# Shape derivative of the Dirichlet energy for a transmission problem

## TL;DR

This paper computes the shape derivative of the Dirichlet energy in a transmission problem with irregular domain boundaries, and applies it to prove the existence of solutions in free boundary electrostatic actuator models.

## Contribution

It provides a rigorous computation of the shape derivative for a complex transmission problem with weak regularity and applies it to a novel free boundary problem in electrostatics.

## Key findings

- Shape derivative is well-defined despite domain irregularities.
- Application to existence of solutions in free boundary transmission problems.
- Addresses challenges from non-smooth interfaces in shape calculus.

## Abstract

For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed. The main difficulties in this context arise from the weak regularity of the domain and the possible non-empty intersection of the graph of u and the transmission interface. The result is applied to establish the existence of a solution to a free boundary transmission problem for an electrostatic actuator.