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In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{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 $$H^{1+\gamma,\frac{1+\gamma}{2}}$$ regularity of the free boundary near such regular points.
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Regularity of solutions to space–time fractional wave equations: A PDE approach
Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.
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- Award ID(s):
- 1720213
- PAR ID:
- 10093769
- Date Published:
- Journal Name:
- Fractional Calculus and Applied Analysis
- Volume:
- 21
- Issue:
- 5
- ISSN:
- 1311-0454
- Page Range / eLocation ID:
- 1262 to 1293
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/fca-2018-0067
