Free Transmission Problems

Abstract

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.

A proof of the Cp′-regularity conjecture in the plane

“…This will allow us to frame the C p ′ conjecture into the formalism of the so called geometric tangential analysis, e.g. [7], [2,1] and [17,18,19,20,21,22].…”

A free boundary optimization problem for the ∞-Laplacian

“…As a consequence of Theorem 4.1 and the Arzelà-Ascoli theorem we obtain the next result. Observe that we can fix p > n, bound the W 1,p -norm by energy considerations and directly obtain the Hölder continuity by embedding (see [4,10,11]). Corollary 4.2.…”

Sharp regularity estimates for quasilinear evolution equations

“…Although weak solutions of (1.1) under the compatibility assumptions (C) are known to be locally of the class C 1+α (in the parabolic sense) for some α ∈ (0, 1), the sharp exponent is known only for some specific cases (see [5,6,26,28,33]). This type of quantitative information plays an essential role in the study of blow-up analysis, related geometric and free boundary problems and for proving Liouville type results (see [3,4,12,16,18,40] for some enlightening examples).…”

“…weak solutions of (1.1) are of class C α for some α ∈ (0, 1). Using compactness and geometric tangential methods (see [3,4,11,12,14,19,36,37]) and intrinsic scaling techniques (see [16,20,24,41] In this work we will solve it in the second scenario. More precisely, our main result reveals that bounded week solutions of (1.1) are locally of the class C 1+α (in the parabolic sense) in the critical zone (i.e.…”