Boundary Regularity and Nontransversal Intersection for the Fully Nonlinear Obstacle Problem

Abstract

In this paper a classification is given of blowup solutions at the intersection of the free and fixed boundary corresponding to obstacle problems generated by fully nonlinear, uniformly elliptic operators. As a consequence, nontransversal intersection is shown to hold in any dimension. Several regularity results are also obtained. © 2019 Wiley Periodicals, Inc.

On $$W^{2,p}$$-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions

“…Meanwhile, for fully nonlinear elliptic equations, there have been a number of noteworthy preceding results, for example, [6,23,34]. And Indrei's recent study [21] provided C 1 -regularity of the free boundary without density assumptions. Regularity results for the elliptic obstacle problem can be found in [2,5,18,19].…”

Free boundary regularity near the fixed boundary for the fully nonlinear obstacle problem

“…In a recent work [Ind19], the author proved that for solutions of (1.1) with zero Dirichlet boundary data, with L replaced by a convex fully nonlinear uniformly elliptic operator F , if x ∈ ∂B 1 ∩ Γ, then Γ can be represented as the graph of a C 1 function in a neighborhood of x. There are two surprising differences between the interior and boundary result: first, there are no density assumptions in the boundary case (in particular, cusp-type singularities do not exist); second, there is an example which generates a free boundary which is C 1 with a specific Dini modulus of continuity for the free normal (see e.g.…”

Non-transversal intersection of the free and fixed boundary in the mean-field theory of superconductivity

“…The linear case has been well-studied via monotonicity formulas and explicit representations involving Green's function [CKS00,SU03,PSU12]; see [PSU12,Chapter 8] for many very interesting references. In the fully nonlinear uniformly elliptic context, non-transversal intersection remained a conjecture and was fully solved in [Ind19b]. In addition, C 1 regularity of the free boundary was obtained in the one-phase problem without density assumptions which completed Caffarelli's regularity theory up-to-the boundary with a novel approach.…”

“…Proposition 3.1 [Ind,. Proposition 3.6] Let {u j } ⊂ P + 1 (0, M, Ω) and suppose {∇u j = 0} ∩ {x n > 0} ⊂ Ω, 0 ∈ {u j = 0}, and ∇u j (0) = 0.…”