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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.
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.
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.
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.
Linear elliptic equations arise in several models describing
various phenomena in the applied sciences, the most famous being
the second order stationary heat eq- tion or,equivalently,the
membraneequation. Forthis intensivelywell-studiedlinear problem
there are two main lines of results. The ?rst line consists of
existence and regularity results. Usually the solution exists and
"gains two orders of differen- ation" with respect to the source
term. The second line contains comparison type results, namely the
property that a positive source term implies that the solution is
positive under suitable side constraints such as homogeneous
Dirichlet bou- ary conditions. This property is often also called
positivity preserving or, simply, maximum principle. These kinds of
results hold for general second order elliptic problems, see the
books by Gilbarg-Trudinger [198] and Protter-Weinberger [347]. For
linear higher order elliptic problems the existence and
regularitytype results - main, as one may say, in their full
generality whereas comparison type results may fail. Here and in
the sequel "higher order" means order at least four. Most
interesting models, however, are nonlinear. By now, the theory of
second order elliptic problems is quite well developed for
semilinear, quasilinear and even for some fully nonlinear problems.
If one looks closely at the tools being used in the proofs, then
one ?nds that many results bene?t in some way from the positivity
preserving property. Techniques based on Harnack's inequality, De
Giorgi-Nash- Moser's iteration, viscosity solutions etc.
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 play a relevant role in many disciplines and
provide powerful tools for analysis and modeling in applied
sciences. The book contains several classical and modern methods
for the study of ordinary and partial differential equations. A
broad space is reserved to Fourier and Laplace transforms together
with their applications to the solution of boundary value and/or
initial value problems for differential equations. Basic
prerequisites concerning analytic functions of complex variable and
Lp spaces are synthetically presented in the first two chapters.
Techniques based on integral transforms and Fourier series are
presented in specific chapters, first in the easier framework of
integrable functions and later in the general framework of
distributions. The less elementary distributional context allows to
deal also with differential equations with highly irregular data
and pulse signals. The theory is introduced concisely, while
learning of miscellaneous methods is achieved step-by-step through
the proposal of many exercises of increasing difficulty. Additional
recap exercises are collected in dedicated sections. Several tables
for easy reference of main formulas are available at the end of the
book. The presentation is oriented mainly to students of Schools in
Engineering, Sciences and Economy. The partition of various topics
in several self-contained and independent sections allows an easy
splitting in at least two didactic modules: one at undergraduate
level, the other at graduate level.
Deep comprehension of applied sciences requires a solid knowledge
of Mathematical Analysis. For most of high level scientific
research, the good understanding of Functional Analysis and weak
solutions to differential equations is essential. This book aims to
deal with the main topics that are necessary to achieve such a
knowledge. Still, this is the goal of many other texts in advanced
analysis; and then, what would be a good reason to read or to
consult this book? In order to answer this question, let us
introduce the three Authors. Alberto Ferrero got his degree in
Mathematics in 2000 and presently he is researcher in Mathematical
Analysis at the Universita del Piemonte Orientale. Filippo Gazzola
got his degree in Mathematics in 1987 and he is now full professor
in Mathematical Analysis at the Politecnico di Milano. Maurizio
Zanotti got his degree in Mechanical Engineering in 2004 and
presently he is structural and machine designer and lecturer
professor in Mathematical Analysis at the Politecnico di Milano.
The three Authors, for the variety of their skills, decided to join
their expertises to write this book. One of the reasons that should
encourage its reading is that the presentation turns out to be a
reasonable compromise among the essential mathematical rigor, the
importance of the applications and the clearness, which is
necessary to make the reference work pleasant to the readers, even
to the inexperienced ones. The range of treated topics is quite
wide and covers the main basic notions of the scientific research
which is based upon mathematical models. We start from vector
spaces and Lebesgue integral to reach the frontier of theoretical
research such as the study of critical exponents for semilinear
elliptic equations and recent problems in fluid dynamics. This long
route passes through the theory of Banach and Hilbert spaces,
Sobolev spaces, differential equations, Fourier and Laplace
transforms, before which we recall some appropriate tools of
Complex Analysis. We give all the proofs that have some didactic or
applicative interest, while we omit the ones which are too
technical or require too high level knowledge. This book has the
ambitious purpose to be useful to a broad variety of readers. The
first possible beneficiaries are of course the second or third year
students of a scientific course of degree: in what follows they
will find the topics that are necessary to approach more advanced
studies in Mathematics and in other fields, especially Physics and
Engineering. This text could be also useful to graduate students
who want to start a Ph.D. course: indeed it contains the matter of
a multidisciplinary Ph.D. course given by Filippo Gazzola for
several years at Politecnico di Milano. Finally, this book could be
addressed also to the ones who have already left education far-back
but occasionally need to use mathematical tools: we refer both to
university professors and their research, and to professionals and
designers who want to model a certain phenomenon, but also to the
nostalgics of the good old days when they were students. ALBERTO
FERRERO got his degree in Mathematics in 2000 and presently he is
researcher in Mathematical Analysis at the Universita del Piemonte
Orientale. FILIPPO GAZZOLA got his degree in Mathematics in 1987
and he is now full professor in Mathematical Analysis at the
Politecnico di Milano. MAURIZIO ZANOTTI got his degree in
Mechanical Engineering in 2004 and presently he is structural and
machine designer and lecturer professor in Mathematical Analysis at
the Politecnico di Milano."
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