Regularity results for a nonlinear elliptic-parabolic system with oscillating coefficients

TL;DR

This paper establishes regularity results for a nonlinear elliptic-parabolic system modeling electrical heating, allowing for highly oscillating conductivity coefficients that can degenerate to zero, by deriving uniform bounds and avoiding traditional regularity assumptions.

Contribution

It introduces a novel approach to prove regularity for the thermistor problem with oscillating coefficients without requiring the elliptic coefficient to be an A2 weight.

Findings

Established existence of weak solutions with bounded gradients

Derived uniform exponential bounds for the solution u

Achieved regularity results without assuming nearly bounded log-conductivity

Abstract

In this paper we study the initial boundary value problem for the system $\mbox{div}(\sigma(u)\nabla\varphi)=0,\ \ u_t-\Delta u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem which models the electrical heating of conductors. Our assumptions on $\sigma(u)$ leave open the possibility that $\liminf_{u\rightarrow\infty}\sigma(u)=0$, while $\limsup_{u\rightarrow\infty}\sigma(u)$ is large. This means that $\sigma(u)$ can oscillate wildly between $0$ and a large positive number as $u\rightarrow \infty$. Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations. We obtain a weak solution $(u, \varphi)$ with $|\nabla \varphi|, |\nabla u|\in L^\infty$ by first establishing a uniform upper bound for $e^{\varepsilon u}$ for some small $\varepsilon$. This leads to an inequality in $\nabla\varphi$, from whence follows the regularity result. This approach enables us to avoid first proving the H\"{o}lder continuity of $\varphi$ in the space variables, which would have required that the elliptic coefficient $\sigma(u)$ be an $A_2$ weight. As it is known, the latter implies that $\ln\sigma(u)$ is "nearly bounded".