A limiting free boundary problem ruled by Aronsson's equation

TL;DR

This paper investigates the limiting behavior of a p-Dirichlet optimal design problem with volume constraints as p approaches infinity, revealing a free boundary problem governed by the infinity-Laplacian and establishing conditions for uniqueness and convergence.

Contribution

It introduces a new limiting free boundary problem governed by the infinity-Laplacian and provides conditions for uniqueness and precise optimal configurations.

Findings

Identifies the limiting free boundary problem as p approaches infinity.

Provides a necessary and sufficient condition for uniqueness.

Establishes convergence of free boundaries.

Abstract

We study the behavior of $p$-Dirichlet optimal design problem with volume constraint for $p$ large. As the limit as $p$ goes to infinity, we find a limiting free boundary problem governed by the infinity-Laplacian operator. We establish a necessary and sufficient condition for uniqueness of the limiting problem and, under such a condition, we determine precisely the optimal configuration for the limiting problem. Finally, we establish convergence results for the free boundaries.