Universality of renormalisable mappings in two dimensions: the case of polar convex integrands
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908474277888000 |
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| author | Irving, Christopher Van Vaerenbergh, Benoît |
| author_facet | Irving, Christopher Van Vaerenbergh, Benoît |
| contents | We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form $\int_Ωf(|\mathrm{D} u|)\,\mathrm{d} x$. The analysis relies on the condition that the vortex map ${x}/{\lvert x\rvert}$ has finite energy and that $t\mapsto f (\sqrt{t})$ is concave. We derive the leading order asymptotics and provide a detailed description of the convergence of $\mathrm{W}^{1,1}$-almost minimisers, leading to a characterization of second-order asymptotics. At the core of the method, we prove a ball merging construction (following Jerrard and Sandier's approach) for a general class of convex integrands. We therefore generalize the approximation by $p$-harmonic mappings when $p\nearrow 2$ and can also cover linearly growing functionals, including those of area-type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17520 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universality of renormalisable mappings in two dimensions: the case of polar convex integrands Irving, Christopher Van Vaerenbergh, Benoît Analysis of PDEs 58E20 (Primary), 49J45 (Secondary) We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form $\int_Ωf(|\mathrm{D} u|)\,\mathrm{d} x$. The analysis relies on the condition that the vortex map ${x}/{\lvert x\rvert}$ has finite energy and that $t\mapsto f (\sqrt{t})$ is concave. We derive the leading order asymptotics and provide a detailed description of the convergence of $\mathrm{W}^{1,1}$-almost minimisers, leading to a characterization of second-order asymptotics. At the core of the method, we prove a ball merging construction (following Jerrard and Sandier's approach) for a general class of convex integrands. We therefore generalize the approximation by $p$-harmonic mappings when $p\nearrow 2$ and can also cover linearly growing functionals, including those of area-type. |
| title | Universality of renormalisable mappings in two dimensions: the case of polar convex integrands |
| topic | Analysis of PDEs 58E20 (Primary), 49J45 (Secondary) |
| url | https://arxiv.org/abs/2411.17520 |