On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies
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| Format: | Preprint |
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2026
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| _version_ | 1866910048133840896 |
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| author | Buttazzo, Giuseppe Horta, Raul Fernandes |
| author_facet | Buttazzo, Giuseppe Horta, Raul Fernandes |
| contents | We consider variational energies of the form \[E_H(u)=\frac12\int_ΩH^2(\nabla u)\,dx\] defined on the Sobolev space $H^1_0(Ω)$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue $λ_H(Ω)$ and the torsional rigidity $T_H(Ω)$ induced by the seminorm $H$. In particular, we focus on functionals of the type \[F_{q,Ω}(H)=λ_H(Ω)\,T_H^q(Ω),\] where $q>0$ is a fixed real parameter. The optimization is performed with respect to the control $H$; we analyze both minimization and maximization problems for $F_{q,Ω}(H)$, as $H$ ranges over a suitable class of seminorms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_09851 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies Buttazzo, Giuseppe Horta, Raul Fernandes Optimization and Control 49J20, 49J30, 49J45, 47J20 We consider variational energies of the form \[E_H(u)=\frac12\int_ΩH^2(\nabla u)\,dx\] defined on the Sobolev space $H^1_0(Ω)$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue $λ_H(Ω)$ and the torsional rigidity $T_H(Ω)$ induced by the seminorm $H$. In particular, we focus on functionals of the type \[F_{q,Ω}(H)=λ_H(Ω)\,T_H^q(Ω),\] where $q>0$ is a fixed real parameter. The optimization is performed with respect to the control $H$; we analyze both minimization and maximization problems for $F_{q,Ω}(H)$, as $H$ ranges over a suitable class of seminorms. |
| title | On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies |
| topic | Optimization and Control 49J20, 49J30, 49J45, 47J20 |
| url | https://arxiv.org/abs/2603.09851 |