Geometric Approach to Nonvariational Singular Elliptic Equations

Abstract

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter (γ − 1), for 0 < γ < 1, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometricmeasure properties of the free boundary F = ∂ {u > 0}. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region {u > 0} and H n−1 a.e. weak differentiability property of F.

Singular perturbation by bending for an adhesive obstacle problem

“…This is not an obstacle problem explicitly but the solutions are automatically constrained above zero, thus it is equivalent to the problem with the constraint u ≥ ψ ≡ 0 (flat obstacle problem). This problem is one of important interface models and has been generalized variously (for example, see [25] generalizing the first term, [3] the second term, or their references). Especially, Yamaura [33] considered a non-linearized case, namely the case that the first term is replaced to the area functional.…”

Geometric regularity estimates for elliptic equations

“…This notion also appears naturally in the study of free boundary problems governed by fully nonlinear operators, see [14,3]. The idea is that the family F µ (M) := µF(µ −1 M) forms a path of uniform elliptic operators (each F µ is elliptic with the same ellipticity constants as F), jointing F and F ⋆ .…”

Sharp Hessian integrability estimates for nonlinear elliptic equations: An asymptotic approach

“…Similarly, u is a viscosity supersolution to (1) if whenever one touches the graph of u from below by a smooth function φ at y 0 ∈ B 1 , there holds…”