Higher Regularity for the Signorini Problem for the Homogeneous, Isotropic Lam\'e System

TL;DR

This paper investigates the higher regularity of solutions and free boundaries in the Signorini problem for the isotropic Lamé system, linking it to the regularity of the obstacle problem for the half-Laplacian.

Contribution

It establishes a connection between the regularity of the Signorini problem for the Lamé system and the obstacle problem for the half-Laplacian, providing new insights into solution smoothness.

Findings

Reduced the regularity question to the half-Laplacian obstacle problem

Linked Lamé system regularity to half-Laplacian obstacle problem

Provided a framework for higher regularity analysis

Abstract

In this note we discuss the (higher) regularity properties of the Signorini problem for the homogeneous, isotropic Lam\'e system. Relying on an observation by Schumann \cite{Schumann1}, we reduce the question of the solution's and the free boundary regularity for the homogeneous, isotropic Lam\'e system to the corresponding regularity properties of the obstacle problem for the half-Laplacian.