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Maximum Principles for vectorial approximate minimizers of nonconvex functionals

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Abstract

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler–Lagrange system of PDE.

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Authors and Affiliations

  1. BCAM—Basque Center for Applied Mathematics, Mazarredo 14, Bilbao, 48009, Spain

    Nikolaos I. Katzourakis

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  1. Nikolaos I. Katzourakis
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Correspondence to Nikolaos I. Katzourakis.

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Communicated by L. Ambrosio.

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Katzourakis, N.I. Maximum Principles for vectorial approximate minimizers of nonconvex functionals. Calc. Var. 46, 505–522 (2013). https://doi.org/10.1007/s00526-012-0491-6

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  • DOI: https://doi.org/10.1007/s00526-012-0491-6

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