Some flows in shape optimization

TL;DR

This paper studies geometric flows in shape optimization involving curvature and nonlocal terms, introducing generalized solutions inspired by viscosity solutions, and proves key properties including existence, uniqueness, stability, and asymptotic convergence.

Contribution

It introduces generalized set solutions for shape flows with curvature and nonlocal terms, establishing fundamental properties and asymptotic behavior.

Findings

Proved inclusion preservation for generalized solutions.

Established existence, uniqueness, and stability of solutions.

Demonstrated convergence to a Bernoulli free boundary problem.

Abstract

Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele-Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed: we prove that the solutions converge to a generalized Bernoulli exterior free boundary problem.