The variable coefficient thin obstacle problem: Carleman inequalities

Abstract

Abstract. In this article we present a new strategy of addressing the (variable coefficient) thin obstacle problem. Our approach is based on a (variable coefficient) Carleman estimate. This yields semi-continuity of the vanishing order, lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow-up procedure. Moreover, the Carleman estimate implies the existence of homogeneous blow-up limits along certain sequences and ultimately leads to an almost optimal regularity statement. As it is a very robust tool, it allows us to consider the problem in the setting of Sobolev metrics, i.e. the coefficients are only W 1,p regular for some p > n + 1. These results provide the basis for our further analysis of the free boundary, the optimal (C 1,1/2 -) regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow-up article, [KRS15], in the framework of W 1,p , p > 2(n + 1), regular coefficients.

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

“…In [2] they give a complete description of the blow-up limits at the points of frequency 3 =2 and prove that the regular free boundary Reg.u/ is locally a .d 2/-dimensional C 1;˛h ypersurface in R d 1 . Later the regular part of the free boundary has been shown to be C 1 in [6] and analytic in [17] (see also [16,18]), and analogous results were extended to more general fractional Laplacian (see [4]), of which the thin obstacle is a particular example. Garofalo and Petrosyan (cf.…”

Higher regularity of the free boundary in the obstacle problem for the fractional Laplacian

“…The original approach to proving higher regularity in obstacle problems, pioneered in [20], was to use the hodograph-Legendre transform. These techniques have been extended to prove higher regularity in the Signorini problem with zero obstacle in [21] and in thin obstacle problems with variable coefficients and inhomogeneities in [22]. On the other hand, at the same time as [21], De Silva and Savin used the higher order boundary Harnack principle to show higher free boundary regularity in the Signorini problem with zero obstacle in [14], as we discussed above.…”

The variable coefficient thin obstacle problem: Optimal regularity and regularity of the regular free boundary

“…Moreover, in this framework we deduce a leading order aymptotic expansion of solutions to (2) with error estimates. Combining the regularity of the regular free boundary with the Carleman estimate from [KRS15], we then show the optimal C 1,min{1− n+1 p , 1 2 } regularity of solutions with W 1,p metrics, a ij , for p > n + 1. In addition to this, we also treat perturbations of the thin obstacle problem including non-flat free boundaries and obstacles, as well as inhomogeneities in the equations and the interior thin obstacle problem.…”

“…We first recall the main results from [KRS15] on free boundary points: All free boundary points x ∈ Γ w are classified by their associated vanishing order κ x (c.f. Section 4 in [KRS15]):…”

“…The remaining set κ≥2 Γ κ (w) consists of all free boundary points with a higher order of vanishing (c.f. Definition 4.1 in [KRS15] or Section 2.2 for the precise definition of κ x ). Moreover, the map Γ w ∋ x → κ x is upper semi-continuous.…”