Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case

Tommaso Bertin; Paulin Huguet

Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case

Analysis of PDEs 2025-10-09 v1 Functional Analysis

Authors: Tommaso Bertin , Paulin Huguet

Abstract

We study integral functionals defined on scalar Sobolev spaces of the form $E[f]:u\mapsto \int_\Omega f(x,u(x),\nabla u(x)) d x,$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev phenomenon. We determine a formulation of the lower semicontinuous envelope of $E[f]$ with respect to various topologies and with fixed Lipschitz Dirichlet boundary conditions.

Keywords

sobolev spaces convex analysis function spaces and inequalities

Cite

@article{arxiv.2510.07085,
  title  = {Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case},
  author = {Tommaso Bertin and Paulin Huguet},
  journal= {arXiv preprint arXiv:2510.07085},
  year   = {2025}
}

Comments

39 pages, 1 figure

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