BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Sabre//Sabre VObject 4.5.8//EN CALSCALE:GREGORIAN BEGIN:VTIMEZONE TZID:Europe/Zurich X-LIC-LOCATION:Europe/Zurich TZURL:http://tzurl.org/zoneinfo/Europe/Zurich BEGIN:DAYLIGHT TZOFFSETFROM:+0100 TZOFFSETTO:+0200 TZNAME:CEST DTSTART:19810329T020000 RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU END:DAYLIGHT BEGIN:STANDARD TZOFFSETFROM:+0200 TZOFFSETTO:+0100 TZNAME:CET DTSTART:19961027T030000 RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU END:STANDARD END:VTIMEZONE BEGIN:VEVENT UID:news474@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181228T203836 DTSTART;TZID=Europe/Zurich:20141112T161500 SUMMARY:Seminar Analysis: Frédéric Robert (University of Lorraine) DESCRIPTION:We investigate the Hardy-Schrödinger operator Lγ=-Δ-γ/|x|2 on domains Ω⊂Rn\, whose boundary contain the singularity 0. The situat ion is quite different from the well-studied case when 0 is in the interio r of Ω. For one\, if 0∈Ω\, then L is positive if and only if γ<(n-2) 2/4\, while if 0∈∂Ω the operator L could be positive for larger value of γ\, potentially reaching the maximal constant n2/4 on convex domains. \\r\\nWe prove optimal regularity and a Hopf-type Lemma for variational so lutions of corresponding linear Dirichlet boundary value problems of the f orm Lγ=a(x)u\, but also for non-linear equations including Lγ=(|u|β-2u) /(|x|s)\, where γ < n2/4\, s∈[0\,2) and β:=2(n-s)/(n-2) is the critica l Hardy-Sobolev exponent. We also provide a Harnack inequality and a compl ete description of the profile of all positive solutions–variational or not– of the corresponding linear equation on the punctured domain. The value γ=(n-1)2/4 turned out to be another critical threshold for the oper ator Lγ\, and our analysis yields a corresponding notion of “Hardy sing ular boundary-mass” mγ(Ω) of a domain Ω having 0∈Ω\, which could b e defined whenever (n2-1)/4 < γ < n2/4.\\r\\nAs a byproduct\, we give a c omplete answer to problems of existence of extremals for Hardy-Sobolev ine qualities of the form\\r\\nC( ∫Ω (uβ)/(|x|s) dx )2/β ≤∫Ω |∇u| 2 dx - γ∫Ω (u2)/(|x|s)dx\\r\\nwhenever γγ= -Δ-γ/|x|2 on domains Ω⊂Rⁿ\, whose boundary contain the singularity 0. The situation is quite different from the well-studied case when 0 is in the interior of Ω. For one\, if 0∈Ω\, then L is po sitive if and only if γ<\;(n-2)²/4\, while if 0∈∂Ω the operator L could be positive for larger value of γ\, potentially reaching the maximal constant n²/4 on convex domains.\nWe prove optimal regularity and a Hopf-type Lemma for variational solutions of correspondi ng linear Dirichlet boundary value problems of the form Lγ=a(x)u\, but al so for non-linear equations including Lγ=(|u|^β-2u)/(|x|^s)\, where γ <\; n²/4\, s∈[0\,2) and β:=2(n-s)/(n-2) is the critical Hardy-Sobolev exponent. We also provide a Harnack inequal ity and a complete description of the profile of all positive solutions– variational or not– of the corresponding linear equation on the puncture d domain. The value γ=(n-1)²/4 turned out to be another criti cal threshold for the operator Lγ\, and our analysis yields a correspondi ng notion of “Hardy singular boundary-mass” mγ(Ω) of a domain Ω hav ing 0∈Ω\, which could be defined whenever (n²-1)/4 <\; γ <\; n²/4.\nAs a byproduct\, we give a complete answer to prob lems of existence of extremals for Hardy-Sobolev inequalities of the form\ nC( ∫_Ω(u^β)/(|x|^s) dx )^{2/β
\;}≤∫_Ω |∇u|² dx - γ∫_Ω&nbs p\;(u²)/(|x|^s)dx\nwhenever γ<\;n²/4\, and in particular\, for those of Caffarelli-Kohn-Nirenberg. These results extend previous contributions by the authors in the case γ=0\, and by Che rn-Lin for the case γ<\;(n-2)²/4. Namely\, if 0≤γ≤(n< sup>2-1)/4\, then the negativity of the mean curvature of ∂ Ω at 0 is sucient for the existence of extremals. This is however not su fficient for (n²-1)/4≤γ≤(n²)/4\, which then req uires the positivity of the Hardy singular boundary-massof the domain unde r consideration.\nJoint work with Nassif Ghoussoub. DTEND;TZID=Europe/Zurich:20141112T171500 END:VEVENT END:VCALENDAR