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 ρ a A ∇ w = ρ a f + div ρ a F in Ω for exponents a > - 1 , where the weight ρ vanishes with non zero gradient on a regular hypersurface Γ , which can be either a part of the boundary of Ω 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 ρ = u , a = 2 and w = v / u ). We prove first the C k , α -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.