On a weighted two-phase boundary obstacle problem

TL;DR

This paper studies a fractional Laplacian-based two-phase obstacle problem, classifying vanishing orders, establishing unique continuation, and analyzing the structure and size of the nodal set using advanced geometric measure theory tools.

Contribution

It introduces a classification of vanishing orders and a stratification of the nodal set for a fractional obstacle problem, employing novel monotonicity formulas and geometric measure theory techniques.

Findings

Classification of vanishing orders for solutions.

Proof of strong unique continuation property.

Hausdorff dimension estimates for the nodal set.

Abstract

In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up analysis, we provide a classification of the possible vanishing orders, which implies the strong unique continuation property. Moreover, we prove a stratification result for the nodal set, together with estimates on its Hausdorff dimensions, for both the regular and the singular part. The main tools come from geometric measure theory and amount to Whitney's Extension Theorem and Federer's Reduction Principle.