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...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2020 |
| Country: | España |
| Institution: | Universidad Autónoma de Madrid |
| Repository: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Language: | English |
| OAI Identifier: | oai:repositorio.uam.es:10486/698560 |
| Online Access: | http://hdl.handle.net/10486/698560 https://dx.doi.org/10.1007/s00526-020-1728-4 |
| Access Level: | Open access |
| Keyword: | Relaxation Nonlocal problems Optimality conditions Young Measures Double-well potential Matemáticas |
| Summary: | 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 function |
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