Higher order boundary Harnack principle via degenerate equations

TL;DR

This paper establishes higher order Schauder estimates for degenerate elliptic equations and extends boundary Harnack principles to ratios of solutions, including across singular sets, using conformal mapping in two dimensions.

Contribution

It introduces higher order regularity results for solutions to degenerate elliptic equations and extends boundary Harnack principles to ratios of solutions across singular sets.

Findings

Proved higher order Schauder estimates for degenerate elliptic equations.

Extended boundary Harnack principle to ratios of solutions near nodal sets.

Provided gradient estimates across singular sets in two dimensions.

Abstract

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \] for exponents $a>-1$, where the weight $\rho$ vanishes in a non degenerate manner on a regular hypersurface $\Gamma$ which can be either a part of the boundary of $\Omega$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $\rho=u$, $a=2$ and $w=v/u$). We prove first $C^{k,\alpha}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set.