Linear Second Order Elliptic Operators
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Linear Second Order Elliptic Operators
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Author : Julián López-Gómez
language : en
Publisher:
Release Date : 2013
Linear Second Order Elliptic Operators written by Julián López-Gómez and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with categories.
Linear Second Order Elliptic Operators
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Author : Julián López-Gómez
language : en
Publisher: World Scientific Publishing Company
Release Date : 2013-04-24
Linear Second Order Elliptic Operators written by Julián López-Gómez and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-24 with Science categories.
The main goal of the book is to provide a comprehensive and self-contained proof of the, relatively recent, theorem of characterization of the strong maximum principle due to Molina-Meyer and the author, published in Diff. Int. Eqns. in 1994, which was later refined by Amann and the author in a paper published in J. of Diff. Eqns. in 1998. Besides this characterization has been shown to be a pivotal result for the development of the modern theory of spatially heterogeneous nonlinear elliptic and parabolic problems; it has allowed us to update the classical theory on the maximum and minimum principles by providing with some extremely sharp refinements of the classical results of Hopf and Protter-Weinberger. By a celebrated result of Berestycki, Nirenberg and Varadhan, Comm. Pure Appl. Maths. in 1994, the characterization theorem is partially true under no regularity constraints on the support domain for Dirichlet boundary conditions. Instead of encyclopedic generality, this book pays special attention to completeness, clarity and transparency of its exposition so that it can be taught even at an advanced undergraduate level. Adopting this perspective, it is a textbook; however, it is simultaneously a research monograph about the maximum principle, as it brings together for the first time in the form of a book, the most paradigmatic classical results together with a series of recent fundamental results scattered in a number of independent papers by the author of this book and his collaborators. Chapters 3, 4, and 5 can be delivered as a classical undergraduate, or graduate, course in Hilbert space techniques for linear second order elliptic operators, and Chaps. 1 and 2 complete the classical results on the minimum principle covered by the paradigmatic textbook of Protter and Weinberger by incorporating some recent classification theorems of supersolutions by Walter, 1989, and the author, 2003. Consequently, these five chapters can be taught at an undergraduate, or graduate, level. Chapters 6 and 7 study the celebrated theorem of Krein–Rutman and infer from it the characterizations of the strong maximum principle of Molina-Meyer and Amann, in collaboration with the author, which have been incorporated to a textbook by the first time here, as well as the results of Chaps. 8 and 9, polishing some recent joint work of Cano-Casanova with the author. Consequently, the second half of the book consists of a more specialized monograph on the maximum principle and the underlying principal eigenvalues.
Second Order Elliptic Equations And Elliptic Systems
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Author : Ya-Zhe Chen
language : en
Publisher: American Mathematical Soc.
Release Date : 1998
Second Order Elliptic Equations And Elliptic Systems written by Ya-Zhe Chen and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Mathematics categories.
There are two parts to the book. In the first part, a complete introduction of various kinds of a priori estimate methods for the Dirichlet problem of second order elliptic partial differential equations is presented. In the second part, the existence and regularity theories of the Dirichlet problem for linear and nonlinear second order elliptic partial differential systems are introduced. The book features appropriate materials and is an excellent textbook for graduate students. The volume is also useful as a reference source for undergraduate mathematics majors, graduate students, professors, and scientists.
Semigroups Of Bounded Operators And Second Order Elliptic And Parabolic Partial Differential Equations
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Author : Luca Lorenzi
language : en
Publisher: CRC Press
Release Date : 2021-01-05
Semigroups Of Bounded Operators And Second Order Elliptic And Parabolic Partial Differential Equations written by Luca Lorenzi and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-05 with Mathematics categories.
Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations aims to propose a unified approach to elliptic and parabolic equations with bounded and smooth coefficients. The book will highlight the connections between these equations and the theory of semigroups of operators, while demonstrating how the theory of semigroups represents a powerful tool to analyze general parabolic equations. Features Useful for students and researchers as an introduction to the field of partial differential equations of elliptic and parabolic types Introduces the reader to the theory of operator semigroups as a tool for the analysis of partial differential equations
Diffusions And Elliptic Operators
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Author : Richard F. Bass
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-11
Diffusions And Elliptic Operators written by Richard F. Bass and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-11 with Mathematics categories.
A discussion of the interplay of diffusion processes and partial differential equations with an emphasis on probabilistic methods. It begins with stochastic differential equations, the probabilistic machinery needed to study PDE, and moves on to probabilistic representations of solutions for PDE, regularity of solutions and one dimensional diffusions. The author discusses in depth two main types of second order linear differential operators: non-divergence operators and divergence operators, including topics such as the Harnack inequality of Krylov-Safonov for non-divergence operators and heat kernel estimates for divergence form operators, as well as Martingale problems and the Malliavin calculus. While serving as a textbook for a graduate course on diffusion theory with applications to PDE, this will also be a valuable reference to researchers in probability who are interested in PDE, as well as for analysts interested in probabilistic methods.
Manifolds With Group Actions And Elliptic Operators
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Author : Vladimir I͡Akovlevich Lin
language : en
Publisher: American Mathematical Soc.
Release Date : 1994
Manifolds With Group Actions And Elliptic Operators written by Vladimir I͡Akovlevich Lin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematics categories.
This work studies equivariant linear second order elliptic operators P on a connected noncompact manifold X with a given action of a group G . The action is assumed to be cocompact, meaning that GV=X for some compact subset V of X . The aim is to study the structure of the convex cone of all positive solutions of Pu= 0. It turns out that the set of all normalized positive solutions which are also eigenfunctions of the given G -action can be realized as a real analytic submanifold *G [0 of an appropriate topological vector space *H . When G is finitely generated, *H has finite dimension, and in nontrivial cases *G [0 is the boundary of a strictly convex body in *H. When G is nilpotent, any positive solution u can be represented as an integral with respect to some uniquely defined positive Borel measure over *G [0 . Lin and Pinchover also discuss related results for parabolic equations on X and for elliptic operators on noncompact manifolds with boundary.
Elliptic Differential Operators And Spectral Analysis
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Author : D. E. Edmunds
language : en
Publisher: Springer
Release Date : 2018-11-20
Elliptic Differential Operators And Spectral Analysis written by D. E. Edmunds and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-20 with Mathematics categories.
This book deals with elliptic differential equations, providing the analytic background necessary for the treatment of associated spectral questions, and covering important topics previously scattered throughout the literature. Starting with the basics of elliptic operators and their naturally associated function spaces, the authors then proceed to cover various related topics of current and continuing importance. Particular attention is given to the characterisation of self-adjoint extensions of symmetric operators acting in a Hilbert space and, for elliptic operators, the realisation of such extensions in terms of boundary conditions. A good deal of material not previously available in book form, such as the treatment of the Schauder estimates, is included. Requiring only basic knowledge of measure theory and functional analysis, the book is accessible to graduate students and will be of interest to all researchers in partial differential equations. The reader will value its self-contained, thorough and unified presentation of the modern theory of elliptic operators.
Non Linear Elliptic Equations In Conformal Geometry
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Author : Sun-Yung A. Chang
language : en
Publisher: European Mathematical Society
Release Date : 2004
Non Linear Elliptic Equations In Conformal Geometry written by Sun-Yung A. Chang and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Computers categories.
Non-linear elliptic partial differential equations are an important tool in the study of Riemannian metrics in differential geometry, in particular for problems concerning the conformal change of metrics in Riemannian geometry. In recent years the role played by the second order semi-linear elliptic equations in the study of Gaussian curvature and scalar curvature has been extended to a family of fully non-linear elliptic equations associated with other symmetric functions of the Ricci tensor. A case of particular interest is the second symmetric function of the Ricci tensor in dimension four closely related to the Pfaffian. In these lectures, starting from the background material, the author reviews the problem of prescribing Gaussian curvature on compact surfaces. She then develops the analytic tools (e.g., higher order conformal invariant operators, Sobolev inequalities, blow-up analysis) in order to solve a fully nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of dimension four. The material is suitable for graduate students and research mathematicians interested in geometry, topology, and differential equations.
Elliptic Operators And Lie Groups
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Author : Derek W. Robinson
language : en
Publisher:
Release Date : 1991
Elliptic Operators And Lie Groups written by Derek W. Robinson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991 with Mathematics categories.
Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural noncommutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subelliptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.
Elliptic Differential Equations And Obstacle Problems
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Author : Giovanni Maria Troianiello
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11
Elliptic Differential Equations And Obstacle Problems written by Giovanni Maria Troianiello and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
In the few years since their appearance in the mid-sixties, variational inequalities have developed to such an extent and so thoroughly that they may now be considered an "institutional" development of the theory of differential equations (with appreciable feedback as will be shown). This book was written in the light of these considerations both in regard to the choice of topics and to their treatment. In short, roughly speaking my intention was to write a book on second-order elliptic operators, with the first half of the book, as might be expected, dedicated to function spaces and to linear theory whereas the second, nonlinear half would deal with variational inequalities and non variational obstacle problems, rather than, for example, with quasilinear or fully nonlinear equations (with a few exceptions to which I shall return later). This approach has led me to omit any mention of "physical" motivations in the wide sense of the term, in spite of their historical and continuing importance in the development of variational inequalities. I here addressed myself to a potential reader more or less aware of the significant role of variational inequalities in numerous fields of applied mathematics who could use an analytic presentation of the fundamental theory, which would be as general and self-contained as possible.