Singular perturbation methods and optimal regularity for degenerate equations.

Abstract

In the first part of this work we prove interior and up to boundary Lipschitz regularity of the viscosity solutions to a singular perturbation problem for a reaction-diffusion equation related to the normalized p-Laplacian equation |∇u | 2−p · div |∇u | p−2 ∇u = β (u ), where the reaction term is of combustion type. We obtain the precise geometric behavior of solutions near -level surfaces, by means of optimal regularity and sharp geometric nondegeneracy. We pass to the limit we investigate Hausdorff measure properties of the limit function. In the second part the aim is to obtain sharp regularity estimates for locally bounded solutions of the degenerate doubly nonlinear equation u t − div(m|u| m−1 |∇u| p−2 ∇u) = f, where m > 1, p > 2 and f ∈ L q,r . More precisely, we show that solutions are locally of class C 0,β , where β depends explicitly only on the optimal H¨older exponent for solutions of the homogeneous case, the integrability of f, the constants p, m and the space dimension n.