On the nonlinear thin obstacle problem

Abstract

The thin obstacle problem, or $n$-dimensional Signorini problem, is a classical variational problem with roots in elasticity theory and wide-ranging applications. The vast literature concerns mostly quadratic energies, whereas only partial results have been proved in the nonlinear case. In this paper, we consider the thin boundary obstacle problem for a general class of nonlinearities and we prove the optimal $C^{1,1/2}$-regularity of the solutions in any space dimension.