Almost Minimizers for the Thin Obstacle Problems

In Chapter 1, we consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their C^1,β regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the C^1,γ regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperimetric inequalities for the solutions of the thin obstacle problem. This chapter is based on a joint work with Arshak Petrosyan [1].

In Chapter 2, we study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their C^1,β regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients (Chapter 1). This chapter is based on recent joint work with Arshak Petrosyan and Mariana Smit Vega Garcia [2].

In Chapter 3, we introduce a notion of almost minimizers for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. In particular, we study almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle. We show that for a certain range of parameters, almost minimizers are almost Lipschitz or C^1,β-regular. This is based on a work in collaboration with Arshak Petrosyan [3].