Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

TL;DR

This paper investigates how the gradient of solutions, representing stress, concentrates around closely spaced inclusions in nonlinear conductivity problems, providing optimal estimates for the blow-up behavior as inclusions approach each other.

Contribution

It establishes optimal $L^ Infty$ estimates for stress concentration in nonlinear, possibly anisotropic media with closely located inclusions, extending understanding of degeneracy in $p$-Laplace type equations.

Findings

Optimal $L^ Infty$ estimates for gradient blow-up as inclusions approach

Analysis applicable to anisotropic and degenerate $p$-Laplace type equations

Quantitative description of stress concentration behavior

Abstract

We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We prove optimal $L^\infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.