Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911198504550400 |
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| author | Bertin, Tommaso Huguet, Paulin |
| author_facet | Bertin, Tommaso Huguet, Paulin |
| contents | We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_Ωf(x,u(x),\nabla u(x)) d x,$$
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07085 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case Bertin, Tommaso Huguet, Paulin Analysis of PDEs Functional Analysis 49-XX We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_Ωf(x,u(x),\nabla u(x)) d x,$$ 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. |
| title | Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case |
| topic | Analysis of PDEs Functional Analysis 49-XX |
| url | https://arxiv.org/abs/2510.07085 |