Regularity of solutions to nonlinear thin and boundary obstacle problems

TL;DR

This paper establishes $C^1$ regularity for solutions to a broad class of nonlinear variational inequalities with thin obstacles and $C^{1, eta}$ regularity for boundary Signorini problems, advancing understanding in nonlinear obstacle problems.

Contribution

It proves $C^1$ regularity for nonlinear thin obstacle problems and $C^{1, eta}$ regularity for boundary Signorini problems, extending known linear results to nonlinear cases.

Findings

Proves $C^1$ regularity for nonlinear thin obstacle solutions.

Establishes $C^{1, eta}$ regularity for Signorini boundary problems.

Advances the theoretical understanding of nonlinear variational inequalities.

Abstract

Variational inequalities with thin obstacles and Signorini-type boundary conditions are classical problems in the calculus of variations, arising in numerous applications. In the linear case many refined results are known, while in the nonlinear setting our understanding is still at a preliminary stage. In this paper we prove $C^1$ regularity for the solutions to a general class of quasi-linear variational inequalities with thin obstacles and $C^{1, \alpha}$ regularity for variational inequalities under Signorini-type conditions on the boundary of a domain.