Radiall symmetry of minimizers to the weighted $p-$Dirichlet energy
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914921048965120 |
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| author | Kalaj, David |
| author_facet | Kalaj, David |
| contents | Let $\mathbb{A}=\{z: r< |z|<R\}$ and $\A^\ast=\{z: r^\ast<|z|<R^\ast\}$ be annuli in the complex plane. Let $p\in[1,2]$ and assume that $\mathcal{H}^{1,p}(\A,\A^*)$ is the class of Sobolev homeomorphisms between $\A$ and $\A^*$, $h:\A\onto \A^*$. Then we consider the following Dirichlet type energy of $h$: $$\mathcal{F}_p[h]=\int_{\A(1,r)}\frac{\|Dh\|^p}{|h|^p}, \ \ 1\le p\le 2.$$ We prove that this energy integral attains its minimum, and the minimum is a certain radial diffeomorphism $h:\A\onto \A^*$, provided a radial diffeomorphic minimizer exists. If $p>1$ then such diffeomorphism exist always. If $p=1$, then the conformal modulus of $\A^\ast$ must not be greater or equal to $π/2$. This curious phenomenon is opposite to the Nitsche type phenomenon known for the standard Dirichlet energy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_00089 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Radiall symmetry of minimizers to the weighted $p-$Dirichlet energy Kalaj, David Analysis of PDEs Complex Variables Let $\mathbb{A}=\{z: r< |z|<R\}$ and $\A^\ast=\{z: r^\ast<|z|<R^\ast\}$ be annuli in the complex plane. Let $p\in[1,2]$ and assume that $\mathcal{H}^{1,p}(\A,\A^*)$ is the class of Sobolev homeomorphisms between $\A$ and $\A^*$, $h:\A\onto \A^*$. Then we consider the following Dirichlet type energy of $h$: $$\mathcal{F}_p[h]=\int_{\A(1,r)}\frac{\|Dh\|^p}{|h|^p}, \ \ 1\le p\le 2.$$ We prove that this energy integral attains its minimum, and the minimum is a certain radial diffeomorphism $h:\A\onto \A^*$, provided a radial diffeomorphic minimizer exists. If $p>1$ then such diffeomorphism exist always. If $p=1$, then the conformal modulus of $\A^\ast$ must not be greater or equal to $π/2$. This curious phenomenon is opposite to the Nitsche type phenomenon known for the standard Dirichlet energy. |
| title | Radiall symmetry of minimizers to the weighted $p-$Dirichlet energy |
| topic | Analysis of PDEs Complex Variables |
| url | https://arxiv.org/abs/2303.00089 |