The Structure of the Free Boundary for Lower Dimensional Obstacle Problems

By I. Athanasopoulos, Luis Caffarelli, Sandro Salsa. 2006

147 citations recorded by the source corpus.

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 C1,α (n - 2)-dimensional surface in Rn-1.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c345897e3771e656f4aafef93a4980bf68e49f72

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Regularity for the nonlinear Signorini problem (2008, background): “The regularity of the free boundary for the Signorini problem was also studied very recently, first in [1] for the case φ = 0 and then in [4] for general obstacles φ.”
Nonlinear Porous Medium Flow with Fractional Potential Pressure (2010, background): “The spatial profile of such solutions is obtained by solving for the pressure p an obstacle problem with a truncated paraboloid as obstacle; the corresponding density u is then the mass of the negative fractional Laplacian of p, supported on the contact set; all this fits perfectly into the elliptic”
On the regularity of the non-dynamic parabolic fractional obstacle problem (2016, methodology): “The quasi-convexity introduced in [5] together with the continuity of the time derivative of §6 allows us to use the elliptic free boundary theory developed in [6] and [10], using the appropriate elliptic Almgren’s frequency formula.”
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian (2007, methodology): “As in [4] or [2], we intend to prove a C1,α regularity estimate for the free boundary applying the boundary Harnack inequality to two positive tangential derivatives uτ .”
An Extension Problem Related to the Fractional Laplacian (2006, background): “An extension of that result to nontangentiably accesible (NTA) domains (as in the last section of Athanasopoulos et al, to appear ) would lead to a boudary Harnack principle for the fractional Laplacian to domains only requiring a uniform capacity condition.”
Nonlinear Diffusion with Fractional Laplacian Operators (2012, background): “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.”
Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem (2013, methodology): “apter 4], the idea of the proof is to show that there is a cone of directions in the thin space, along which v ’is increasing. This approach was successfully used in the elliptic Signorini problem in [ACS08], [CSS08], see also [PSU12, Chapter 9], and in the arguments below we generalize the construc”
Asymptotic behaviour of a porous medium equation with fractional diffusion (2010, background): “ we prove that the stable conﬁgurations of this modiﬁed equation are just the solutions of an obstacle problem with fractional Laplacian, as studied by Athanasopoulos, Caﬀarelli, Salsa and Silvestre, [2] and [7]. This connection, described in Section 3, is quite surprising. In the original variables”
The obstacle problem for nonlinear integro-differential operators (2016, methodology): “in [1,7,42], via the Dirichlet-to-Neumann extension.”
Obstacle problems for integro-differential operators: Higher regularity of free boundaries (2019, methodology): “and choosing k ∈ N such that 2kr ∈ [1, 2), we deduce”
An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients (2015, unknown): “ne has B′ η0(x0)∩ Γ(v) ⊂ Γ3/2(v), and B′ η0 ∩Λ(v) = B ′ η0 ∩{xn−1 ≤ g(x1,...,xn−2)} for g∈ C1,β(Rn−2) with a universal exponent β∈ (0,1). Thisresult is known and well-understood in the case L= ∆, see [ACS08] or Chapter 9 in [PSU12]. However, the existing proofs are based on diﬀerentiating the equati”
Boundary Harnack estimates in slit domains and applications to thin free boundary problems (2014, background): “e free boundary if u(X) = o(|X −X0| 3 2). Otherwise, X0 is called a regular point of the free boundary. Concerning the regularity of the free boundary Γ, Athanasopoulos, Caﬀarelli and Salsa showed in [ACS] that if X0 is a regular point, then Γ is given locally by the graph of a C1,α function g in so”