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Algebra i Analiz, 2020, Volume 32, Issue 3, Pages 84–126 (Mi aa1701)  

This article is cited in 3 scientific papers (total in 3 papers)

Research Papers

The regular free boundary in the thin obstacle problem for degenerate parabolic equations

A. Banerjeea, D. Daniellib, N. Garofaloc, A. Petrosyanb

a TIFR CAM, Bangalore-560065
b Department of Mathematics, Purdue University, 47907 West Lafayette, IN
c Dipartimento di Ingegneria Civile, Edile e Ambientale (DICEA), Università di Padova, 35131 Padova, ITALY
Full-text PDF (330 kB) Citations (3)
References:
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 $(\partial_t - \Delta_x)^s$ for $s \in (0,1)$. 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 $H^{1+\gamma,\frac{1+\gamma}{2}}$ regularity of the free boundary near such regular points.
Keywords: Signorini complementary conditions, elastostatics, problems with unilateral constraints, fractional heat equation.
Funding agency Grant number
National Science Foundation DMS-1800527
Science and Engineering Research Board Matrix grant MTR/2018/000267
Università di Padova
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.
Received: 16.06.2019
English version:
St. Petersburg Mathematical Journal, 2021, Volume 32, Issue 3, Pages 449–480
DOI: https://doi.org/10.1090/spmj/1656
Document Type: Article
Language: English
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
Citation in format AMSBIB
\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}
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Алгебра и анализ St. Petersburg Mathematical Journal
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    References:36
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