Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition
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| Format: | Preprint |
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2026
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| _version_ | 1866917364568686592 |
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| author | Barbato, Rosa Masiello, Alba Lia Sannipoli, Rossano |
| author_facet | Barbato, Rosa Masiello, Alba Lia Sannipoli, Rossano |
| contents | In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin Pólya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition Barbato, Rosa Masiello, Alba Lia Sannipoli, Rossano Analysis of PDEs 35P15, 49Q10, 35J05, 35J25 In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin Pólya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains. |
| title | Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition |
| topic | Analysis of PDEs 35P15, 49Q10, 35J05, 35J25 |
| url | https://arxiv.org/abs/2603.26582 |