Interior gradient and Hessian estimates for the Dirichlet problem of semi-linear degenerate elliptic systems: a probabilistic approach

TL;DR

This paper develops probabilistic methods to derive interior gradient and Hessian estimates for semi-linear degenerate elliptic PDE systems, advancing understanding of their regularity properties.

Contribution

It introduces a novel probabilistic approach combining backward stochastic differential equations and quasi-derivatives for estimating derivatives of degenerate elliptic systems.

Findings

Established interior gradient estimates for semi-linear degenerate elliptic systems

Derived Hessian bounds using stochastic techniques

Enhanced regularity understanding of degenerate elliptic PDEs

Abstract

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.