A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients

TL;DR

This paper introduces a probabilistic method to establish the existence and uniqueness of solutions for Dirichlet boundary value problems in semilinear elliptic PDEs with singular coefficients, leveraging stochastic processes and backward SDEs.

Contribution

It develops a novel probabilistic framework for solving semilinear elliptic PDEs with singular coefficients, extending existing methods to more general cases.

Findings

Proved existence and uniqueness of solutions for a broad class of PDEs.

Utilized Dirichlet processes and backward stochastic differential equations.

Established a new probabilistic approach for PDE boundary value problems.

Abstract

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.