Higher order boundary Harnack principles in Dini type domains

Abstract

Aim of this paper is to provide higher order boundary Harnack principles (De Silva and Savin, 2015 [13]) for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in Terracini et al. (2024) [36], the ratio v/u of two solutions vanishing on a common portion Γ of a regular boundary solves a degenerate elliptic equation whose coefficients behave as u2 at Γ. Hence, for any k≥1 we provide Ck estimates for solutions to the auxiliary degenerate equation under double Dini conditions, actually for general powers of the weight a>−1, and we imply Ck estimates for the ratio v/u under triple Dini conditions, as a corollary in the case a=2.