Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below
We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by and dimension bounded above by in a synthetic sense, the so called spaces. We first establish a Polya-Szego type inequality stating that the -Sobolev norm d...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2019
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| _version_ | 1826271414795108352 |
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| author | Mondino, A Semola, D |
| author_facet | Mondino, A Semola, D |
| author_sort | Mondino, A |
| collection | OXFORD |
| description | We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by and dimension bounded above by in a synthetic sense, the so called spaces. We first establish a Polya-Szego type inequality stating that the -Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the p-Laplace operator with Dirichlet boundary conditions (on open subsets), for every . This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41]; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci , finite dimensional Alexandrov spaces with curvature, Finsler manifolds with Ricci . In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of spaces, which are interesting even for smooth Riemannian manifolds with Ricci. |
| first_indexed | 2024-03-06T21:56:18Z |
| format | Journal article |
| id | oxford-uuid:4d08c982-f376-4618-ae84-cc035f8a0912 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:56:18Z |
| publishDate | 2019 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:4d08c982-f376-4618-ae84-cc035f8a09122022-03-26T15:53:01ZPolya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded belowJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4d08c982-f376-4618-ae84-cc035f8a0912EnglishSymplectic Elements at OxfordElsevier2019Mondino, ASemola, DWe study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by and dimension bounded above by in a synthetic sense, the so called spaces. We first establish a Polya-Szego type inequality stating that the -Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the p-Laplace operator with Dirichlet boundary conditions (on open subsets), for every . This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41]; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci , finite dimensional Alexandrov spaces with curvature, Finsler manifolds with Ricci . In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of spaces, which are interesting even for smooth Riemannian manifolds with Ricci. |
| spellingShingle | Mondino, A Semola, D Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below |
| title | Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below |
| title_full | Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below |
| title_fullStr | Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below |
| title_full_unstemmed | Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below |
| title_short | Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below |
| title_sort | polya szego inequality and dirichlet p spectral gap for non smooth spaces with ricci curvature bounded below |
| work_keys_str_mv | AT mondinoa polyaszegoinequalityanddirichletpspectralgapfornonsmoothspaceswithriccicurvatureboundedbelow AT semolad polyaszegoinequalityanddirichletpspectralgapfornonsmoothspaceswithriccicurvatureboundedbelow |