Mostra el registre d'ítem simple

dc.contributor.authorRos Oton, Xavier
dc.contributor.authorSerra Montolí, Joaquim
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
dc.date.accessioned2014-07-14T11:48:18Z
dc.date.created2014-07-01
dc.date.issued2014-07-01
dc.identifier.citationRos, X.; Serra, J. The extremal solution for the fractional Laplacian. "Calculus of variations and partial differential equations", 01 Juliol 2014, vol. 50, núm. 3-4, p. 723-750.
dc.identifier.issn0944-2669
dc.identifier.urihttp://hdl.handle.net/2117/23499
dc.description.abstractWe study the extremal solution for the problem (-¿)su=¿f(u) in O , u=0 in Rn\O , where ¿>0 is a parameter and s¿(0,1) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions n<4s . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever n<10s . In the limit s¿1 , n<10 is optimal. In addition, we show that the extremal solution is Hs(Rn) in any dimension whenever the domain is convex. To obtain some of these results we need Lq estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with Lp data. We prove optimal Lq and Cß estimates, depending on the value of p . These estimates follow from classical embedding results for the Riesz potential in Rn . Finally, to prove the Hs regularity of the extremal solution we need an L8 estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.
dc.format.extent28 p.
dc.language.isoeng
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica aplicada a les ciències
dc.subject.lcshCalculus of variations and optimal control
dc.subject.otherSEMILINEAR ELLIPTIC-EQUATIONS
dc.subject.otherVARIATIONAL-METHODS
dc.subject.otherPOSITIVE SOLUTIONS
dc.subject.otherDIMENSION 4
dc.subject.otherREGULARITY
dc.subject.otherBOUNDEDNESS
dc.subject.otherMINIMIZERS
dc.subject.otherOPERATORS
dc.subject.otherINEQUALITIES
dc.titleThe extremal solution for the fractional Laplacian
dc.typeArticle
dc.subject.lemacCàlcul de variacions
dc.identifier.doi10.1007/s00526-013-0653-1
dc.description.peerreviewedPeer Reviewed
dc.relation.publisherversionhttp://link.springer.com/article/10.1007%2Fs00526-013-0653-1
dc.rights.accessRestricted access - publisher's policy
local.identifier.drac14977828
dc.description.versionPostprint (published version)
dc.date.lift10000-01-01
local.citation.authorRos, X.; Serra, J.
local.citation.publicationNameCalculus of variations and partial differential equations
local.citation.volume50
local.citation.number3-4
local.citation.startingPage723
local.citation.endingPage750


Fitxers d'aquest items

Imatge en miniatura

Aquest ítem apareix a les col·leccions següents

Mostra el registre d'ítem simple