16 research outputs found
Lower semicontinuity and relaxation via young measures for nonlocal variational problems and applications to peridynamics
âFirst Published in SIAM Journal of Mathematical Analysis in [50, 1, 2018], published by the Society for Industrial and Applied Mathematics (SIAM)â and the copyright notice as stated in the article itself (e.g., âCopyright © SIAM. Unauthorized reproduction of this article is prohibited"We study nonlocal variational problems in Lp, like those that appear in peridynamics. The functional object of our study is given by a double integral. We establish characterizations of weak lower semicontinuity of the functional in terms of nonlocal versions of either a convexity notion of the integrand or a Jensen inequality for Young measures. Existence results, obtained through the direct method of the calculus of variations, are also established. We cover different boundary conditions, for which the coercivity is obtained from nonlocal PoincarĂ© inequalities. Finally, we analyze the relaxation (that is, the computation of the lower semicontinuous envelope) for this problem when the lower semicontinuity fails. We state a general relaxation result in terms of Young measures and show, by means of two examples, the difficulty of having a relaxation in Lp in an integral form. At the root of this difficulty lies the fact that, contrary to what happens for local functionals, nonpositive integrands may give rise to positive nonlocal functionals.Supported by the Spanish Ministerio de EconomĂa y Competitividad through grants
MTM2011-28198 and RYC-2010-06125 (RamĂłn y Cajal programme), and
the ERC Starting Grant 30717
A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case
In this note we present a sufficient condition ensuring lower semicontinuity
for nonlocal supremal functionals of the type where is a bounded open subset of and .Comment: to appear in European Journal of Mathematic
Relaxation of a scalar nonlocal variational problem with a double-well potential
We consider nonlocal variational problems in Lp, like those that appear in peridynamics, where the functional object of the study is given by a double integral. It is known that convexity of the integrand implies the lower semicontinuity of the functional in the weak topology of Lp. If the integrand is not convex, a usual approach is to compute the relaxation, which is the lower semicontinuous envelope in the weak topology. In this paper we compute such a relaxation for a scalar problem with a double-well integrand. The relaxation is non-trivial, and, contrary to the local case, it cannot be represented as a double integral, as the original problem. Nonetheless, we show that, as for the local case, the relaxation can be expressed in terms of the energy of a suitable truncation of the considered functionThis work has been supported by the Spanish Ministry of Economy and Competitivity through project MTM2017-85934-C3-2-P (C.M.-C.) and project PGC2018-097104-B-100 and Juan de la Cierva Incorporation fellowship IJCI-2015-25084 (A.T.
Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation
Based on recent developments in the theory of fractional Sobolev spaces, an
interesting new class of nonlocal variational problems has emerged in the
literature. These problems, which are the focus of this work, involve integral
functionals that depend on Riesz fractional gradients instead of ordinary
gradients and are considered subject to a complementary-value condition. With
the goal of establishing a comprehensive existence theory, we provide a full
characterization for the weak lower semicontinuity of these functionals under
suitable growth assumptions on the integrands. In doing so, we surprisingly
identify quasiconvexity, which is intrinsic to the standard vectorial calculus
of variations, as the natural notion also in the fractional setting. In the
absence of quasiconvexity, we determine a representation formula for the
corresponding relaxed functionals, obtained via partial quasiconvexification
outside the region where complementary values are prescribed. Thus, in contrast
to classical results, the relaxation process induces a structural change in the
functional, turning the integrand from a homogeneous into an inhomogeneous one.
Our proofs rely crucially on an inherent relation between classical and
fractional gradients, which we extend to Sobolev spaces, enabling us to
transition between the two settings.Comment: 25 page