Higher order boundary Harnack principles in Dini type domains

Aim of this paper is to provide higher order boundary Harnack principles (De Silva and Savin, 2015) 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), the ratio v/u of two solutions vanishing on a common portion of a regular boundary solves a degenerate elliptic equation whose coefficients behave as u^2 . 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.