Generic regularity of free boundaries for the thin obstacle problem

2022/09/02 by Xavier Fernández‐Real, Fernández-Real, Xavier, Damià Torres-Latorre +1 · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2209.01063

Abstract

The free boundary for the Signorini problem in ℝⁿ⁺¹ is smooth outside of a degenerate set, which can have the same dimension (n-1) as the free boundary itself. In [FR21] it was shown that generically, the set where the free boundary is not smooth is at most (n-2)-dimensional. Our main result establishes that, in fact, the degenerate set has zero H^n-3-α₀ measure for a generic solution. As a by-product, we obtain that, for n+1 ≤ 4, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in ℝ³ and ℝ⁴ for the thin obstacle problem.

Cited by

• Linearized equation and generic regularity in the Alt-Caffarelli problem

Related

• Generic regularity of free boundaries for the obstacle problem
• The Thin Obstacle Problem: A Survey
• The structure of the free boundary in the fully nonlinear thin obstacle problem
• The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary
• The structure of the free boundary for lower dimensional obstacle problems