Boundary Harnack estimates in slit domains and applications to thin free boundary problems

Silva, Daniela De; Savin, Ovidiu

doi:10.4171/rmi/902

Cited by 57 publications

(66 citation statements)

References 5 publications

(15 reference statements)

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“…Relying neither on frequency functions nor on comparison arguments, Andersson [And13] has shown that similar results also hold for the full Lamé system. Recently, Koch, Petrosyan and Shi [KPS14] as well as De Silva and Savin [DSS14] have proved smoothness ([KPS14] proved analyticity) of the regular free boundary. Moreover, Garofalo and Petrosyan [GP09] give a structure theorem for the singular set of the thin obstacle problem.…”

Section: Resultsmentioning

confidence: 98%

The variable coefficient thin obstacle problem: Carleman inequalities

Koch

Rüland

Shi

2016

Advances in Mathematics

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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.

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Section: Resultsmentioning

confidence: 98%

The variable coefficient thin obstacle problem: Carleman inequalities

Koch

Rüland

Shi

2016

Advances in Mathematics

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“…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.…”

Section: State Of the Artmentioning

confidence: 84%

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

Colombo

Spolaor

Velichkov

2019

Comm Pure Appl Math

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We introduce a new logarithmic epiperimetric inequality for the 2m‐Weiss energy in any dimension, and we recover with a simple direct approach the usual epiperimetric inequality for the 3/2‐Weiss energy. In particular, even in the latter case, unlike the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension 2, we also prove for the first time the classical epiperimetric inequality for the (2m − 1/2)‐Weiss energy, thus covering all the admissible energies. As a first application, we classify the global λ‐homogeneous minimizers of the thin obstacle problem, with λ∈[]3/2,2+c∪⋃m∈ℕ‍(),2m−cm−2m+cm+, showing as a consequence that the frequencies 3/2 and 2m are isolated and thus improving on the previously known results. Moreover, we give an example of a new family of (2m − 1/2)‐homogeneous minimizers in dimension higher than 2. Second, we give a short and self‐contained proof of the regularity of the free boundary of the thin obstacle problem, previously obtained by Athanasopoulos, Caffarelli, and Salsa (2008) for regular points and Garofalo and Petrosyan (2009) for singular points. In particular, we improve the C1 regularity of the singular set with frequency 2m by an explicit logarithmic modulus of continuity. © 2019 Wiley Periodicals, Inc.

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“…Remark 1.4. Besides its own interest, sharp boundary regularity estimates find usually applications in free boundary problems; see for example [DS16]. We think that the ideas introduced in this paper could be used in order to establish the higher regularity of free boundaries in the obstacle problem for the fractional Laplacian, at least in case s > 1/2; see [BFR17].…”

Section: Introductionmentioning

confidence: 92%

Higher-order boundary regularity estimates for nonlocal parabolic equations

Ros‐Oton

Vivas

2018

Calc. Var.

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We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form ∂ t u − Lu = f (t, x) in I × Ω where I ⊂ R, Ω ⊂ R n and f is Hölder continuous. The nonlocal operators L that we consider are those arising in stochastic processes with jumps, such as the fractional Laplacian (−∆) s , s ∈ (0, 1).Our main result establishes that, if f is C γ is space and C γ/2s in time, and Ω is a C 2,γ domain, then u/d s is C s+γ up to the boundary in space and u is C 1+γ/2s up the boundary in time, where d is the distance to ∂Ω. This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in C ∞ domains.

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Boundary Harnack estimates in slit domains and applications to thin free boundary problems

Abstract: Abstract. We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the C ∞ regularity of the free boundary in the Signorini problem near non-degenerate points.

Cited by 57 publications

References 5 publications

The variable coefficient thin obstacle problem: Carleman inequalities

The variable coefficient thin obstacle problem: Carleman inequalities

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

Higher-order boundary regularity estimates for nonlocal parabolic equations

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