Superlinear free-discontinuity models: relaxation and phase-field approximation

Mathematics > Analysis of PDEs [Submitted on 1 May 2025 (v1), last revised 16 Jun 2026 (this version, v3)] Title:Superlinear free-discontinuity models: relaxation and phase-field approximation View PDF HTML (experimental)Abstract:In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on $SBV$ functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase-field models. Submission history From: Matteo Focardi [view email][v1] Thu, 1 May 2025 20:33:41 UTC (113 KB) [v2] Mon, 3 Nov 2025 16:28:07 UTC (109 KB) [v3] Tue, 16 Jun 2026 09:10:39 UTC (109 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.