Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds |
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Authors: | Eduardo V. Teixeira Lei Zhang |
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Affiliation: | aUniversidade Federal do Ceará, Departamento de Matemática, Av. Humberto Monte, s/n, Campus do Pici – Bloco 914 Fortaleza-CE, CEP 60.455-760, Brazil;bUniversity of Florida, 358 Little Hall, P.O. Box 118105, Gainesville, FL 32611-8105, United States |
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Abstract: | For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt–Caffarelli–Friedman and Caffarelli–Jerison–Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace–Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds. |
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Keywords: | MSC: primary, 58J05, 35B65 secondary, 35J05 |
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