An Elliptic Free Boundary Arising From the Jump of Conductivity

TL;DR

This paper studies a quasilinear elliptic PDE with a jump in conductivity across a level surface of the solution, establishing Lipschitz regularity and introducing a novel ACF-monotonicity formula involving two operators.

Contribution

It introduces a new ACF-monotonicity formula for elliptic PDEs with jumps in conductivity, advancing understanding of solution regularity and level surface behavior.

Findings

Proved Lipschitz regularity for solutions.

Analyzed regularity of level surfaces with conductivity jumps.

Developed a new ACF-monotonicity formula involving two operators.

Abstract

In this paper we consider a quasilinear elliptic PDE, $\text{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns Lipschitz regularity for the solution $u$, and the regularity of the level surfaces, where $A(x,u)$ has a jump and the solution $u$ does not degenerate. In proving Lipschitz regularity of solutions, we introduce a new and unexpected type of ACF-monotonicity formula with two different operators, that might be of independent interest, and surely can be applied in other related situations. The proof of the monotonicity formula is done through careful computations, and (as a byproduct) a slight generalization to a specific type of variable matrix-valued conductivity is presented.