A transmission problem on a polygonal partition: regularity and shape differentiability

TL;DR

This paper investigates the regularity and shape differentiability of solutions to a transmission problem on polygonal partitions, providing explicit formulas for shape derivatives and analyzing gradient behavior near vertices.

Contribution

It establishes the exact behavior of solutions' gradients near vertices and proves shape differentiability with explicit formulas for the shape derivative.

Findings

Gradient behavior near vertices characterized

Shape differentiability proven with explicit formulas

Applicable to two-dimensional conductivity problems

Abstract

We consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes of the partition. This allows to prove shape differentiability of solutions and to establish an explicit formula for the shape derivative.