%0 Journal Article
%J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450
%D 2013
%T A note on KAM theory for quasi-linear and fully nonlinear forced KdV
%A P Baldi
%A Massimiliano Berti
%A Riccardo Montalto
%K KAM for PDEs
%X We present the recent results in [3] concerning quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities the solutions are linearly stable. The proofs are based on a combination of di erent ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a di erential operator with constant coe cients plus a bounded remainder. These transformations are obtained by changes of variables induced by di eomorphisms of the torus and pseudo-di erential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant coe cients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.
%B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450
%I European Mathematical Society
%G en
%1 7268
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-12-09T08:16:04Z
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%R 10.4171/RLM/660
%0 Journal Article
%J Duke Mathematical Journal
%D 2011
%T Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces
%A Massimiliano Berti
%A Michela Procesi
%X We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.
%B Duke Mathematical Journal
%V 159
%8 2011
%G eng
%N 3
%& 479
%R 10.1215/00127094-1433403
%0 Journal Article
%J J. Funct. Anal. 180 (2001) 210-241
%D 2001
%T Non-compactness and multiplicity results for the Yamabe problem on Sn
%A Massimiliano Berti
%A Andrea Malchiodi
%B J. Funct. Anal. 180 (2001) 210-241
%I Elsevier
%G en
%U http://hdl.handle.net/1963/1345
%1 3110
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T12:56:35Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999
%R 10.1006/jfan.2000.3699