The Structure of the Free Boundary for Lower Dimensional Obstacle Problems

Athanasopoulos, Ioannis; Caffarelli, Luis A.; Salsa, Sandro

doi:10.1353/ajm.2008.0016

Cited by 123 publications

(238 citation statements)

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“…In [8], the boundary Harnack principle is proven for A 2 weights and Lipschitz domains. An extension of that result to nontangentiably accesible (NTA) domains (as in the last section of [2]) would lead to a boudary Harnack principle for the fractional Laplacian to domains only requiring a uniform capacity condition.…”

Section: Corollary 54 Let F and G Be As In Theorem 53 Thenmentioning

confidence: 99%

“…We can apply the boundary Harnack inequality of [8] to the functions u (1) 3 and u (2) 3 to obtain ϕ 1 (x 1 , . .…”

Section: Theorem 53 (Boundary Harnackmentioning

confidence: 99%

“…In the proof we can see that what we need to apply the result of [8] is that the corresponding extensions u (1) and u (2) are nonnegative in the unit ball of n + 1 dimensions. The advantage is that this is a local condition, therefore we can obtain the standard corollary of boundary Harnack saying that u (1) u (2) is C α in a neighborhood of the origin. In particular f g is C α , which is not strictly a corollary of Theorem 5.3 but of its proof.…”

Section: Theorem 53 (Boundary Harnackmentioning

confidence: 99%

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An Extension Problem Related to the Fractional Laplacian

Caffarelli

Silvestre

2007

Communications in Partial Differential Equations

2,519

2,827

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The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

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Section: Corollary 54 Let F and G Be As In Theorem 53 Thenmentioning

confidence: 99%

“…We can apply the boundary Harnack inequality of [8] to the functions u (1) 3 and u (2) 3 to obtain ϕ 1 (x 1 , . .…”

Section: Theorem 53 (Boundary Harnackmentioning

confidence: 99%

Section: Theorem 53 (Boundary Harnackmentioning

confidence: 99%

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An Extension Problem Related to the Fractional Laplacian

Caffarelli

Silvestre

2007

Communications in Partial Differential Equations

2,519

2,827

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“…The interest in using fractional Laplacians in modeling diffusive processes has a wide literature, especially when one wants to model long-range diffusive interaction, and this interest has been activated by the recent progress in the mathematical theory as represented in [6], [8], [24], the thesis work by Silvestre [60], and many others. A variant of the proposed model was studied by Lions and Mas-Gallic [54] They study the regularization of the velocity field in the standard porous medium equation by means of a convolution kernel to get a system like ours, with a difference, namely that they assume the kernel to be smooth and integrable.…”

Section: Nonlocal Diffusion Model Of Porous Medium Typementioning

confidence: 99%

“…The systematic study of the corresponding PDE models is more recent and many of the results have arisen in the last decade. The linear or quasilinear elliptic theory has been actively studied recently in the works of Caffarelli and collaborators [6,8,24], Kassmann [44], Silvestre [60] and many others. The standard linear evolution equation involving fractional diffusion is…”

mentioning

confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

2012

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We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that they describe the asymptotic behaviour of a wide class of solutions. The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1 -contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity. Nonlinear diffusion and fractional diffusionSince the work by Einstein [39] and Smoluchowski [62] at the beginning of the last century (cf. also Bachelier [9]), we possess an explantation of diffusion and Brownian motion in terms of the heat equation, and in particular of the Laplace operator. This explanation has had an enormous success both in Mathematics and Physics. In the decades that followed, the Laplace operator has been often replaced by more general types of so-called elliptic operators with variable coefficients, and later by nonlinear differential operators; a huge body of theory is now available, both for the evolution equations [49] and for the stationary states, described by elliptic equations of different kinds [50,42].

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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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The Structure of the Free Boundary for Lower Dimensional Obstacle Problems

Abstract: We study the regularity of the "free surface" in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a C 1,α (n − 2)-dimensional surface in R n−1 .

Cited by 123 publications

References 6 publications

An Extension Problem Related to the Fractional Laplacian

An Extension Problem Related to the Fractional Laplacian

Nonlinear Diffusion with Fractional Laplacian Operators

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

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