Gradient integrability and rigidity results for two-phase conductivities in dimension two

TL;DR

This paper investigates the optimal higher gradient integrability for two-phase conductivity problems in two dimensions, identifying worst-case geometries and exact solutions that attain these bounds.

Contribution

It determines the optimal integrability exponent for gradients in two-phase conductivities and characterizes the worst-case configurations among fixed ellipticity.

Findings

Identified the optimal integrability exponent for gradient fields.

Characterized the microgeometries that achieve worst-case integrability.

Proved that exact solutions attain the optimal integrability bounds.

Abstract

This paper deals with higher gradient integrability for $\sigma$-harmonic functions $u$ with discontinuous coefficients $\sigma$, i.e. weak solutions of $\div(\sigma \nabla u) = 0$. We focus on two-phase conductivities, and study the higher integrability of the corresponding gradient field $|\nabla u|$. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement. We find the optimal integrability exponent of the gradient field corresponding to any pair $\{\sigma_1,\sigma_2\}$ of positive definite matrices, i.e., the worst among all possible microgeometries. We also show that it is attained by so-called exact solutions of the corresponding PDE. Furthermore, among all two-phase conductivities with fixed ellipticity, we characterize those that correspond to the worse integrability.