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On a special class of non-local variational problems

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Abstract

Non-local variational problems where non-locality is expressed in a special form are considered. In addition to introducing non-local spaces as the natural ambient spaces for such variational problems, we establish two surprising facts.

  1. (1)

    A way to clearly show the difference with local problems as linear functions are shown not to be minimizers for the Dirichlet, non-local integral under boundary conditions determined by them, in sharp contrast with the classic, local situation.

  2. (2)

    The fact that no convexity whatsoever is required to prove existence of minimizers, again in clear contrast with the local situation.

The former is explored with the help of optimality conditions, which in this non-local context adopt the form of some very special integral equations. The latter is actually valid for scalar and vector problems alike, with a special significance for models of hyper-elasticity where other types of non-locality have been recently shown to be in need of usual convexity conditions for weak lower semicontinuity.

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  1. INEI, U. de Castilla-La Mancha, 13071, Ciudad Real, Spain

    Pablo Pedregal

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  1. Pablo Pedregal
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Pablo Pedregal: Supported by grants PID2020-116207GB-I00, and SBPLY/19/180501/000110.

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Pedregal, P. On a special class of non-local variational problems. Rev Mat Complut 37, 237–251 (2024). https://doi.org/10.1007/s13163-022-00454-x

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