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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.06749 |
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| _version_ | 1866916385106427904 |
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| author | Dengler, Marcel Bevan, Jonathan J. |
| author_facet | Dengler, Marcel Bevan, Jonathan J. |
| contents | In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,ξ)$ is quadratic and convex in $ξ$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,ξ)$, depending smoothly on $ξ$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_06749 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A uniqueness criterion and a counterexample to regularity in an incompressible variational problem Dengler, Marcel Bevan, Jonathan J. Analysis of PDEs In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,ξ)$ is quadratic and convex in $ξ$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,ξ)$, depending smoothly on $ξ$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$. |
| title | A uniqueness criterion and a counterexample to regularity in an incompressible variational problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2205.06749 |