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This work provides a detailed and up-to-the-minute survey of the various stability problems that can affect suspension bridges. In order to deduce some experimental data and rules on the behavior of suspension bridges, a number of historical events are first described, in the course of which several questions concerning their stability naturally arise. The book then surveys conventional mathematical models for suspension bridges and suggests new nonlinear alternatives, which can potentially supply answers to some stability questions. New explanations are also provided, based on the nonlinear structural behavior of bridges. All the models and responses presented in the book employ the theory of differential equations and dynamical systems in the broader sense, demonstrating that methods from nonlinear analysis can allow us to determine the thresholds of instability.

Mathematics. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Mathematical Modeling and Industrial Mathematics. --- Structural Mechanics. --- Appl.Mathematics/Computational Methods of Engineering. --- Differential Equations. --- Differential equations, partial. --- Engineering mathematics. --- Mechanical engineering. --- Mathématiques --- Mathématiques de l'ingénieur --- Génie mécanique --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Suspension bridges --- Mathematical models. --- Bridges, Suspension --- Differential equations. --- Partial differential equations. --- Applied mathematics. --- Structural mechanics. --- Bridges --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Models, Mathematical --- Simulation methods

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This book develops a full theory for hinged beams and degenerate plates with multiple intermediate piers with the final purpose of understanding the stability of suspension bridges. New models are proposed and new tools are provided for the stability analysis. The book opens by deriving the PDE’s based on the physical models and by introducing the basic framework for the linear stationary problem. The linear analysis, in particular the behavior of the eigenvalues as the position of the piers varies, enables the authors to tackle the stability issue for some nonlinear evolution beam equations, with the aim of determining the “best position” of the piers within the beam in order to maximize its stability. The study continues with the analysis of a class of degenerate plate models. The torsional instability of the structure is investigated, and again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out by means of both analytical tools and numerical experiments. Several open problems and possible future developments are presented. The qualitative analysis provided in the book should be seen as the starting point for a precise quantitative study of more complete models, taking into account the action of aerodynamic forces. This book is intended for a two-fold audience. It is addressed both to mathematicians working in the field of Differential Equations, Nonlinear Analysis and Mathematical Physics, due to the rich number of challenging mathematical questions which are discussed and left as open problems, and to Engineers interested in mechanical structures, since it provides the theoretical basis to deal with models for the dynamics of suspension bridges with intermediate piers. More generally, it may be enjoyable for readers who are interested in the application of Mathematics to real life problems. .

Differential equations, Partial. --- Partial differential equations --- Differential equations. --- Partial differential equations. --- Ordinary Differential Equations. --- Partial Differential Equations. --- 517.91 Differential equations --- Differential equations

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This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on “near positivity.” The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the first part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenberg-type symmetry result is included. Finally, some recent progress on the Dirichlet problem for Willmore surfaces under symmetry assumptions is discussed.

Boundary value problems --- Mathematics --- Mathematical Theory --- Calculus --- Physical Sciences & Mathematics --- Boundary value problems. --- Boundary conditions (Differential equations) --- Mathematics. --- Functional analysis. --- Differential geometry. --- Continuum mechanics. --- Mathematics, general. --- Functional Analysis. --- Differential Geometry. --- Continuum Mechanics and Mechanics of Materials. --- Mechanics of continua --- Elasticity --- Mechanics, Analytic --- Field theory (Physics) --- Differential geometry --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Global differential geometry. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Geometry, Differential

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This book collects recent research papers by respected specialists in the field. It presents advances in the field of geometric properties for parabolic and elliptic partial differential equations, an area that has always attracted great attention. It settles the basic issues (existence, uniqueness, stability and regularity of solutions of initial/boundary value problems) before focusing on the topological and/or geometric aspects. These topics interact with many other areas of research and rely on a wide range of mathematical tools and techniques, both analytic and geometric. The Italian and Japanese mathematical schools have a long history of research on PDEs and have numerous active groups collaborating in the study of the geometric properties of their solutions. .

Mathematics. --- Functional analysis. --- Differential equations. --- Partial differential equations. --- Convex geometry. --- Discrete geometry. --- Calculus of variations. --- Partial Differential Equations. --- Functional Analysis. --- Ordinary Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Convex and Discrete Geometry. --- Differential equations, Parabolic --- Geometric analysis --- Differential equations, Elliptic --- Geometric analysis PDEs (Geometric partial differential equations) --- Parabolic differential equations --- Parabolic partial differential equations --- Geometry --- Mathematical analysis --- Differential equations, Partial --- Differential equations, partial. --- Differential Equations. --- Mathematical optimization. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Discrete mathematics --- Convex geometry . --- Isoperimetrical problems --- Variations, Calculus of --- Combinatorial geometry