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dc.contributor.author
Figalli, Alessio
dc.contributor.author
Glaudo, Federico
dc.date.accessioned
2020-12-24T06:17:09Z
dc.date.available
2020-12-24T06:17:09Z
dc.date.issued
2020-07
dc.identifier.issn
0003-9527
dc.identifier.issn
1432-0673
dc.identifier.other
10.1007/s00205-020-01506-6
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/458524
dc.description.abstract
Given n≥ 3 , consider the critical elliptic equation Δu+u2∗-1=0 in Rn with u> 0. This equation corresponds to the Euler–Lagrange equation induced by the Sobolev embedding H1(Rn)↪L2∗(Rn), and it is well-known that the solutions are uniquely characterized and are given by the so-called “Talenti bubbles”. In addition, thanks to a fundamental result by Struwe (Math Z 187(4):511–517, 1984), this statement is “stable up to bubbling”: if u:Rn→(0,∞)almost solves Δu+u2∗-1=0 then u is (nonquantitatively) close in the H1(Rn) -norm to a sum of weakly-interacting Talenti bubbles. More precisely, if δ(u) denotes the H1(Rn) -distance of u from the manifold of sums of Talenti bubbles, Struwe proved that δ(u) → 0 as [InlineEquation not available: see fulltext.]. In this paper we investigate the validity of a sharp quantitative version of the stability for critical points: more precisely, we ask whether under a bound on the energy [InlineEquation not available: see fulltext.] (that controls the number of bubbles) it holds that [Equation not available: see fulltext.]A recent paper by the first author together with Ciraolo and Maggi (Int Math Res Not 2018(21):6780–6797, 2017) shows that the above result is true if u is close to only one bubble. Here we prove, to our surprise, that whenever there are at least two bubbles then the estimate above is true for 3 ≤ n≤ 5 while it is false for n≥ 6. To our knowledge, this is the first situation where quantitative stability estimates depend so strikingly on the dimension of the space, changing completely behavior for some particular value of the dimension n. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
en_US
dc.language.iso
en
en_US
dc.publisher
Springer
en_US
dc.title
On the Sharp Stability of Critical Points of the Sobolev Inequality
en_US
dc.type
Journal Article
dc.date.published
2020-03-12
ethz.journal.title
Archive for Rational Mechanics and Analysis
ethz.journal.volume
237
en_US
ethz.journal.issue
1
en_US
ethz.journal.abbreviated
Arch. ration. mech. anal.
ethz.pages.start
201
en_US
ethz.pages.end
258
en_US
ethz.grant
Regularity and Stability in Partial Differential Equations
en_US
ethz.identifier.wos
ethz.identifier.scopus
ethz.publication.place
Berlin
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02003 - Mathematik Selbständige Professuren::09565 - Figalli, Alessio / Figalli, Alessio
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02003 - Mathematik Selbständige Professuren::09565 - Figalli, Alessio / Figalli, Alessio
en_US
ethz.grant.agreementno
721675
ethz.grant.fundername
EC
ethz.grant.funderDoi
10.13039/501100000780
ethz.grant.program
H2020
ethz.date.deposited
2020-04-30T02:40:22Z
ethz.source
SCOPUS
ethz.source
FORM
ethz.eth
yes
en_US
ethz.availability
Metadata only
en_US
ethz.rosetta.installDate
2021-02-15T22:50:14Z
ethz.rosetta.lastUpdated
2021-02-15T22:50:14Z
ethz.rosetta.versionExported
true
dc.identifier.olduri
http://hdl.handle.net/20.500.11850/412432
dc.identifier.olduri
http://hdl.handle.net/20.500.11850/457790
ethz.COinS
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