The Neumann Boundary Problem for Elliptic Partial Differential Equations with Nonlinear Divergence Terms

TL;DR

This paper establishes existence and uniqueness of weak solutions for a Neumann boundary problem involving elliptic PDEs with nonlinear divergence terms, using probabilistic methods via backward stochastic differential equations.

Contribution

It introduces a novel probabilistic approach to handle elliptic PDEs with singular divergence terms in Neumann boundary problems.

Findings

Proved existence of weak solutions under singular divergence conditions

Established uniqueness of solutions in the weak sense

Linked PDE solutions to backward stochastic differential equations

Abstract

We prove the existence and uniqueness of weak solution of a Neumann boundary problem for an elliptic partial differential equation (PDE for short) with a singular divergence term which can only be understood in a weak sense. A probabilistic approach is applied by studying the backward stochastic differential equation (BSDE for short) corresponding to the PDE.