We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

%B C. R. Math. Acad. Sci. Paris %I Elsevier %V 352 %P 603-607 %G en %U http://urania.sissa.it/xmlui/handle/1963/35067 %N 7-8 %1 35302 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-11-30T15:30:40Z (GMT) No. of bitstreams: 0 %R 10.1016/j.crma.2014.04.012 %0 Journal Article %J Arch. Ration. Mech. Anal. %D 2014 %T KAM for Reversible Derivative Wave Equations %A Massimiliano Berti %A Luca Biasco %A Michela Procesi %XWe prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

%B Arch. Ration. Mech. Anal. %I Springer %V 212 %P 905-955 %G en %U http://urania.sissa.it/xmlui/handle/1963/34646 %N 3 %1 34850 %2 Mathematics %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-10-14T16:55:00Z No. of bitstreams: 1 preprint2014.pdf: 620515 bytes, checksum: d1d981f350e63c9906f793bcfe66e972 (MD5) %R 10.1007/s00205-014-0726-0 %0 Conference Paper %B The 24th International Ocean and Polar Engineering Conference %D 2014 %T Potential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures %A Andrea Mola %A Luca Heltai %A Antonio DeSimone %A Massimiliano Berti %B The 24th International Ocean and Polar Engineering Conference %I International Society of Offshore and Polar Engineers %V 4 %P 815–822 %G eng %0 Journal Article %J Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni %D 2013 %T Existence and stability of quasi-periodic solutions for derivative wave equations %A Massimiliano Berti %A Luca Biasco %A Michela Procesi %K Constant coefficients %K Dynamical systems %K Existence and stability %K Infinite dimensional %K KAM for PDEs %K Linearized equations %K Lyapunov exponent %K Lyapunov methods %K Quasi-periodic solution %K Small divisors %K Wave equations %X In this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*. %B Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni %V 24 %P 199-214 %G eng %R 10.4171/RLM/652 %0 Journal Article %J Annales Scientifiques de l'Ecole Normale Superieure %D 2013 %T KAM theory for the Hamiltonian derivative wave equation %A Massimiliano Berti %A Luca Biasco %A Michela Procesi %XWe prove an infinite dimensional KAM theorem which implies the existence of Can- tor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. © 2013 Société Mathématique de France.

%B Annales Scientifiques de l'Ecole Normale Superieure %V 46 %P 301-373 %G eng %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 No. of bitstreams: 1 Baldi-Berti-Montalto-Lincei-Note-KAM-forced-KdV.pdf: 322654 bytes, checksum: ddae153eff7ad17e2be36cb3ba1af9bf (MD5) %R 10.4171/RLM/660 %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 Communications in Mathematical Physics %D 2011 %T Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs %A Massimiliano Berti %A Luca Biasco %X We consider infinite dimensional Hamiltonian systems. We prove the existence of "Cantor manifolds" of elliptic tori-of any finite higher dimension-accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are "branching" points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation. © 2011 Springer-Verlag. %B Communications in Mathematical Physics %V 305 %P 741-796 %G eng %R 10.1007/s00220-011-1264-3 %0 Journal Article %J Journal of Differential Equations %D 2011 %T Degenerate KAM theory for partial differential equations %A Dario Bambusi %A Massimiliano Berti %A Elena Magistrelli %X This paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given. © 2010 Elsevier Inc. %B Journal of Differential Equations %V 250 %P 3379-3397 %G eng %R 10.1016/j.jde.2010.11.002 %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 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 SIAM J. Math. Anal. 40 (2008) 382-412 %D 2008 %T Forced Vibrations of a Nonhomogeneous String %A P Baldi %A Massimiliano Berti %X We prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov-Schmidt reduction and a Nash-Moser iteration scheme. %B SIAM J. Math. Anal. 40 (2008) 382-412 %G en_US %U http://hdl.handle.net/1963/2643 %1 1480 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-05-07T08:29:52Z\\nNo. of bitstreams: 1\\nBaldiBerti06-1.pdf: 277037 bytes, checksum: 6abb75a412da123c87879c25714e41b2 (MD5) %R 10.1137/060665038 %0 Journal Article %J Communications on Pure and Applied Analysis %D 2008 %T On periodic elliptic equations with gradient dependence %A Massimiliano Berti %A Matzeu, M %A Enrico Valdinoci %X We construct entire solutions of Δu = f(x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope. %B Communications on Pure and Applied Analysis %V 7 %P 601-615 %G eng %0 Journal Article %J NATO Science for Peace and Security Series B: Physics and Biophysics %D 2008 %T Variational methods for Hamiltonian PDEs %A Massimiliano Berti %X We present recent existence results of periodic solutions for completely resonant nonlinear wave equations in which both "small divisor" difficulties and infinite dimensional bifurcation phenomena occur. These results can be seen as generalizations of the classical finite-dimensional resonant center theorems of Weinstein-Moser and Fadell-Rabinowitz. The proofs are based on variational bifurcation theory: after a Lyapunov-Schmidt reduction, the small divisor problem in the range equation is overcome with a Nash-Moser implicit function theorem for a Cantor set of non-resonant parameters. Next, the infinite dimensional bifurcation equation, variational in nature, possesses minimax mountain-pass critical points. The big difficulty is to ensure that they are not in the "Cantor gaps". This is proved under weak non-degeneracy conditions. Finally, we also discuss the existence of forced vibrations with rational frequency. This problem requires variational methods of a completely different nature, such as constrained minimization and a priori estimates derivable from variational inequalities. © 2008 Springer Science + Business Media B.V. %B NATO Science for Peace and Security Series B: Physics and Biophysics %P 391-420 %@ 9781402069628 %G eng %R 10.1007/978-1-4020-6964-2-16 %0 Journal Article %J SIAM J. Math. Anal. 37 (2006) 83-102 %D 2006 %T A Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs %A Dario Bambusi %A Massimiliano Berti %X In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schr\\\\\\\"odinger equation with smoothing nonlinearity. %B SIAM J. Math. Anal. 37 (2006) 83-102 %G en_US %U http://hdl.handle.net/1963/2159 %1 2085 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-02T09:24:09Z\\nNo. of bitstreams: 1\\n0310182v1.pdf: 221749 bytes, checksum: 1ca47fecc44576d771b14ead5f53db0e (MD5) %R 10.1137/S0036141003436107 %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 Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 439-474 %D 2006 %T Forced vibrations of wave equations with non-monotone nonlinearities %A Massimiliano Berti %A Luca Biasco %X We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz. %B Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 439-474 %G en_US %U http://hdl.handle.net/1963/2160 %1 2084 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-02T09:35:53Z\\nNo. of bitstreams: 1\\n0410619v1.pdf: 401724 bytes, checksum: 1aeb5616e38d96fffc8efa0b0e6cdc14 (MD5) %R 10.1016/j.anihpc.2005.05.004 %0 Journal Article %J Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni %D 2006 %T Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies %A P Baldi %A Massimiliano Berti %X We prove existence and multiplicity of small amplitude periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for asymptotically full measure sets of frequencies, extending the results of [7] to new types of nonlinearities. %B Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni %V 17 %P 257-277 %G eng %0 Journal Article %J Comm. Partial Differential Equations 31 (2006) 959 - 985 %D 2006 %T Quasi-periodic solutions of completely resonant forced wave equations %A Massimiliano Berti %A Michela Procesi %X We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number. %B Comm. Partial Differential Equations 31 (2006) 959 - 985 %G en_US %U http://hdl.handle.net/1963/2234 %1 2010 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-16T07:48:45Z\\nNo. of bitstreams: 1\\n0504406v1.pdf: 330239 bytes, checksum: 5dbf59bdd590a6876ea206f70cf0ecc9 (MD5) %R 10.1080/03605300500358129 %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124 %D 2005 %T Periodic solutions of nonlinear wave equations with non-monotone forcing terms %A Massimiliano Berti %A Luca Biasco %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124 %I Accademia Nazionale dei Lincei %G en %U http://hdl.handle.net/1963/4581 %1 4349 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T10:23:29Z\\nNo. of bitstreams: 1\\nBertiBiasco04-1.pdf: 267180 bytes, checksum: 2e0e4b98f4985c1e79dc4c03fb30618e (MD5) %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %D 2005 %T Quasi-periodic oscillations for wave equations under periodic forcing %A Massimiliano Berti %A Michela Procesi %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %I Accademia Nazionale dei Lincei %G en %U http://hdl.handle.net/1963/4583 %1 4350 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T10:27:05Z\\nNo. of bitstreams: 1\\nBertiProcesi05-1.pdf: 211758 bytes, checksum: b6c3ae059191cddb5c025aee61a23799 (MD5) %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 Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138 %D 2004 %T Periodic orbits close to elliptic tori and applications to the three-body problem %A Massimiliano Berti %A Luca Biasco %A Enrico Valdinoci %X We prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.) %B Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138 %I Scuola Normale Superiore di Pisa %G en_US %U http://hdl.handle.net/1963/2985 %1 1348 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-30T09:03:01Z\\nNo. of bitstreams: 1\\n0304103v1.pdf: 482533 bytes, checksum: 5da7f32109202edb44f004885b665b48 (MD5) %0 Journal Article %J Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661 %D 2004 %T Soluzioni periodiche di PDEs Hamiltoniane %A Massimiliano Berti %B Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661 %I Unione Matematica Italiana %G en %U http://hdl.handle.net/1963/4582 %1 4351 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T10:33:48Z\\nNo. of bitstreams: 1\\nberti2004.pdf: 249023 bytes, checksum: bab6b2d956c1de46de694dfca3bc4cec (MD5) %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 Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002 %D 2002 %T Arnold diffusion: a functional analysis approach %A Massimiliano Berti %X We present, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to study the “splitting of the whiskers” and the “shadowing problem” developed in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the problem of Arnold diffusion for nearly integrable partially isochronous systems improving known results. %B Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002 %I Natsīonal. Akad. Nauk Ukraïni %G en %1 7269 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-12-09T08:37:48Z No. of bitstreams: 1 Berti712-719.pdf: 135161 bytes, checksum: 75afb14b28171db9cab374b3ccbbc265 (MD5) %0 Journal Article %J Nonlinear Anal. 48 (2002) 481-504 %D 2002 %T Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems %A Massimiliano Berti %A Carlo Carminati %B Nonlinear Anal. 48 (2002) 481-504 %I Elsevier %G en %U http://hdl.handle.net/1963/1279 %1 3176 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:55:42Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1999 %R 10.1016/S0362-546X(00)00200-5 %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 %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 %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