Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations

TL;DR

This paper proves Schauder estimates and regularity results for solutions to boundary-degenerate elliptic PDEs, which are important in finance, biology, and porous media modeling.

Contribution

It establishes new Schauder a priori estimates and regularity results for a class of boundary-degenerate elliptic PDEs, including boundary regularity up to degenerate regions.

Findings

Regularity of solutions up to degenerate boundary

Schauder estimates for boundary-degenerate operators

Applications to finance, biology, and porous media

Abstract

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical biology, and the study of porous media.