Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian

AbstractWe consider the SQG equation with dissipation given by a fractional Laplacian of order $$\alpha <\frac{1}{2}$$ α < 1 2 . We introduce a notion of suitable weak solution, which exists for every $$L^2$$ L 2 initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most $$\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) $$ 1 2 α 1 + α α ( 1 - 2 α ) + 2 .

Download Full-text

Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian

Nonlinear Analysis ◽

10.1016/j.na.2021.112473 ◽

2021 ◽

Vol 211 ◽

pp. 112473

Author(s):

Simon Eberle ◽

Xavier Ros-Oton ◽

Georg S. Weiss

Keyword(s):

Obstacle Problem ◽

Fractional Laplacian ◽

Thin Obstacle

Download Full-text

Finite element approximation of an obstacle problem for a class of integro–differential operators

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019058 ◽

2020 ◽

Vol 54 (1) ◽

pp. 229-253 ◽

Cited By ~ 1

Author(s):

Andrea Bonito ◽

Wenyu Lei ◽

Abner J. Salgado

Keyword(s):

Finite Element ◽

Obstacle Problem ◽

Fractional Laplacian ◽

Differential Operators ◽

Finite Element Approximation ◽

Element Approximation ◽

Finite Element Scheme ◽

Order Elliptic Operator ◽

Nonlocal Part ◽

Differential Part

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

Download Full-text

Existence of Solutions for Implicit Obstacle Problems of Fractional Laplacian Type Involving Set-Valued Operators

Journal of Optimization Theory and Applications ◽

10.1007/s10957-020-01752-4 ◽

2020 ◽

Vol 187 (2) ◽

pp. 391-407

Author(s):

Dumitru Motreanu ◽

Van Thien Nguyen ◽

Shengda Zeng

Keyword(s):

Fixed Point ◽

Existence Theorem ◽

Nonsmooth Analysis ◽

Obstacle Problem ◽

Fractional Laplacian ◽

Existence Of Solutions ◽

Obstacle Problems ◽

Generalized Gradient ◽

Fixed Point Principle ◽

Set Valued Mappings

Abstract The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis.

Download Full-text

Convergence rates for the classical, thin and fractional elliptic obstacle problems

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2014.0449 ◽

2015 ◽

Vol 373 (2050) ◽

pp. 20140449 ◽

Cited By ~ 10

Author(s):

Ricardo H. Nochetto ◽

Enrique Otárola ◽

Abner J. Salgado

Keyword(s):

Obstacle Problem ◽

Fractional Laplacian ◽

Convergence Rates ◽

Finite Element Approximation ◽

Obstacle Problems ◽

Element Approximation ◽

Optimal Error ◽

Optimal Convergence Rates ◽

Regularity Results ◽

Thin Obstacle

We review the finite-element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results, we present an optimal error analysis for the thin obstacle problem. Finally, we discuss the localization of the obstacle problem for the fractional Laplacian and prove quasi-optimal convergence rates.

Download Full-text