Non-linear ground state representations and sharp Hardy inequalities

By Rupert L. Frank, Robert Seiringer. 2008

497 citations recorded by the source corpus.

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.

Subfields: Applied Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/1481b3765003845e4fd974d60701ac1e644ec457

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In Applied Mathematics

1. Giampiero Palatucci
2. Enrico Valdinoci
3. Lorenzo Brasco
4. Rupert L. Frank
5. Vicentiu D. Rădulescu
6. Boumediene Abdellaoui
7. Patrizia Pucci
8. Binlin Zhang

In Mechanics of Materials

1. Boumediene Abdellaoui
2. Rupert L. Frank
3. Michael Loss
4. Bartłomiej Dyda
5. Lorenzo Brasco
6. Rachid Bentifour
7. Michael Ruzhansky
8. Durvudkhan Suragan

Citations in the corpus

• Fractional Orlicz-Sobolev embeddings (2020, background): “stant Csuch that (1.2) kuk L np n−sp (Rn) ≤ C|u|s,p,Rn for every measurable function udecaying to 0 (in a suitable sense) near infinity. An improved version of inequality (1.2) has been established in [54]. It asserts that, in fact, (1.3) kuk L np n−sp,p(Rn) ≤ C|u|s,p,Rn , Date: January 17, 2020. 200”
• Anisotropic fractional perimeters (2013, unknown): “Using a symmetrization result by Almgren & Lieb [1], Frank & Seiringer [14] proved that there is equality in (5) precisely for balls (up to sets of measure zero).”
• Fractional Caffarelli–Kohn–Nirenberg inequalities (2017, unknown): “Assertion ii) was established in [6] for a = 1, τ = p, α1 = α2 = 0, and γ = −s.”
• Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces (2013, background): “1 where 2∗ = 2N/(N − 2s) is the critical Sobolev exponent, so that 2∗ ∈ (2,∞) as s ∈ (0, N/2), which allows to define the fractional Laplacian e. g. as a bounded linear operator (−∆)s/2 : Ḣs(RN )→ L2(RN ).”
• Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications (2016, methodology): “In the whole of the paper we will use the next elementary inequality, see for instance [16].”
• Pitt's inequality and the fractional Laplacian: Sharp error estimates (2007, unknown): “Then 1 4 n=2 .n=2/ Z RC RC jg.s/ g.t/j2 ' hs t C t s 2 i ds s dt t 2 n Z RC ln jgj jgj2 dt t CEn (3.1) and Theorem 3.1.”
• Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R (2010, background): “4) follows directly from the ground state uniqueness and the strict rearrangement inequalities in [14]; see also Section 5 below.”
• On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups (2016, background): “To give an idea for obtaining more general improved weighted Hardy type inequalities let us conclude this paper with the following very short discussion of techniques from [47] (see also [24] and [42]), now in the setting of stratified groups.”
• Uniqueness of Radial Solutions for the Fractional Laplacian (2013, background): “0; 1/ (see [8, 21]) we deduce that any optimizer v 2 H .”
• The quantitative isoperimetric inequality and related topics (2015, background): “42) can be deduced from a symmetrization result due to Almgren and Lieb [4], while the cases of equality have been determined in [71].”