On approximation of the Dirichlet problem for divergence form operator by Robin problems

TL;DR

This paper demonstrates that solutions to Dirichlet problems for elliptic divergence form operators can be approximated pointwise by solutions to Robin problems, using stochastic methods and reflected diffusions.

Contribution

It introduces a novel approximation approach for Dirichlet problems via Robin problems, leveraging stochastic representations and properties of reflected diffusions.

Findings

Dirichlet solutions can be approximated by Robin solutions

Stochastic representation is key to the approximation

Reflected diffusions relate to divergence form operators

Abstract

We show, that under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operator can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.