Aproximación de homeomorfismos continuos Hölder por homeomorfismos afines a trozos

En esta comunicación tratamos el problema de la aproximación de homeomorfismos por homeomorfismos afines a trozos. El resultado principal es el siguiente: cualquier homeomorfismo continuo H¨older de exponente α ∈ (0, 1] definido en un dominio de R2 con frontera poligonal, y cuyo inverso tambi´en es continuo H¨older de exponente α, puede ser aproximado en la norma H¨older de exponente β, para un cierto β < α, por homeomorfismos afines a trozos sobre triangulaciones

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.

Lower semicontinuity and relaxation via young measures for nonlocal variational problems and applications to peridynamics

“First Published in SIAM Journal of Mathematical Analysis in [50, 1, 2018], published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © 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 Poincaré 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 Economía y Competitividad through grants MTM2011-28198 and RYC-2010-06125 (Ramón y Cajal programme), and the ERC Starting Grant 30717

Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings

In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our motivation is to develop the proper functional analysis framework in order to tackle nonlocal models in Continuum Mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s-fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the Lp norms of the function and its nonlocal gradient. Among the results showed in this investigation we highlight a nonlocal version of the Fundamental Theorem of Calculus (namely, a representation formula where a function can be recovered from its nonlocal gradient), which allows us to prove inequalities in the spirit of Poincar\'e, Morrey, Trudinger and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this nonlocal situation are also established, and those can be viewed as a new class of nonlocal partial differential equations in bounded domains