A microscopic convexity theorem of level sets for solutions to elliptic equations

DOI10.1007/s00526-010-0333-3zbMath1205.35111OpenAlexW2147907963MaRDI QIDQ618746

Lu Xu

Publication date: 17 January 2011

Published in: Calculus of Variations and Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00526-010-0333-3

zbMATH Keywords

\(p\)-Laplacianelliptic equationmicroscopic level-set convexity theorem

Mathematics Subject Classification ID

Inequalities and extremum problems involving convexity in convex geometry (52A40) Geometric theory, characteristics, transformations in context of PDEs (35A30) Convexity properties of solutions to PDEs with constant coefficients (35E10) Quasilinear elliptic equations (35J62)

Related Items

The convexity of the level sets of maximal strictly space-like hypersurfaces defined on 2-dimensional space forms, On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions, Convexity of level lines of maximal space-like hypersurfaces in Minkowski space, Curvature estimate of steepest descents of 2-dimensional maximal space-like hypersurfaces on space forms, Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds, Weak Harnack inequalities for eigenvalues and a constant rank theorem for level sets, The convexity and the Gaussian curvature estimates for the level sets of harmonic functions on convex rings in space forms, Curvature estimates for the level sets of spatial quasiconcave solutions to a class of parabolic equations, Convexity of level sets of minimal graph on space form with nonnegative curvature, A quantitative constant rank theorem for quasiconcave solutions to fully nonlinear elliptic equations, Some properties of minimizers of a variational problem involving the total variation functional

Cites Work

• Unnamed Item
• Convexity of solutions of semilinear elliptic equations
• Convex solutions of certain elliptic equations have constant rank Hessians
• On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
• Capacitary functions in convex rings
• The Christoffel-Minkowski problem. I: Convexity of solutions of a Hessian equation
• Quasi-concavity for semilinear elliptic equations with non-monotone and anisotropic nonlinearities
• A Result Concerning Convex Level Surfaces of 3-Dimensional Harmonic Functions
• Quasiconcave solutions to elliptic problems in convex rings
• On new isoperimetric inequalities and symmetrization.
• Quasi–concave envelope of a function and convexity of level sets of solutions to elliptic equations
• Convexity properties of solutions to some classical variational problems
• Conexity Of Level Sets For Solutions To elliptic Ring Problems
• A constant rank theorem for solutions of fully nonlinear elliptic equations