Stable solutions of elliptic equations on Riemannian manifolds

DOI10.1007/s12220-011-9278-9zbMath1273.53029arXiv0809.3025OpenAlexW1978797040WikidataQ115376885 ScholiaQ115376885MaRDI QIDQ363202

Yannick Sire, Alberto Farina, Enrico Valdinoci

Publication date: 2 September 2013

Published in: The Journal of Geometric Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0809.3025

zbMATH Keywords

Riemannian manifoldselliptic PDEsrigidity results

Mathematics Subject Classification ID

Elliptic equations on manifolds, general theory (58J05) Rigidity results (53C24) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Methods of local Riemannian geometry (53B21) PDEs on manifolds (35R01)

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Cites Work

• Unnamed Item
• Unnamed Item
• Function theory, random paths and covering spaces
• On subharmonic functions and differential geometry in the large
• On a Poincaré type formula for solutions of singular and degenerate elliptic equations
• Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems
• A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems
• Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result
• On a semilinear diffusion equation on a Riemannian manifold and its stable equilibrium solutions
• Positive solutions of elliptic equations
• Riemannian geometry and geometric analysis.
• Connectivity of phase boundaries in strictly convex domains
• The Poisson kernel of positively curved manifolds
• The Kelvin-Nevanlinna-Royden criterion for \(p\)-parabolicity
• Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
• Riemann Surfaces of Class A
• A Hyperbolic Surface in 3-Space
• Bernstein and De Giorgi type problems: new results via a geometric approach
• Instability results for an elliptic equation on compact Riemannian manifolds with non-negative Ricci curvature
• Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space
• Unique continuation in geometry
• The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature
• Differential equations on riemannian manifolds and their geometric applications
• On Deciding Whether a Surface is Parabolic or Hyperbolic
• Harmonic Functions on Open Riemann Surfaces
• On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property

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