Existence and improved regularity for a nonlinear system with collapsing ellipticity

TL;DR

This paper proves the existence of solutions and establishes improved regularity estimates for a complex nonlinear elliptic system combining singular, degenerate, and Poisson equations, advancing understanding of thermistor-like models.

Contribution

It introduces novel existence proofs and regularity results for a nonlinear elliptic system with mixed singular and degenerate features, applicable in various dimensions.

Findings

Existence of weak solutions in any space dimension.

Derivation of $ ext{C}^{1, ext{log-Lip}}$ regularity estimates.

Results are new even for simplified models.

Abstract

We study a nonlinear system made up of an elliptic equation of blended singular/degenerate type and Poisson's equation with a lowly integrable source. We prove the existence of a weak solution in any space dimension and, chiefly, derive an improved $\mathcal{C}^{1,\text{log-Lip}}$-regularity estimate using tangential analysis methods. The system illustrates a sophisticated version of the proverbial thermistor problem and our results are new even in simpler modelling scenarios.