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

2025/10/08 by Bertin, Tommaso, Huguet, Paulin
#49-XX #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2510.07085

Abstract

We study integral functionals defined on scalar Sobolev spaces of the form E[f]:u↦ ∫_Ωf(x,u(x),∇ 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.

Citations

• Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians
• Scalar minimizers with maximal singular sets and lack of Meyers property
• Non-occurrence of gap for one-dimensional non autonomous functionals
• New Examples on Lavrentiev Gap Using Fractals

Related

• Relaxation of one-dimensional nonlocal supremal functionals in the Sobolev setting
• Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians
• On the relaxation of unbounded multiple integrals
• Relaxation for highly discontinuous, possibly unbounded, integral functionals
• A simplified counterexample to the integral representation of the relaxation of double integrals