Free transmission problems

TL;DR

This paper investigates free boundary transmission problems involving different PDEs in separate phases, establishing existence, bounds, non-degeneracy, and regularity of solutions across random media interfaces.

Contribution

It introduces a comprehensive analysis of transmission problems with free interfaces, including existence, bounds, non-degeneracy, and regularity results for solutions.

Findings

Existence and $L^{ abla}$ bounds of solutions established.

Solutions are proven to be non-degenerate.

Regularity theory for free boundary problems developed.

Abstract

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