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arXiv · 2510.07085

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

Tommaso Bertin·Paulin Huguet
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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.

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Tommaso Bertin, Paulin Huguet. 2025-10-08. Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case. https://arxiv.org/abs/2510.07085

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