A short note on nowhere smooth critical points of polyconvex functionals in arbitrary dimension
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913364866760704 |
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| author | Mazowiecka, Katarzyna Schikorra, Armin |
| author_facet | Mazowiecka, Katarzyna Schikorra, Armin |
| contents | For any $M, n \geq 2$ and any open set $Ω\subset \mathbb{R}^n$ we find a smooth, strongly polyconvex function $F\colon \mathbb{R}^{M\times n}\to \mathbb{R}$ and a Lipschitz map $u\colon \mathbb{R}^n \to \mathbb{R}^M$ that is a weak local minimizer of the energy \[
\int_Ω F(Du). \] but with nowhere continuous partial derivatives. This extends celebrated results by Müller-Sverák and Székelyhidi to higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A short note on nowhere smooth critical points of polyconvex functionals in arbitrary dimension Mazowiecka, Katarzyna Schikorra, Armin Analysis of PDEs For any $M, n \geq 2$ and any open set $Ω\subset \mathbb{R}^n$ we find a smooth, strongly polyconvex function $F\colon \mathbb{R}^{M\times n}\to \mathbb{R}$ and a Lipschitz map $u\colon \mathbb{R}^n \to \mathbb{R}^M$ that is a weak local minimizer of the energy \[ \int_Ω F(Du). \] but with nowhere continuous partial derivatives. This extends celebrated results by Müller-Sverák and Székelyhidi to higher dimensions. |
| title | A short note on nowhere smooth critical points of polyconvex functionals in arbitrary dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.17084 |