework for analyzing nonlinear Dirichlet-to-Neumann maps via integro-differential operators, including detailed Lévy measure characterizations.

Findings

New results on linear and nonlinear Dirichlet-to-Neumann mappings

Detailed information about Lévy measures for these operators

Extension of classical theory to fully nonlinear elliptic equations

Abstract

Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a \emph{nonlinear} Dirichlet-to-Neumann mapping that is constructed using a solution to a \emph{fully nonlinear} elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the L\'evy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the L\'evy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.