Search for: All records | NSF Public Access

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

Almost minimizers for the thin obstacle problem

https://doi.org/10.1007/s00526-021-01986-8

Jeon, Seongmin ; Petrosyan, Arshak ( August 2021 , Calculus of Variations and Partial Differential Equations)
null (Ed.)
Full Text Available
The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

https://doi.org/10.1007/s00526-021-01938-2

Banerjee, Agnid ; Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ( April 2021 , Calculus of Variations and Partial Differential Equations)

Abstract
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$ ${| y |}^{a}$ for$$a \in (-1,1)$$ $a \in (- 1, 1)$ . Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$ ${(\partial_{t} - Δ_{x})}^{s}$ for$$s \in (0,1)$$ $s \in (0, 1)$ . 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 ($$a=0$$ $a = 0$ ).

more » « less
Almost minimizers for certain fractional variational problems

Jeon, Seongmin ; Petrosyan, Arshak ( July 2020 , Algebra i analiz)

In this paper we introduce a notion of almost minimizers for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. In particular, we study almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle. We show that for a certain range of parameters, almost minimizers are almost Lipschitz or C1,β-regular.
more » « less
Full Text Available