The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

2019/02/20 by Agnid Banerjee, Donatella Danielli, Banerjee, Agnid +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1902.07457

Revised version

openalex publication_date 2019/02/20 · arxiv created 2019/07/27 · arxiv updated 2019/07/30 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight |y|^a for a ∈ (-1,1). Such problem arises as the local extension of the obstacle problem for the fractional heat operator (∂_t - Δ_x)^s for s ∈ (0,1). Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation (a=0).

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