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The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter \\center Motivés par un problème d’approximation à un mode pour une évolution avec accumulation de charge dans des puits de potentiel, nous considérons un problème d’évolution linéaire pour un opérateur de Schrödinger avec un potentiel dépendant du temps avec un puits dans une île. En particular, nous montrons que nous pouvons choisir le paramètre adiabatique

This volume contains the proceedings of the conference on Spectral Theory and Geometric Analysis, held at Northeastern University, Boston, MA, from July 29-August 2, 2009. The papers cover important topics in spectral theory and geometric analysis, such as resolutions of smooth group actions, spectral asymptotics, solutions of the Ginzburg-Landau equation, scattering theory, Riemann surfaces of infinite genus, tropical mathematics and geometric methods in the analysis of flows in porous media.

This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers. The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.

We develop a (global) solvability theory for Euler type linear partial differential equations P(θ)P(\\theta ) on C∞(Ω)C^\\infty (\\Omega ), with Ω\\Omega an open subset of Rd\\mathbb {R}^d, i.e., for a special type of linear partial differential equation with polynomial coefficients. There is a natural closed upper bound CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ) for the range of P(θ)P(\\theta ) on C∞(Ω)C^\\infty (\\Omega ). We characterize by P(θ)P(\\theta )-convexity type conditions those Ω\\Omega such that P(θ)P(\\theta ) is surjective on CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ). We also clarify when all shifted operators P(c+θ)P(c+\\theta ) are surjective on CI(P(c+ ⋅ ))∞(Ω)C^\\infty _{I(P(c+\\ \\cdot \\ ))}(\\Omega ). We classify in geometric terms those Ω\\Omega with 0∈Ω0\\in \\Omega such that every Euler operator P(θ)P(\\theta ) is surjective on CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ). Moreover, we determine the operators P(θ)P(\\theta ) which are surjective onto CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ) for every open set Ω⊆Rd\\Omega \\subseteq \\mathbb {R}^d. Under some mild assumptions on Ω\\Omega, we characterize those Euler operators which are invertible on C∞(Ω)C^\\infty (\\Omega ). Under the same assumptions we also calculate the spectrum of P(θ)P(\\theta ) on C∞(Ω)C^\\infty (\\Omega ). The results follow from the solvability theory for Hadamard type operators on the space of smooth functions and from a new general Mellin transform, both developed in this paper.

We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of the third order on a two dimensional manifold and show their application to the equivalence problem of such operators.

Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics. This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models. Global Carleman estimates are a priori estimates in weighted Sobolev norms for solutions of linear partial differential equations subject to boundary conditions. Such estimates proved to be extremely useful for several kinds of uniformly parabolic equations and systems. This is the first work where such estimates are derived for degenerate parabolic operators in dimension higher than one. Applications to null controllability with locally distributed controls and inverse source problems are also developed in full detail. Compared to nondegenerate parabolic problems, the current context requires major technical adaptations and a frequent use of Hardy type inequalities. On the other hand, the treatment is essentially self-contained, and only calls upon standard results in Lebesgue measure theory, functional analysis and ordinary differential equations.