Analysis And Computation Of Multiple Unstable Solutions To Nonlinear Elliptic Systems

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Analysis and Computation of Multiple Unstable Solutions to Nonlinear Elliptic Systems

Author : Xianjin Chen
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (69 download)

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Book Synopsis Analysis and Computation of Multiple Unstable Solutions to Nonlinear Elliptic Systems by : Xianjin Chen

Download or read book Analysis and Computation of Multiple Unstable Solutions to Nonlinear Elliptic Systems written by Xianjin Chen and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We study computational theory and methods for finding multiple unstable solutions (corresponding to saddle points) to three types of nonlinear variational elliptic systems: cooperative, noncooperative, and Hamiltonian. We first propose a new Lorthogonal selection in a product Hilbert space so that a solution manifold can be defined. Then, we establish, respectively, a local characterization for saddle points of finite Morse index and of infinite Morse index. Based on these characterizations, two methods, called the local min-orthogonal method and the local min-max-orthogonal method, are developed and applied to solve those three types of elliptic systems for multiple solutions. Under suitable assumptions, a subsequence convergence result is established for each method. Numerical experiments for different types of model problems are carried out, showing that both methods are very reliable and efficient in computing coexisting saddle points or saddle points of infinite Morse index. We also analyze the instability of saddle points in both single and product Hilbert spaces. In particular, we establish several estimates of the Morse index of both coexisting and non-coexisting saddle points via the local min-orthogonal method developed and propose a local instability index to measure the local instability of both degenerate and nondegenerate saddle points. Finally, we suggest two extensions of an L-orthogonal selection for future research so that multiple solutions to more general elliptic systems such as nonvariational elliptic systems may also be found in a stable way.

The Analysis of Solutions of Elliptic Equations

Author : Nikolai Tarkhanov
Publisher : Springer Science & Business Media
ISBN 13 : 940158804X
Total Pages : 496 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis The Analysis of Solutions of Elliptic Equations by : Nikolai Tarkhanov

Download or read book The Analysis of Solutions of Elliptic Equations written by Nikolai Tarkhanov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended as a continuation of my book "Parametrix Method in the Theory of Differential Complexes" (see [291]). There, we considered complexes of differential operators between sections of vector bundles and we strived more than for details. Although there are many applications to for maximal generality overdetermined systems, such an approach left me with a certain feeling of dissat- faction, especially since a large number of interesting consequences can be obtained without a great effort. The present book is conceived as an attempt to shed some light on these new applications. We consider, as a rule, differential operators having a simple structure on open subsets of Rn. Currently, this area is not being investigated very actively, possibly because it is already very highly developed actively (cf. for example the book of Palamodov [213]). However, even in this (well studied) situation the general ideas from [291] allow us to obtain new results in the qualitative theory of differential equations and frequently in definitive form. The greater part of the material presented is related to applications of the L- rent series for a solution of a system of differential equations, which is a convenient way of writing the Green formula. The culminating application is an analog of the theorem of Vitushkin [303] for uniform and mean approximation by solutions of an elliptic system. Somewhat afield are several questions on ill-posedness, but the parametrix method enables us to obtain here a series of hitherto unknown facts.

Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

Author : Messoud Efendiev
Publisher : Springer
ISBN 13 : 3319984071
Total Pages : 273 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations by : Messoud Efendiev

Download or read book Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations written by Messoud Efendiev and published by Springer. This book was released on 2018-10-17 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with a systematic study of a dynamical system approach to investigate the symmetrization and stabilization properties of nonnegative solutions of nonlinear elliptic problems in asymptotically symmetric unbounded domains. The usage of infinite dimensional dynamical systems methods for elliptic problems in unbounded domains as well as finite dimensional reduction of their dynamics requires new ideas and tools. To this end, both a trajectory dynamical systems approach and new Liouville type results for the solutions of some class of elliptic equations are used. The work also uses symmetry and monotonicity results for nonnegative solutions in order to characterize an asymptotic profile of solutions and compares a pure elliptic partial differential equations approach and a dynamical systems approach. The new results obtained will be particularly useful for mathematical biologists.

Contributions to Nonlinear Elliptic Equations and Systems

Author : Alexandre N. Carvalho
Publisher : Birkhäuser
ISBN 13 : 3319199021
Total Pages : 434 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis Contributions to Nonlinear Elliptic Equations and Systems by : Alexandre N. Carvalho

Download or read book Contributions to Nonlinear Elliptic Equations and Systems written by Alexandre N. Carvalho and published by Birkhäuser. This book was released on 2015-11-14 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume of contributions pays tribute to the life and work of Djairo Guedes de Figueiredo on the occasion of his 80th birthday. The articles it contains were born out of the ICMC Summer Meeting on Differential Equations – 2014 Chapter, also dedicated to de Figueiredo and held at the Universidade de São Paulo at São Carlos, Brazil from February 3-7, 2014. The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems. Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles. His influence on Brazilian mathematics has made him one of the pillars of the subject in that country. He had a major impact on the development of analysis, especially in its application to nonlinear elliptic partial differential equations and systems throughout the entire world. The articles collected here pay tribute to him and his legacy and are intended for graduate students and researchers in mathematics and related areas who are interested in nonlinear elliptic partial differential equations and systems.

Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations

Author : Vicentiu D. Radulescu
Publisher : Hindawi Publishing Corporation
ISBN 13 : 9774540395
Total Pages : 205 pages
Book Rating : 4.7/5 (745 download)

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Book Synopsis Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations by : Vicentiu D. Radulescu

Download or read book Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations written by Vicentiu D. Radulescu and published by Hindawi Publishing Corporation. This book was released on 2008 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.

Singular Solutions of Nonlinear Elliptic and Parabolic Equations

Author : Alexander A. Kovalevsky
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110332248
Total Pages : 448 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Singular Solutions of Nonlinear Elliptic and Parabolic Equations by : Alexander A. Kovalevsky

Download or read book Singular Solutions of Nonlinear Elliptic and Parabolic Equations written by Alexander A. Kovalevsky and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-03-21 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph looks at several trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations. It discusses results on the existence and properties of weak and entropy solutions for elliptic second-order equations and some classes of fourth-order equations with L1-data and questions on the removability of singularities of solutions to elliptic and parabolic second-order equations in divergence form. It looks at localized and nonlocalized singularly peaking boundary regimes for different classes of quasilinear parabolic second- and high-order equations in divergence form. The book will be useful for researchers and post-graduate students that specialize in the field of the theory of partial differential equations and nonlinear analysis. Contents: Foreword Part I: Nonlinear elliptic equations with L^1-data Nonlinear elliptic equations of the second order with L^1-data Nonlinear equations of the fourth order with strengthened coercivity and L^1-data Part II: Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order Removability of singularities of the solutions of quasilinear elliptic equations Removability of singularities of the solutions of quasilinear parabolic equations Quasilinear elliptic equations with coefficients from the Kato class Part III: Boundary regimes with peaking for quasilinear parabolic equations Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations Method of functional inequalities in peaking regimes for parabolic equations of higher orders Nonlocalized regimes with singular peaking Appendix: Formulations and proofs of the auxiliary results Bibliography

Elliptic Systems of Phase Transition Type

Author : Nicholas D. Alikakos
Publisher : Springer
ISBN 13 : 3319905724
Total Pages : 343 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Elliptic Systems of Phase Transition Type by : Nicholas D. Alikakos

Download or read book Elliptic Systems of Phase Transition Type written by Nicholas D. Alikakos and published by Springer. This book was released on 2019-01-21 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes – non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that played a major role in Savin's proof. It also introduces an alternative method for obtaining pointwise estimates. Key features and topics of this self-contained, systematic exposition include: • Resolution of the structure of minimal solutions in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions. • Preliminary material beginning with the stress-energy tensor, via which monotonicity formulas, and Hamiltonian and Pohozaev identities are developed, including a self-contained exposition of the existence of standing and traveling waves. • Tools that allow the derivation of general properties of minimizers, without any assumptions of symmetry, such as a maximum principle or density and pointwise estimates. • Application of the general tools to equivariant solutions rendering exponential estimates, rigidity theorems and stratification results. This monograph is addressed to readers, beginning from the graduate level, with an interest in any of the following: differential equations – ordinary or partial; nonlinear analysis; the calculus of variations; the relationship of minimal surfaces to diffuse interfaces; or the applied mathematics of materials science.

Nonlinear Second Order Elliptic Equations

Author : Mingxin Wang
Publisher : Springer Nature
ISBN 13 : 9819986923
Total Pages : 319 pages
Book Rating : 4.8/5 (199 download)

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Book Synopsis Nonlinear Second Order Elliptic Equations by : Mingxin Wang

Download or read book Nonlinear Second Order Elliptic Equations written by Mingxin Wang and published by Springer Nature. This book was released on with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Quasilinear Elliptic Equations with Degenerations and Singularities

Author : Pavel Drábek
Publisher : Walter de Gruyter
ISBN 13 : 9783110154900
Total Pages : 240 pages
Book Rating : 4.1/5 (549 download)

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Book Synopsis Quasilinear Elliptic Equations with Degenerations and Singularities by : Pavel Drábek

Download or read book Quasilinear Elliptic Equations with Degenerations and Singularities written by Pavel Drábek and published by Walter de Gruyter. This book was released on 1997 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-Chief J rgen Appell, W rzburg, Germany Honorary and Advisory Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Alfonso Vignoli, Rome, Italy Editorial Board Manuel del Pino, Bath, UK, and Santiago, Chile Mikio Kato, Nagano, Japan Wojciech Kryszewski, Toruń, Poland Vicenţiu D. Rădulescu, Krak w, Poland Simeon Reich, Haifa, Israel Please submit book proposals to J rgen Appell. Titles in planning include Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019) Tomasz W. Dlotko and Yejuan Wang, Critical Parabolic-Type Problems (2019) Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019) Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy-Leray Potential (2020) Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020) Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

Fully Nonlinear Elliptic Equations

Author : Luis A. Caffarelli
Publisher : American Mathematical Soc.
ISBN 13 : 0821804375
Total Pages : 114 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Fully Nonlinear Elliptic Equations by : Luis A. Caffarelli

Download or read book Fully Nonlinear Elliptic Equations written by Luis A. Caffarelli and published by American Mathematical Soc.. This book was released on 1995 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Regularity Results for Nonlinear Elliptic Systems and Applications

Author : Alain Bensoussan
Publisher : Springer
ISBN 13 : 9783642087264
Total Pages : 443 pages
Book Rating : 4.0/5 (872 download)

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Book Synopsis Regularity Results for Nonlinear Elliptic Systems and Applications by : Alain Bensoussan

Download or read book Regularity Results for Nonlinear Elliptic Systems and Applications written by Alain Bensoussan and published by Springer. This book was released on 2010-12-18 with total page 443 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects many helpful techniques for obtaining regularity results for solutions of nonlinear systems of partial differential equations. These are applied in various cases to provide useful examples and relevant results, particularly in such fields as fluid mechanics, solid mechanics, semiconductor theory and game theory.

On the Rugularity of Weak Solutions to Nonlinear Elliptic Systems of Partial Differential Equations

Author : Jindrich Nec̆as
Publisher :
ISBN 13 :
Total Pages : 56 pages
Book Rating : 4.:/5 (758 download)

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Book Synopsis On the Rugularity of Weak Solutions to Nonlinear Elliptic Systems of Partial Differential Equations by : Jindrich Nec̆as

Download or read book On the Rugularity of Weak Solutions to Nonlinear Elliptic Systems of Partial Differential Equations written by Jindrich Nec̆as and published by . This book was released on 1979 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Nonlinear Analysis and Semilinear Elliptic Problems

Author : Antonio Ambrosetti
Publisher : Cambridge University Press
ISBN 13 : 1139460633
Total Pages : 239 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Nonlinear Analysis and Semilinear Elliptic Problems by : Antonio Ambrosetti

Download or read book Nonlinear Analysis and Semilinear Elliptic Problems written by Antonio Ambrosetti and published by Cambridge University Press. This book was released on 2007-01-04 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in science and engineering are described by nonlinear differential equations, which can be notoriously difficult to solve. Through the interplay of topological and variational ideas, methods of nonlinear analysis are able to tackle such fundamental problems. This graduate text explains some of the key techniques in a way that will be appreciated by mathematicians, physicists and engineers. Starting from elementary tools of bifurcation theory and analysis, the authors cover a number of more modern topics from critical point theory to elliptic partial differential equations. A series of Appendices give convenient accounts of a variety of advanced topics that will introduce the reader to areas of current research. The book is amply illustrated and many chapters are rounded off with a set of exercises.

Methods for Analysis of Nonlinear Elliptic Boundary Value Problems

Author : I. V. Skrypnik
Publisher : American Mathematical Soc.
ISBN 13 : 9780821897560
Total Pages : 370 pages
Book Rating : 4.8/5 (975 download)

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Book Synopsis Methods for Analysis of Nonlinear Elliptic Boundary Value Problems by : I. V. Skrypnik

Download or read book Methods for Analysis of Nonlinear Elliptic Boundary Value Problems written by I. V. Skrypnik and published by American Mathematical Soc.. This book was released on 1994-01-01 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of nonlinear elliptic equations is currently one of the most actively developing branches of the theory of partial differential equations. This book investigates boundary value problems for nonlinear elliptic equations of arbitrary order. In addition to monotone operator methods, a broad range of applications of topological methods to nonlinear differential equations is presented: solvability, estimation of the number of solutions, and the branching of solutions of nonlinear equations. Skrypnik establishes, by various procedures, a priori estimates and the regularity of solutions of nonlinear elliptic equations of arbitrary order. Also covered are methods of homogenization of nonlinear elliptic problems in perforated domains. The book is suitable for use in graduate courses in differential equations and nonlinear functional analysis.

Optimization of Elliptic Systems

Author : Pekka Neittaanmaki
Publisher : Springer Science & Business Media
ISBN 13 : 0387272364
Total Pages : 514 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Optimization of Elliptic Systems by : Pekka Neittaanmaki

Download or read book Optimization of Elliptic Systems written by Pekka Neittaanmaki and published by Springer Science & Business Media. This book was released on 2007-01-04 with total page 514 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present monograph is intended to provide a comprehensive and accessible introduction to the optimization of elliptic systems. This area of mathematical research, which has many important applications in science and technology. has experienced an impressive development during the past two decades. There are already many good textbooks dealing with various aspects of optimal design problems. In this regard, we refer to the works of Pironneau [1984], Haslinger and Neittaanmaki [1988], [1996], Sokolowski and Zolksio [1992], Litvinov [2000], Allaire [2001], Mohammadi and Pironneau [2001], Delfour and Zolksio [2001], and Makinen and Haslinger [2003]. Already Lions [I9681 devoted a major part of his classical monograph on the optimal control of partial differential equations to the optimization of elliptic systems. Let us also mention that even the very first known problem of the calculus of variations, the brachistochrone studied by Bernoulli back in 1696. is in fact a shape optimization problem. The natural richness of this mathematical research subject, as well as the extremely large field of possible applications, has created the unusual situation that although many important results and methods have already been est- lished, there are still pressing unsolved questions. In this monograph, we aim to address some of these open problems; as a consequence, there is only a minor overlap with the textbooks already existing in the field.

Variational Methods for the Numerical Solution of Nonlinear Elliptic Problem

Author : Roland Glowinski
Publisher : SIAM
ISBN 13 : 1611973783
Total Pages : 473 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Variational Methods for the Numerical Solution of Nonlinear Elliptic Problem by : Roland Glowinski

Download or read book Variational Methods for the Numerical Solution of Nonlinear Elliptic Problem written by Roland Glowinski and published by SIAM. This book was released on 2015-11-04 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems?addresses computational methods that have proven efficient for the solution of a large variety of nonlinear elliptic problems. These methods can be applied to many problems in science and engineering, but this book focuses on their application to problems in continuum mechanics and physics. This book differs from others on the topic by presenting examples of the power and versatility of operator-splitting methods; providing a detailed introduction to alternating direction methods of multipliers and their applicability to the solution of nonlinear (possibly nonsmooth) problems from science and engineering; and showing that nonlinear least-squares methods, combined with operator-splitting and conjugate gradient algorithms, provide efficient tools for the solution of highly nonlinear problems. The book provides useful insights suitable for advanced graduate students, faculty, and researchers in applied and computational mathematics as well as research engineers, mathematical physicists, and systems engineers.

Non-linear Elliptic Systems with Measure-valued Right Hand Side

Author : Georg Dolzmann
Publisher :
ISBN 13 :
Total Pages : 33 pages
Book Rating : 4.:/5 (558 download)

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Book Synopsis Non-linear Elliptic Systems with Measure-valued Right Hand Side by : Georg Dolzmann

Download or read book Non-linear Elliptic Systems with Measure-valued Right Hand Side written by Georg Dolzmann and published by . This book was released on 1996 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We prove existence of a solution u for the nonlinear elliptic system -div [sigma](x, u, Du) = [mu] in D(́[omega]), u = O on [delta omega] where [mu] is Radon measure on [omega] with finite mass. In particular we show that if the coercivity rate of [delta] lies in the range (1,2 - 1/n] then u is approximately differentiable and the equation holds with Du replaced by ap Du. The proof relies on an approximation of [mu] by smooth functions f[subscript k] and a compactness result for the corresponding solutions u[subscript k]. This follows from a detailed analysis of the Young measure [[delta subscript u(x)][circled x] v[subscript x]] generated by the sequence [(u[subscript k], Du[subscript k])] and the div-curl type inequality v[subscript x, [sigma](x, u, ·)· [or =] [−sigma](x)