An overdetermined problem with non constant boundary condition

TL;DR

This paper studies an overdetermined torsion problem with a non-constant boundary condition, establishing existence, regularity, and geometric properties of solutions through shape optimization techniques.

Contribution

It introduces a novel overdetermined problem with non-constant boundary conditions and proves existence, regularity, and geometric characteristics of solutions.

Findings

Existence of solutions established

Regularity results proven

Geometric properties characterized

Abstract

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and regularity of a solution. Moreover several geometric properties of the solution are shown.