No abstract

We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy S(u) := {X : Du(X) = 0}. Our main Theorem reveals that along S(u), u is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set S(u), Du enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that H 1 -weak solutions to div (aij (X)Du) = 0, with aij elliptic and Dini-continuous are actually C 1,1 − along S(u). The results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class C p ′ = C 1, 1 p−1 ; this regularity is optimal.

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F (X, D 2 u) = f (X), based on weakest integrability properties of f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L n norm of f , which corresponds to optimal regularity bounds for the critical threshold case. Optimal C 1,α regularity estimates are delivered when f ∈ L n+ǫ . The limiting upper borderline case, f ∈ L ∞ , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on F , that u ∈ C 1,Log−Lip , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the C 0, n−2ε n−ε norm of u based on the L n−ε norm of f , where ε is the Escauriaza universal constant. The exponent n−2ε n−ε is optimal. When the source function f lies in L q , n > q > n − ε, we also obtain the exact, improved sharp Hölder exponent of continuity.MSC: 35B65, 35J60.

We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, L + and L − , within their positive and negative sets respectively. A corresponding flux balance from one phase to another is also imposed. We establish existence and L ∞ bounds of solutions. We also prove that variational solutions are non-degenerate and develop the regularity theory for solutions of such free boundary problems.