A miscellanea of qualitative and symmetry properties of the solutions to the two-phase Serrin's problem

TL;DR

This paper explores the symmetry and geometric properties of solutions to the two-phase Serrin's problem, revealing that only concentric balls satisfy the conditions for distinct conductivities, with no local minimizers existing.

Contribution

It proves the uniqueness of concentric balls as solutions and rules out the existence of local minimizers and certain boundary features in the problem.

Findings

Only concentric balls solve the two-phase Serrin's problem for distinct conductivities.

Solutions do not have flat boundary parts or tentacle-like extensions.

No local minimizers exist for the shape optimization problem.

Abstract

This paper investigates the solutions to the two-phase Serrin's problem, an overdetermined boundary value problem motivated by shape optimization. Specifically, we study the torsional rigidity of composite beams, where two distinct materials interact, and examine the properties of the optimal configurations (critical shapes) under volume constraints. We first show that such a shape optimization problem admits no local minimizers. Then, using the method of moving planes, we show that the solutions exhibit no extended or narrow branches ("tentacles") away from the core. We then show that the outer boundary of a solution cannot exhibit flat parts and that the only configuration whose outer boundary contains a portion of a sphere is the one given by concentric balls. Finally, we establish that concentric balls are the only admissible configurations that solve the two-phase Serrin's problem for two distinct sets of conductivity values.