Higher Order Boundary Harnack Principle via Degenerate Equations

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type

$$\begin{aligned} -\textrm{div}\left( \rho ^aA\nabla w\right) =\rho ^af+\textrm{div}\left( \rho ^aF\right) \quad \text {in}\; \Omega \end{aligned}$$

for exponents \(a>-1\), where the weight \(\rho \) vanishes with non zero gradient 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 the \(C^{k,\alpha }\)-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 in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). 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.