A two phase boundary obstacle-type problem for the bi-Laplacian

TL;DR

This paper investigates a two-phase boundary obstacle problem for the bi-Laplacian in the upper unit ball, establishing well-posedness, optimal regularity, and analyzing free boundary structure using monotonicity formulas.

Contribution

It introduces a novel two-phase boundary obstacle problem for the bi-Laplacian, extending to an obstacle problem for the fractional Laplacian (-Δ)^{3/2} and provides new regularity and free boundary results.

Findings

Proved well-posedness of the problem

Established optimal regularity of solutions

Analyzed free boundary structure using monotonicity formulas

Abstract

In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian $(-\Delta )^{3/2}$, as first observed in \cite{Y}. We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type.