Optimal regularity of the thin obstacle problem by an epiperimetric inequality

2023/07/24 by Matteo Carducci, Carducci, Matteo
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2307.12658

openalex publication_date 2023/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The key point to prove the optimal C^1,\frac12 regularity of the thin obstacle problem is that the frequency at a point of the free boundary x₀∈Γ(u), say N^x₀(0⁺,u), satisfies the lower bound N^x₀(0⁺,u)≥\frac32. In this paper we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies W_\frac32. It allows to say that there are not λ-homogeneous global solutions with λ∈ (1,\frac32), and by this frequancy gap, we obtain the desired lower bound, thus a new self contained proof of the optimal regularity.

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