Optimal regularity for the Signorini problem

TL;DR

This paper establishes the optimal regularity $C^{1,1/2}$ for solutions to the Signorini problem in any dimension, extending previous results to more general obstacles and elliptic operators.

Contribution

It proves the optimal regularity for the Signorini problem with obstacles on $C^{1,eta}$ hypersurfaces and for general linear elliptic operators, broadening the scope of prior results.

Findings

Solutions achieve $C^{1,1/2}$ regularity.

Regularity holds for obstacles on $C^{1,eta}$ hypersurfaces, $eta>1/2$.

Results apply to any linear elliptic divergence form operator with smooth coefficients.

Abstract

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity $C^{1,1/2}$. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary $C^{1,\beta}$ hypersurface, $\beta>1/2$, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren's monotonicity formula and the optimal regularity of global solutions.