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 -div (rho(a)A del w) = rho(a)f + div (rho F-a) in Omega 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,C-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