*preprint*

**Inserted:** 1 nov 2023

**Last Updated:** 1 nov 2023

**Year:** 2022

**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.*