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Recent rigidity results for graphs with prescribed mean curvature

1 Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Trieste 63, I-35121 Padova, Italy
2 Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, I-20133 Milano, Italy
3 Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici - Bloco 914, Av. Humberto Monte s/n, 60.455-760 Fortaleza, Brazil
4 Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, I-10123 Torino, Italy
5 Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I-06123 Perugia, Italy

This contribution is part of the Special Issue: Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday
Guest Editors: Serena Dipierro; Luca Lombardini
Link: www.aimspress.com/mine/article/5752/special-articles

Special Issues: Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs u : M → $\mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain Ω ⊂ M, namely, for CMC graphs satisfying an overdetermined boundary condition.
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