A flow approach to the monotonicity of shape functionals
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911448884576256 |
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| author | Huang, Yong Li, Qinfeng Xie, Shuangquan Yang, Hang |
| author_facet | Huang, Yong Li, Qinfeng Xie, Shuangquan Yang, Hang |
| contents | We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching flows, leg-stretching flows, and angle-bisector flows, we prove new monotonicity properties for these functionals under deformations of triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, we introduce a mean curvature flow approach to the Saint-Venant inequality, providing a new geometric proof for smooth convex domains. We establish a weak monotonicity property along the flow and characterize the equality case, which leads to the discovery of an intriguing new functional whose extremal properties suggest a further conjecture. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A flow approach to the monotonicity of shape functionals Huang, Yong Li, Qinfeng Xie, Shuangquan Yang, Hang Analysis of PDEs We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching flows, leg-stretching flows, and angle-bisector flows, we prove new monotonicity properties for these functionals under deformations of triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, we introduce a mean curvature flow approach to the Saint-Venant inequality, providing a new geometric proof for smooth convex domains. We establish a weak monotonicity property along the flow and characterize the equality case, which leads to the discovery of an intriguing new functional whose extremal properties suggest a further conjecture. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles. |
| title | A flow approach to the monotonicity of shape functionals |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.09485 |