Aperiodic fractional obstacle problems

Abstract

We determine the asymptotic behaviour of (bilateral) obstacle problems for fractional energies in rather general aperiodic settings via Γ-convergence arguments. As further developments we consider obstacles with random sizes and shapes located on points of standard lattices, and the case of random homothetics obstacles centered on random Delone sets of points.Obstacle problems for non-local energies occur in several physical phenomenona, for which our results provide a description of the first order asympotitc behaviour. IntroductionNon-local energies and operators have been actively investigated over recent years. They arise in problems from different fields, the most celebrated being Signorini's problem in contact mechanics:finding the equilibria of an elastic body partially laying on a surface and acted upon part of its boundary by unilateral shear forces (see [43], [31]). In the anti-plane setting the elastic energy can be then expressed in terms of the seminorm of a H 1/2 function, or equivalently as the boundary trace energy of a W 1,2 displacement.As further examples we mention applications in elasticity, for instance in phase field theories for dislocations (see [34] and the references therein); in heat transfer for optimal control of temperature across a surface [33], [6]; in equilibrium statistical mechanics to model free energies of Ising spin systems with Kac potentials on lattices (see [2] and the references therein); in fluid dynamics to describe flows through semi-permeable membranes [30]; in financial mathematics in pricing models for American options [4]; and in probability in the theory of Markov processes (see [10], [11] and the references therein).Many efforts have been done to extend the existing theories for (fully non-linear) second order elliptic equations to non-local equations. Regularity has been developed for integro-differential operators[10], [11], [22], and for obstacle problems for the fractional laplacian (see [20], [44] and the references therein). Periodic homogenization has been studied for a quite general class of non-linear, non-local uniformly elliptic equations [41] and for obstacle problems for the fractional laplacian [19], [32].

The obstacle problem for nonlinear integro-differential operators

“…since the first term is nonpositive by (16), whereas in the second term, L(v(x), v(y)) < L(u(x), u(y)) and u(x) > v(x). Consequently, |{u > v}| = 0.…”

Hölder continuity up to the boundary for a class of fractional obstacle problems

“…The obstacle problem involving the fractional Laplacian operator indeed appears in many contexts, such as in the analysis of anomalous diffusion, in the quasi-geostrophic flow problem, and in pricing of American options regulated by assets evolving in relation to jump processes; in particular, this important application in Financial Mathematics made the obstacle problem very important in recent times. A large treatment of the fractional obstacle problem can be found in the important papers by Caffarelli, Figalli, Salsa, and Silvestre (see, e. g., [1][2][3]30]); see also [10] for the analysis of families of bilateral obstacle problems involving fractional type energies in aperiodic settings; and the paper [26] for the fractional obstacle problems with drift. However, despite its relatively short history, this problem has already evolved into an elaborate theory with several connections to other branches; the literature is too wide to attempt any reasonable comprehensive treatment in a single paper.…”

Γ‐convergence: a tool to investigate physical phenomena across scales

“…The interest in the problem is motivated by several applications in different fields, running from the classical Signorini's problem in contact mechanics and diffusion through semipermeable membranes (corresponding to the case s = 1 ∕ 2, see ), to stock option pricing models in Finance (see the papers for an overview).…”