C 1, α Regularity for Solutions to the p-harmonic Thin Obstacle Problem

Andersson, John; Mikayelyan, Hayk

doi:10.1093/imrn/rnq061

Cited by 5 publications

(5 citation statements)

References 8 publications

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“…We now prove the lemma for the case r = 1. From the proof of Theorem 4.3 in [4], we can obtain that sup…”

Section: Hölder Continuity Of the Gradientmentioning

confidence: 99%

“…2 ) result was achieved by Athanasopoulos and Caffarelli [1]. For the case p(x) ≡ p ∈ (1, ∞), the C 0,α -regularity and the gradient estimates for a minimizer of the obstacle problem were established by Bögelein, F. Duzaar and Mingione [8], and the C 1,α -regularity for a minimizer of the thin obstacle problem was obtained by Andersson and Mikayelyan [4].…”

Section: Introductionmentioning

confidence: 96%

“…The aim of this paper is to extend the C 1,α -regularity obtained in [4] to the thin obstacle problem for the p(x)-Laplacian. In the process, we find a minimal regularity requirement on the variable exponent p(x) to ensure the higher integrability, Theorem 3.3, and the C 1,α -regularity, Theorem 4.4, for the thin obstacle problem (1.1)- (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Regularity Results of the Thin Obstacle Problem for the $p(x)$-Laplacian

Byun¹,

Lee²,

Oh³

et al. 2018

Preprint

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“…We now prove the lemma for the case r = 1. From the proof of Theorem 4.3 in [4], we can obtain that sup…”

Section: Hölder Continuity Of the Gradientmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Regularity Results of the Thin Obstacle Problem for the $p(x)$-Laplacian

Byun¹,

Lee²,

Oh³

et al. 2018

Preprint

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“…In principle, the former condition implies that normal stress acts on the crack only when its both sides are in contact, and the latter means that slip velocity across it occurs only when tangential stress reaches a threshold (see also remark 2.1 below). We leave two side remarks regarding this model: optimal regularity of weak solutions for the Signorini problem is obtained by Andersson [ 6 ] (see also [ 7 ]), and non-monotone friction laws that lead to hemi-variational inequalities can also be employed in place of the Coulomb law (see [ 8 ]).…”

Section: Introductionmentioning

confidence: 99%

Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body

Itou

Kashiwabara

2022

Phil. Trans. R. Soc. A.

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We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type , and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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“…Obstacle problems are free boundary problems: the principal part of their study is the structure and regularity of the boundary of the contact set of the solution and the obstacle, the free boundary. The lower dimensional obstacle problem we consider -the thin obstacle problem with weight |x n+1 | a -has garnered much interest and attention (see [AC04, CS07, ACS08, GP09, KRS16, FoSp18, CSV19, JN17]); it is a model setting, and has motivated the study of many other types of lower dimensional obstacle problems (see [MS08,AM11,Fer16,RS17,RuSh17,FeSe18,FoSp18b,GR18,BLOP19]).…”

Section: Introductionmentioning

confidence: 99%

On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem

Fernández-Real,

Jhaveri

2018

Preprint

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In this work, we consider the singular set in the thin obstacle problem with weight |xn+1| a for a ∈ (−1, 1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a non-local problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single C 2 manifold, up to a lower dimensional subset, and the top stratum is locally contained in a C 1,α manifold for some α > 0 if a < 0.In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension two lower dimensional obstacle problem (or fractional thin obstacle problem) when a < 0, whereas second blow-ups at singular points in the top stratum are global, homogeneous, and aharmonic polynomials when a ≥ 0. To do so, we establish regularity results for this codimension two problem, what we call the very thin obstacle problem.Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many C 2 manifolds, up to a lower dimensional subset.

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C 1, α Regularity for Solutions to the p-harmonic Thin Obstacle Problem

Abstract: We prove C 1,α regularity for a thin obstacle problem for the p-laplace equation. Due to the nonlinearity of the p-laplace operator we can not use the same methods used for the Laplace case, instead we use techniques developed by E. de Giorgi.

Cited by 5 publications

References 8 publications

Regularity Results of the Thin Obstacle Problem for the $p(x)$-Laplacian

Regularity Results of the Thin Obstacle Problem for the $p(x)$-Laplacian

Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body

On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem

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