The local Burkholder functional, quasiconvexity and Geometric Function Theory
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911757891534848 |
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| author | Astala, Kari Faraco, Daniel Guerra, André Koski, Aleksis Kristensen, Jan |
| author_facet | Astala, Kari Faraco, Daniel Guerra, André Koski, Aleksis Kristensen, Jan |
| contents | We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant.
In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03495 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The local Burkholder functional, quasiconvexity and Geometric Function Theory Astala, Kari Faraco, Daniel Guerra, André Koski, Aleksis Kristensen, Jan Analysis of PDEs Complex Variables We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers. |
| title | The local Burkholder functional, quasiconvexity and Geometric Function Theory |
| topic | Analysis of PDEs Complex Variables |
| url | https://arxiv.org/abs/2309.03495 |