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

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 C∞-smooth source function, we prove C∞-regularity of solutions up to the portion of the boundary where the operator is degenerate. Boundary-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 [10,25], generators of diffusion processes arising in mathematical biology [3,11], and the study of porous media [7,8].