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Relaxation of a scalar nonlocal variational problem with a double-well potential

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Abstract

We consider nonlocal variational problems in \(L^p\), 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 \(L^p\). 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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Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Carlos Mora-Corral

  2. Departamento de Matemática Aplicada a la Ingeniería Industrial, Escuela Técnica Superior de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, 28012, Madrid, Spain

    Andrea Tellini

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  1. Carlos Mora-Corral
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Correspondence to Andrea Tellini.

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Communicated by J. Ball.

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This 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.).

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Mora-Corral, C., Tellini, A. Relaxation of a scalar nonlocal variational problem with a double-well potential. Calc. Var. 59, 67 (2020). https://doi.org/10.1007/s00526-020-1728-4

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