Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators

In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers [C77, C98], regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of C 1 manifolds of varying dimension. In two dimensions, this C 1 result has been improved to C 1,α by Weiss [W99].In this paper we prove that, for n = 2 singular points are locally contained in a C 2 curve. In higher dimension n ≥ 3, we show that the same result holds with C 1,1 manifolds (or with countably many C 2 manifolds), up to the presence of some "anomalous" points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a C 1,α manifold, thus extending to every dimension the result in [W99].We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for n ≥ 3 we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.

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“…We will also need the following version of Whitney's extension theorem (see for instance [Fef09] and the references therein):…”

Section: 2mentioning

confidence: 99%

On the fine structure of the free boundary for the classical obstacle problem

Figalli

Serra

2018

Invent. math.

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“…For C m,ω (R n ), these problems are completely solved, although we do not discuss them here; see [13], [14], [15]. In the setting of W m,p (R n ), work is just beginning; see Shvartsman [31].…”

Section: Define a Banach Spacementioning

confidence: 99%

“…Along the way, we solved the analogous problems with C m (R n ) replaced by C m,ω (R n ), the space of functions whose m th derivatives have a given modulus of continuity ω. (See [14], [15].) It is natural also to consider Sobolev spaces W m,p (R n ), for which work on the above problems is just beginning (see Shvartsman [31]).…”

mentioning

confidence: 99%

Whitney’s extension problems and interpolation of data

Fefferman

2008

Bull. Amer. Math. Soc.

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Abstract. Given a function f : E → R with E ⊂ R n , we explain how to decide whether f extends to a C m function F on R n . If E is finite, then one can efficiently compute an F as above, whose C m norm has the least possible order of magnitude (joint work with B. Klartag).Let f : E → R be a function defined on a given (arbitrary) set E ⊂ R n , and let m ≥ 1 be a given integer. How can we decide whether f extends to a function The finite, effective versions of the above problems are basic questions about interpolation of data: Let E ⊂ R n be a finite set, and let f : E → R be given. Fix m ≥ 1. We want to find an interpolant, i.e., a function F ∈ C m (R n ) such that F = f on E. How small can we take the C m norm of an interpolant? How can we compute an interpolant whose C m norm is close to least possible? What if we require only that F and f agree approximately on E? What if we are allowed to discard a few points from E? An efficient solution to these problems would likely have practical applications. Joint work [19], [20] of B. Klartag and the author shows how to compute efficiently an interpolant F whose C m norm is within a factor C of least possible, where C is a constant depending only on m and n. Unfortunately, we do not know whether that constant C is absurdly large; we suspect that it is. To remedy this defect, and hopefully obtain results with practical applications, we pose the following sharper version of the interpolation problem.

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“…Recently bounded linear extensions operators of depth d depending on k and n only were constructed by Luli [Lu] for all spaces C k,ω b (S); their norms are bounded by C ω(1) , where C ∈ (1, ∞) is a constant depending on k and n only. (Earlier such extension operators were constructed for finite sets S by C. Fefferman [F2,Th. 8].…”

Section: Formulation Of Main Resultsmentioning

confidence: 99%