TY - JOUR
T1 - A note on KAM theory for quasi-linear and fully nonlinear forced KdV
JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450
Y1 - 2013
A1 - P Baldi
A1 - Massimiliano Berti
A1 - Riccardo Montalto
KW - KAM for PDEs
AB - 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.
PB - European Mathematical Society
U1 - 7268
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -
TY - JOUR
T1 - Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces
JF - Duke Mathematical Journal
Y1 - 2011
A1 - Massimiliano Berti
A1 - Michela Procesi
AB - 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.
VL - 159
IS - 3
ER -
TY - JOUR
T1 - Non-compactness and multiplicity results for the Yamabe problem on Sn
JF - J. Funct. Anal. 180 (2001) 210-241
Y1 - 2001
A1 - Massimiliano Berti
A1 - Andrea Malchiodi
PB - Elsevier
UR - http://hdl.handle.net/1963/1345
U1 - 3110
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -