Abstract:
In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator (∂t−Δx)s for s∈(0,1). Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as the H1+γ,1+γ2 regularity of the free boundary near such regular points.
The first author was supported in part by SERB Matrix grant MTR/2018/000267.
The third author was supported in part by a Progetto SID (Investimento Strategico di Dipartimento) “Non-local operators in geometry and in free boundary problems, and their connection with the applied sciences,” University of Padova, 2017.
The fourth author was supported in part by NSF Grant DMS-1800527.
Citation:
A. Banerjee, D. Danielli, N. Garofalo, A. Petrosyan, “The regular free boundary in the thin obstacle problem for degenerate parabolic equations”, Algebra i Analiz, 32:3 (2020), 84–126; St. Petersburg Math. J., 32:3 (2021), 449–480
\Bibitem{BanDanGar20}
\by A.~Banerjee, D.~Danielli, N.~Garofalo, A.~Petrosyan
\paper The regular free boundary in the thin obstacle problem for degenerate parabolic equations
\jour Algebra i Analiz
\yr 2020
\vol 32
\issue 3
\pages 84--126
\mathnet{http://mi.mathnet.ru/aa1701}
\transl
\jour St. Petersburg Math. J.
\yr 2021
\vol 32
\issue 3
\pages 449--480
\crossref{https://doi.org/10.1090/spmj/1656}
Linking options:
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https://www.mathnet.ru/eng/aa/v32/i3/p84
This publication is cited in the following 3 articles:
Alessandro Audrito, Susanna Terracini, “On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations”, Memoirs of the AMS, 301:1512 (2024)
Agnid Banerjee, Nicola Garofalo, “On the space-like analyticity in the extension problem for nonlocal parabolic equations”, Proc. Amer. Math. Soc., 151:3 (2022), 1235
Agnid Banerjee, Donatella Danielli, Nicola Garofalo, Arshak Petrosyan, “The structure of the singular set in the thin obstacle problem for degenerate parabolic equations”, Calc. Var., 60:3 (2021)