Quasiconvex bulk and surface energies with subquadratic growth

Menita Carozza; Luca Esposito; Lorenzo Lamberti; Menita Carozza; Luca Esposito; Lorenzo Lamberti

doi:10.3934/mine.2025011

Mathematics in Engineering

2025, Volume 7, Issue 3: 228-263. doi: 10.3934/mine.2025011

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Quasiconvex bulk and surface energies with subquadratic growth

1.
Dipartimento di Ingegneria, Università del Sannio, Corso Garibaldi 107, Benevento, 82100, Italy
2.
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo Ⅱ 132, Fisciano 84084, Italy
3.
Institut Élie Cartan, Université de Lorraine, CNRS, IECL, F-54000 Nancy, France

Received: 09 December 2024 Revised: 05 May 2025 Accepted: 08 May 2025 Published: 15 May 2025

We establish partial Hölder continuity of the gradient for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with $ p $-growth, $ 1 < p < 2 $, without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand $ \Phi $ not necessarily regular.
- regularity,
- nonlinear variational problem,
- free boundary,
- quasiconvexity,
- subquadratic growth,
- perimeter penalization
Citation: Menita Carozza, Luca Esposito, Lorenzo Lamberti. Quasiconvex bulk and surface energies with subquadratic growth[J]. Mathematics in Engineering, 2025, 7(3): 228-263. doi: 10.3934/mine.2025011

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Abstract

We establish partial Hölder continuity of the gradient for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with $ p $-growth, $ 1 < p < 2 $, without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand $ \Phi $ not necessarily regular.

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