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Titolo:
Concentration of low energy extremals
Autore:
Flucher, M; Muller, S;
Indirizzi:
GFAI AG, Banking Syst, CH-4503 Solothurn, Switzerland GFAI AG Solothurn Switzerland CH-4503 st, CH-4503 Solothurn, Switzerland Max Planck Inst Math Nat Wissensch, D-04103 Leipzig, Germany Max Planck Inst Math Nat Wissensch Leipzig Germany D-04103 pzig, Germany
Titolo Testata:
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
fascicolo: 3, volume: 16, anno: 1999,
pagine: 269 - 298
SICI:
0294-1449(199905/06)16:3<269:COLEE>2.0.ZU;2-9
Fonte:
ISI
Lingua:
ENG
Keywords:
variational problem; concentration; critical Sobolev exponent; flee-boundary problem;
Tipo documento:
Article
Natura:
Periodico
Settore Disciplinare:
Physical, Chemical & Earth Sciences
Citazioni:
12
Recensione:
Indirizzi per estratti:
Indirizzo: Flucher, M GFAIndG, Banking Syst, Gludz Blotzheimerstr 1, CH-4503 Solothurn, Switzerla GFAI AG Gludz Blotzheimerstr 1 Solothurn Switzerland CH-4503 a
Citazione:
M. Flucher e S. Muller, "Concentration of low energy extremals", ANN IHP-AN, 16(3), 1999, pp. 269-298

Abstract

We study variational problems of the formsup{integral(Omega)F(u) : integral(Omega)\del u\(2) less than or equal to epsilon(2), u = 0 on partial derivative Omega}with small epsilon and 0 less than or equal to F(t) less than or equal to c\t\(2n/n-2) on a domain of dimension n greater than or equal to 3. The corresponding Euler Lagrange equation is a semilinear Dirichlet problem-Delta u = lambda f(u) in Omega,u = 0 on partial derivative Omegawith f = F' and a large Lagrange multiplier lambda. Our goal is to obtain qualitative information on the extremals u(epsilon) for small epsilon. The integrand F can be nonconvex and discontinuous, Thus our results apply to nonlinear eigenvalue problems as well as to certain free-boundary problems. Our starting point is a generalized Sobolev inequality that covers the classical Sobolev inequality and the isoperimetric inequality relating capacity and volume as special cases. Using a local version of this inequality we prove a generalized concentration-compactness alternative and show that as epsilon --> 0 the extremals concentrate at a single point, The local behaviour of the extremals near the concentration point depends only on F. On a microscopic scale they tend to an extremal for the generalized Sobolev constant on IRn provided that F satisfies certain growth conditions at 0 and infinity. (C) Elsevier, Paris.

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Documento generato il 08/04/20 alle ore 12:05:33