Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

By Nicola Garofalo, Arshak Petrosyan. 2008

127 citations recorded by the source corpus.

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set.Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/246a094b9841513080e8c560b46eafa1d80fa4a5

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In Applied Mathematics

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Citations in the corpus

Boundary Harnack estimates in slit domains and applications to thin free boundary problems (2014, background): “ndary 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 some direction, say the en direction (see also [GP] concerning the singular set). Moreover, the derivatives ui, i = 1,2,..,n, are harmonic in the s”
Higher regularity of the free boundary in the elliptic Signorini problem (2014, methodology): “The following Weiss-type monotonicity formula, whose proof can be found in [14], implies that u0 is a homogeneous global solution of the Signorini problem of degree 3/2.”
Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem (2013, methodology): “e caloric in an innite strip S ˆ = Rn 2( ˆ;0]. In Section 6, we establish a truncated version of that formula for the solutions of the parabolic Signorini problem, similar to the ones in [CSS08] and [GP09]. The time dependent case presents, however, substantial novel challenges with respect to the e”
An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients (2015, methodology): “ WL(v,r) + Cr1/2 is nondecreasing for a universal constant C. Here, we were inspired by [Wei99] and [Wei98], where Weiss introduced related monotonicity formulas in the classical obstacle problem. In [GP09] two of us also proved a similar monotonicity formula in the Signorini setting, in the case of”
A geometric approach to generalized Stokes conjectures (2009, background): “Almgren’s formula has subsequently been extended to various perturbations (see [14] for a recent extension).”
Obstacle problems for integro-differential operators: Higher regularity of free boundaries (2019, result): “After the results in [6], many more results have been obtained concerning the set of degenerate/singular points [2,14,16,17,19], the case of the fractional Laplacian with a drift [15,18,27], and also the parabolic version of the problem [3,4].”