Interacting free boundaries in obstacle problems

TL;DR

This paper analyzes obstacle problems with two interacting free boundaries governed by different diffusion operators, revealing regularity properties, coupling effects, and geometric structure of the free boundaries.

Contribution

It provides a comprehensive analysis of the regularity and structure of interacting free boundaries in obstacle problems, including coupling properties and singularity characterization.

Findings

Free boundary is analytic near regular points of coordinate functions.

Singular points form a smooth manifold.

Uncoupled free boundary points are singular.

Abstract

We study obstacle problems governed by two distinct types of diffusion operators involving interacting free boundaries. We obtain a somewhat surprising coupling property, leading to a comprehensive analysis of the free boundary. More precisely, we show that near regular points of a coordinate function, the free boundary is analytic, whereas singular points lie on a smooth manifold. Additionally, we prove that uncoupled free boundary points are singular, indicating that regular points lie exclusively on the coupled free boundary. Furthermore, optimal regularity, non-degeneracy, and lower dimensional Hausdorff measure estimates are obtained. Explicit examples illustrate the sharpness of assumptions.