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# Interfaces and Free Boundaries

Volume 20, Issue 1, 2018, pp. 1–24
DOI: 10.4171/IFB/395

Published online: 2018-05-03

Fractional elliptic quasi-variational inequalities: Theory and numerics

Harbir Antil[1] and Carlos N. Rautenberg[2]

(1) George Mason University, Fairfax, USA
(2) Humboldt-Universität zu Berlin, Germany

This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order $s \in (0, 1)$, studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter  tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.

Keywords: Quasivariational inequality, QVI, fractional derivatives, fractional diffusion, free boundary problem, Caffarelli–Silvestre and Stinga–Torrea extension, weighted Sobolev spaces, Mosco convergence, fixed point algorithm, finite element method

Antil Harbir, Rautenberg Carlos: Fractional elliptic quasi-variational inequalities: Theory and numerics. Interfaces Free Bound. 20 (2018), 1-24. doi: 10.4171/IFB/395