Lower Semicontinuity and Relaxation Via Young Measures for Nonlocal Variational Problems and Applications to Peridynamics
| Autor: | Carlos Mora-Corral, José C. Bellido |
|---|---|
| Přispěvatelé: | UAM. Departamento de Matemáticas |
| Rok vydání: | 2018 |
| Předmět: |
Relaxation
Peridynamics Matemáticas Applied Mathematics Multiple integral 010102 general mathematics Mathematical analysis State (functional analysis) 01 natural sciences Convexity 010101 applied mathematics Computational Mathematics Cover (topology) Young measures Nonlocal variational problems Boundary value problem Relaxation (approximation) 0101 mathematics Lower semicontinuity Jensen's inequality Analysis Mathematics |
| Zdroj: | Biblos-e Archivo. Repositorio Institucional de la UAM instname |
| ISSN: | 1095-7154 0036-1410 |
| DOI: | 10.1137/17m1114181 |
| Popis: | “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 307179 |
| Databáze: | OpenAIRE |
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