|
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
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.
Received: 16.06.2019
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
Linking options:
https://www.mathnet.ru/eng/aa1701 https://www.mathnet.ru/eng/aa/v32/i3/p84
|
Statistics & downloads: |
Abstract page: | 187 | Full-text PDF : | 13 | References: | 36 | First page: | 8 |
|