Mixed Boundary Value Problems of Semilinear Elliptic PDEs and BSDEs with Singular Coefficients

TL;DR

This paper establishes the existence and uniqueness of weak solutions for a class of semilinear elliptic PDEs with singular coefficients using probabilistic methods involving Dirichlet forms and BSDEs.

Contribution

It introduces a probabilistic approach to solve mixed boundary value problems for semilinear elliptic PDEs with singular coefficients, expanding the theoretical framework.

Findings

Proved existence and uniqueness of weak solutions

Developed a probabilistic method using Dirichlet forms and BSDEs

Addressed PDEs with singular coefficients and mixed boundary conditions

Abstract

In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.