Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent

Braides, Andrea; Sigalotti, Laura

doi:10.1016/j.crma.2008.01.010

Cited by 1 publication

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“…If ps > n points have positive capacity and W s,p is embedded into C 0 . In this case it is well known that the homogenized obstacle problem turns out to be trivial, this is the reason why we required ps < n. The borderline case ps = n deserves an analysis similar to that we will perform but different in some details, so that its study is not dealt with in this paper (see for instance [24] and [17] in the local framework).…”

Section: Fractional Capacitiesmentioning

confidence: 99%

Aperiodic fractional obstacle problems

Focardi¹

2010

Advances in Mathematics

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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].

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Section: Fractional Capacitiesmentioning

confidence: 99%

Aperiodic fractional obstacle problems

Focardi¹

2010

Advances in Mathematics

View full text Add to dashboard Cite

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Asymptotic analysis of periodically-perforated nonlinear media at and close to the critical exponent

Cited by 1 publication

References 17 publications

Aperiodic fractional obstacle problems

Aperiodic fractional obstacle problems

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