Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

DOI10.1080/03605302.2013.795969zbMath1287.58011arXiv1210.5720OpenAlexW2013003815MaRDI QIDQ5746974

Luciano Mari, Enrico Valdinoci, Alberto Farina

Publication date: 11 February 2014

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1210.5720

zbMATH Keywords

rigidity and classification resultspartial differential equations on manifolds

Mathematics Subject Classification ID

Elliptic equations on manifolds, general theory (58J05) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Boundary value problems on manifolds (58J32) Symmetries, invariants, etc. in context of PDEs (35B06) PDEs on manifolds (35R01) Overdetermined boundary value problems for PDEs and systems of PDEs (35N25)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Stable solutions of elliptic equations on Riemannian manifolds
• Diffusion-type operators, Liouville theorems and gradient estimates on complete manifolds
• Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems
• Stable solutions of \(-\Delta u = f(u)\) in \(\mathbb R^N\)
• On a semilinear diffusion equation on a Riemannian manifold and its stable equilibrium solutions
• Four-manifolds with positive curvature operator
• A theorem on elliptic differential inequalities with an application to gradient bounds
• A relation between growth and the spectrum of the Laplacian
• Hypersurfaces with constant scalar curvature
• Positive solutions of elliptic equations
• Connectivity of phase boundaries in strictly convex domains
• Correction of a technical point and some remarks about the paper appeared in Arch. Ration. Mech. Anal. 195(2010), no. 3, 1025-1058
• Geometry and topology of some overdetermined elliptic problems
• Isometric immersions of Riemannian manifolds
• An upper bound to the spectrum of \(\Delta\) on a manifold of negative curvature
• The splitting theorem for manifolds of nonnegative Ricci curvature
• REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS
• The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature
• Differential equations on riemannian manifolds and their geometric applications
• Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
• The principal eigenvalue and maximum principle for second‐order elliptic operators in general domains
• Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi
• On the method of moving planes and the sliding method
• On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property

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