Singular solutions of fully nonlinear elliptic equations and applications

Abstract

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of ${\mathbb{R}}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.

41 pages, 2 figures

References in corpus (2)

• Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations
• Fundamental solutions of homogeneous fully nonlinear elliptic equations

Cited by in corpus (12)

• A free boundary problem with facets
• Inhomogeneous Hopf-Oleĭnik Lemma and Applications. Part IV: Sharp Krylov Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with unbounded coefficients and $C^{1,Dini}$ boundary data
• Generalized principal eigenvalues of convex nonlinear elliptic operators in ${\mathbb{R}}^N$
• Overdetermined problems for fully nonlinear elliptic equations
• Boundary weak Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE
• Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
• Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations
• Explicit subsolutions and a Liouville theorem for fully nonlinear uniformly elliptic inequalities in halfspaces
• Homogeneous solutions of extremal Pucci's equations in planar cones
• Overdetermined problems for fully nonlinear equations in space forms
• On Homogenization Problems with Oscillating Dirichlet Conditions in Space-Time Domains
• Unique continuation for fully nonlinear elliptic equations