A Microscopic Convexity Principle for Nonlinear Partial Differential Equations

Abstract

We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.

Cited by in corpus (32)

• A fully-nonlinear flow and quermassintegral inequalities in the sphere
• Complete translating solitons to the mean curvature flow in ${\mathbb{R}}^3$ with nonnegative mean curvature
• Curvature estimates for the level set of spatial quasiconcave solutions to a class of parabolic equations
• Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space
• The J-flow and stability
• A variational method for second order shape derivatives
• Constant rank theorems for curvature problems via a viscosity approach
• Convexity of 2-convex translating solitons to the mean curvature flow in ${\mathbb{R}}^{n+1}$
• Convexity of level sets and a two-point function
• On the Maximal Rank Problem for the Complex Homogeneous Monge-Ampère Equation
• A mean curvature type flow in space forms
• Counterexamples to quasiconcavity for the heat equation
• On the microscopic spacetime convexity principle for fully nonlinear parabolic equations I: Spacetime convex solutions
• Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations
• On Concavity of Solutions of the Nonlinear Poisson Equation
• Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations
• Existence and convexity of local solutions to degenerate hessian equations
• A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow
• On the evolution of convex hypersurfaces by the $Q_k$ flow
• Contracting convex hypersurfaces by curvature
• A constant rank theorem for partial convex solutions of partial differential equations
• Concavity and perturbed concavity for $p$-Laplace equations
• Weak Harnack inequalities for eigenvalues and constant rank theorems
• Strong space-time convexity and the heat equation
• On a constant rank theorem for nonlinear elliptic PDEs
• Complex Hessian Equations on Compact Kahler Manifold
• Stochastic ordering by g-expectations
• A rigidity theorem for codimension one shrinking gradient Ricci solitons in ${\mathbb{R}}^{n+1}$
• $C^{2,\alpha }$ estimates and existence results for certain nonconcave PDE
• Parabolic power concavity and parabolic boundary value problems
• Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in ${\mathbb{R}}^3$
• Convexity of constant mean curvature graphs in ${\mathbb{R}}^{n+1}$ with planar boundary