Higher order boundary Harnack principles in Dini type domains

Aim of this paper is to provide higher order boundary Harnack principles [De Silva-Savin 15] for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in [Terracini-Tortone-Vita 22], the ratio $v\mathrm{/}u$ of two solutions vanishing on a common portion $\mathrm{\Gamma }$ of a regular boundary solves a degenerate elliptic equation whose coefficients behave as $u^2$ at $\mathrm{\Gamma }$. Hence, for any $k\ge 1$ we provide $C^k$ estimates for solutions to the auxiliary degenerate equation under double Dini conditions, actually for general powers of the weight $a>-1$, and we imply $C^k$ estimates for the ratio $v\mathrm{/}u$ under triple Dini conditions, as a corollary in the case $a=2$.