%0 Journal Article
%J Journal of the European Mathematical Society
%D 2013
%T Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential
%A Massimiliano Berti
%A Philippe Bolle
%X We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td , d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates. © European Mathematical Society 2013.
%B Journal of the European Mathematical Society
%V 15
%P 229-286
%G eng
%R 10.4171/JEMS/361
%0 Journal Article
%J Nonlinearity
%D 2012
%T Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential
%A Massimiliano Berti
%A Philippe Bolle
%X We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T d , d ≥ 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the 'separation properties' of the small divisors assuming weaker non-resonance conditions than in [11]. © 2012 IOP Publishing Ltd.
%B Nonlinearity
%V 25
%P 2579-2613
%G eng
%R 10.1088/0951-7715/25/9/2579
%0 Journal Article
%J Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
%D 2010
%T An abstract Nash-Moser theorem with parameters and applications to PDEs
%A Massimiliano Berti
%A Philippe Bolle
%A Michela Procesi
%K Abstracting
%K Aircraft engines
%K Finite dimensional
%K Hamiltonian PDEs
%K Implicit function theorem
%K Invariant tori
%K Iterative schemes
%K Linearized operators
%K Mathematical operators
%K Moser theorem
%K Non-Linearity
%K Nonlinear equations
%K Nonlinear wave equation
%K Periodic solution
%K Point of interest
%K Resonance phenomena
%K Small divisors
%K Sobolev
%K Wave equations
%X We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. © 2009 Elsevier Masson SAS. All rights reserved.
%B Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
%V 27
%P 377-399
%G eng
%R 10.1016/j.anihpc.2009.11.010
%0 Journal Article
%J Archive for Rational Mechanics and Analysis
%D 2010
%T Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions
%A Massimiliano Berti
%A Philippe Bolle
%X We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash-Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large "clusters of small divisors". © Springer-Verlag (2009).
%B Archive for Rational Mechanics and Analysis
%V 195
%P 609-642
%G eng
%R 10.1007/s00205-008-0211-8
%0 Journal Article
%J Frontiers of Mathematics in China
%D 2008
%T Cantor families of periodic solutions for completely resonant wave equations
%A Massimiliano Berti
%A Philippe Bolle
%X We present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods. © 2008 Higher Education Press.
%B Frontiers of Mathematics in China
%V 3
%P 151-165
%G eng
%R 10.1007/s11464-008-0011-3
%0 Journal Article
%J Advances in Mathematics
%D 2008
%T Cantor families of periodic solutions for wave equations via a variational principle
%A Massimiliano Berti
%A Philippe Bolle
%X We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation-variational in nature-defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to "small divisors" phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical "Arnold non-degeneracy condition" of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities. © 2007 Elsevier Inc. All rights reserved.
%B Advances in Mathematics
%V 217
%P 1671-1727
%G eng
%R 10.1016/j.aim.2007.11.004
%0 Journal Article
%J Nonlinear Differential Equations and Applications
%D 2008
%T Cantor families of periodic solutions of wave equations with C k nonlinearities
%A Massimiliano Berti
%A Philippe Bolle
%X We prove bifurcation of Cantor families of periodic solutions for wave equations with nonlinearities of class C k . It requires a modified Nash-Moser iteration scheme with interpolation estimates for the inverse of the linearized operators and for the composition operators. © 2008 Birkhaueser.
%B Nonlinear Differential Equations and Applications
%V 15
%P 247-276
%G eng
%R 10.1007/s00030-007-7025-5
%0 Journal Article
%J Duke Math. J. 134 (2006) 359-419
%D 2006
%T Cantor families of periodic solutions for completely resonant nonlinear wave equations
%A Massimiliano Berti
%A Philippe Bolle
%X We prove the existence of small amplitude, $2\\\\pi \\\\slash \\\\om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \\\\om $ belonging to a Cantor-like set of positive measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem. In spite of the complete resonance of the equation we show that we can still reduce the problem to a {\\\\it finite} dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows to deal also with nonlinearities which are not odd and with finite spatial regularity.
%B Duke Math. J. 134 (2006) 359-419
%G en_US
%U http://hdl.handle.net/1963/2161
%1 2083
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-02T09:53:35Z\\nNo. of bitstreams: 1\\n0410618v1.pdf: 427752 bytes, checksum: 08750b5fadd830fbe76aa48d54824f17 (MD5)
%R 10.1215/S0012-7094-06-13424-5
%0 Journal Article
%J Boll. Unione Mat. Ital. Sez. B 7 (2004) 519-528
%D 2004
%T Bifurcation of free vibrations for completely resonant wave equations
%A Massimiliano Berti
%A Philippe Bolle
%X We prove existence of small amplitude, 2 pi/omega -periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency omega belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.
%B Boll. Unione Mat. Ital. Sez. B 7 (2004) 519-528
%G en_US
%U http://hdl.handle.net/1963/2245
%1 1999
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-17T09:55:20Z\\nNo. of bitstreams: 1\\n0409052v1.pdf: 145050 bytes, checksum: 68913e2c100ffa7488447933554c30e7 (MD5)
%0 Journal Article
%J Nonlinear Anal. 56 (2004) 1011-1046
%D 2004
%T Multiplicity of periodic solutions of nonlinear wave equations
%A Massimiliano Berti
%A Philippe Bolle
%B Nonlinear Anal. 56 (2004) 1011-1046
%I Elsevier
%G en_US
%U http://hdl.handle.net/1963/2974
%1 1359
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-29T12:46:01Z\\nNo. of bitstreams: 1\\nBertiBolle04.pdf: 387346 bytes, checksum: 6374d2229ca34f3936e690a01f4b4eab (MD5)
%R 10.1016/j.na.2003.11.001
%0 Journal Article
%J J. Math. Pures Appl. 82 (2003) 613-664
%D 2003
%T Drift in phase space: a new variational mechanism with optimal diffusion time
%A Massimiliano Berti
%A Luca Biasco
%A Philippe Bolle
%X We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\\\\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ \\\\mu) \\\\log (1/ \\\\mu))$ by a variational method which does not require the existence of ``transition chains of tori\\\'\\\' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.
%B J. Math. Pures Appl. 82 (2003) 613-664
%I Elsevier
%G en_US
%U http://hdl.handle.net/1963/3020
%1 1313
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-02T15:07:59Z\\nNo. of bitstreams: 1\\n0205307v1.pdf: 505505 bytes, checksum: 2dab01ff574df56912b2fe8d3c56108a (MD5)
%R 10.1016/S0021-7824(03)00032-1
%0 Journal Article
%J Comm.Math.Phys. 243 (2003) no.2, 315
%D 2003
%T Periodic solutions of nonlinear wave equations with general nonlinearities
%A Massimiliano Berti
%A Philippe Bolle
%B Comm.Math.Phys. 243 (2003) no.2, 315
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1648
%1 2470
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T13:05:51Z (GMT). No. of bitstreams: 1\\nmath.AP0211310.pdf: 363476 bytes, checksum: 15bbb8f96ff0c106ea4dc1343e19afa4 (MD5)\\n Previous issue date: 2002
%R 10.1007/s00220-003-0972-8
%0 Journal Article
%J Discrete Contin. Dyn. Syst. 8 (2002) 795-811
%D 2002
%T Fast Arnold diffusion in systems with three time scales
%A Massimiliano Berti
%A Philippe Bolle
%X We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the \\\"splitting determinant\\\" is exponentially small.
%B Discrete Contin. Dyn. Syst. 8 (2002) 795-811
%I American Institute of Mathematical Sciences
%G en_US
%U http://hdl.handle.net/1963/3058
%1 1275
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-09T16:26:36Z\\nNo. of bitstreams: 1\\n0103065v1.pdf: 229503 bytes, checksum: eb476d4f1629873c6d0688c3a4a9dc61 (MD5)
%0 Journal Article
%J Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450
%D 2002
%T A functional analysis approach to Arnold diffusion
%A Massimiliano Berti
%A Philippe Bolle
%X We discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the \\\"splitting of separatrices\\\" and to the \\\"shadowing problem\\\". As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.
%B Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450
%I Elsevier
%G en_US
%U http://hdl.handle.net/1963/3151
%1 1182
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-20T14:35:29Z\\nNo. of bitstreams: 1\\nArnold_diffusion.pdf: 397794 bytes, checksum: 16e32aa6198ef17e8f55e7292307ee09 (MD5)
%R 10.1016/S0294-1449(01)00084-1
%0 Journal Article
%D 2002
%T An optimal fast-diffusion variational method for non isochronous system
%A Luca Biasco
%A Massimiliano Berti
%A Philippe Bolle
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1579
%1 2539
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T13:04:50Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2002
%0 Journal Article
%J Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84
%D 2002
%T Optimal stability and instability results for a class of nearly integrable Hamiltonian systems
%A Massimiliano Berti
%A Luca Biasco
%A Philippe Bolle
%B Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1596
%1 2522
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T13:05:04Z (GMT). No. of bitstreams: 1\\nmath.DS0203188.pdf: 131832 bytes, checksum: 468589170f7f79086f08c3a5902e94d7 (MD5)\\n Previous issue date: 2002
%0 Journal Article
%D 2000
%T Arnold's Diffusion in nearly integrable isochronous Hamiltonian systems
%A Massimiliano Berti
%A Philippe Bolle
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1554
%1 2564
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T13:04:03Z (GMT). No. of bitstreams: 1\\nmath.DS0011017.pdf: 394514 bytes, checksum: 1c7ce3b0641b2ada647963d5a00697e0 (MD5)\\n Previous issue date: 2000
%0 Journal Article
%J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 2000, 11, 235
%D 2000
%T Diffusion time and splitting of separatrices for nearly integrable
%A Massimiliano Berti
%A Philippe Bolle
%B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 2000, 11, 235
%I SISSA Library
%G en
%U http://hdl.handle.net/1963/1547
%1 2571
%2 Mathematics
%3 Functional Analysis and Applications
%$ Made available in DSpace on 2004-09-01T13:03:57Z (GMT). No. of bitstreams: 1\\nmath.DS0009176.pdf: 135686 bytes, checksum: 0da5be3a4a495a53dbf6cadb69d7cb2e (MD5)\\n Previous issue date: 2000