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On the nonlinear thin obstacle problem

Anna Abbatiello, Giovanna Andreucci, Emanuele Spadaro

Abstract

The thin obstacle problem or $n$-dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. 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 nonlineraities and we prove the optimal $C^{1, \frac{1}{2}}$-regularity of the solutions in any space dimension.

On the nonlinear thin obstacle problem

Abstract

The thin obstacle problem or -dimensional Signorini problem is a classical variational problem arising in several applications, starting with its first introduction in elasticity theory. 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 nonlineraities and we prove the optimal -regularity of the solutions in any space dimension.
Paper Structure (10 sections, 129 equations)

This paper contains 10 sections, 129 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Main result: Optimal $C^{1, \frac{1}{2}}$ regularity
  4. Frequency function
  5. Schauder estimates
  6. Frequency function
  7. Boundedness of the frequency.
  8. Quasi-monotonicity of the frequency.
  9. Blowup sequence
  10. Proof of the optimal $C^{1, \frac{1}{2}}$ regularity