Partial differential equations with variable exponents : variational methods and qualitative analysis PDF
Preview Partial differential equations with variable exponents : variational methods and qualitative analysis
Mathematics MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS P Partial Differential a Partial Differential Equations with Variable Exponents: Variational r Methods and Qualitative Analysis provides you with a thorough ti a introduction to the theory of nonlinear partial differential equations (PDEs) l Equations with D with a variable exponent, particularly those of elliptic type. i f The book presents the most important variational methods for elliptic PDEs fe described by nonhomogeneous differential operators and containing one or re Variable Exponents n more power-type nonlinearities with a variable exponent. The authors give t a systematic treatment of the basic mathematical theory and constructive ia l methods for these classes of nonlinear elliptic equations as well as their Variational Methods E applications to various processes arising in the applied sciences. q u and Qualitative Analysis The analysis developed in the book is based on the notion of a generalized a t or weak solution. This approach leads not only to the fundamental results of i o existence and multiplicity of weak solutions but also to several qualitative n s properties, including spectral analysis, bifurcation, and asymptotic w analysis. i t h The book examines the equations from different points of view while V using the calculus of variations as the unifying theme. You will see how a all of these diverse topics are connected to other important parts of r i a mathematics, including topology, differential geometry, mathematical b physics, and potential theory. le E Features x • Provides a modern, unified approach to analyzing PDEs p o • Presents elliptic equations with variable exponents from different n viewpoints e n • Demonstrates the power of Sobolev spaces in analysis t s • Reveals a number of surprising interactions among various topics • Covers applications in elasticity, heat diffusion, and other physical areas RR eă pd ou vl še Vicenţiu D. Rădulescu s c u Dušan D. Repovš K24661 www.crcpress.com K24661_cover.indd 1 4/24/15 8:53 AM Partial Differential Equations with Variable Exponents Variational Methods and Qualitative Analysis MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Series Editors John A. Burns Thomas J. Tucker Miklos Bona Michael Ruzhansky Chi-Kwong Li Published Titles Application of Fuzzy Logic to Social Choice Theory, John N. Mordeson, Davender S. Malik and Terry D. Clark Blow-up Patterns for Higher-Order: Nonlinear Parabolic, Hyperbolic Dispersion and Schrödinger Equations, Victor A. Galaktionov, Enzo L. Mitidieri, and Stanislav Pohozaev Difference Equations: Theory, Applications and Advanced Topics, Third Edition, Ronald E. Mickens Dictionary of Inequalities, Second Edition, Peter Bullen Iterative Optimization in Inverse Problems, Charles L. Byrne Modeling and Inverse Problems in the Presence of Uncertainty, H. T. Banks, Shuhua Hu, and W. Clayton Thompson Monomial Algebras, Second Edition, Rafael H. Villarreal Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Vicenţiu D. Rădulescu and Dušan D. Repovš Signal Processing: A Mathematical Approach, Second Edition, Charles L. Byrne Sinusoids: Theory and Technological Applications, Prem K. Kythe Special Integrals of Gradshetyn and Ryzhik: the Proofs – Volume l, Victor H. Moll Forthcoming Titles Actions and Invariants of Algebraic Groups, Second Edition, Walter Ferrer Santos and Alvaro Rittatore Analytical Methods for Kolmogorov Equations, Second Edition, Luca Lorenzi Complex Analysis: Conformal Inequalities and the Bierbach Conjecture, Prem K. Kythe Computational Aspects of Polynomial Identities: Volume l, Kemer’s Theorems, 2nd Edition Belov Alexey, Yaakov Karasik, Louis Halle Rowen Cremona Groups and Icosahedron, Ivan Cheltsov and Constantin Shramov Geometric Modeling and Mesh Generation from Scanned Images, Yongjie Zhang Groups, Designs, and Linear Algebra, Donald L. Kreher Handbook of the Tutte Polynomial, Joanna Anthony Ellis-Monaghan and Iain Moffat Lineability: The Search for Linearity in Mathematics, Juan B. Seoane Sepulveda, Richard W. Aron, Luis Bernal-Gonzalez, and Daniel M. Pellegrinao Forthcoming Titles (continued) Line Integral Methods and Their Applications, Luigi Brugnano and Felice Iaverno Microlocal Analysis on Rˆn and on NonCompact Manifolds, Sandro Coriasco Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory Under Weak Topology for Nonlinear Operators and Block Operators with Applications Aref Jeribi and Bilel Krichen Practical Guide to Geometric Regulation for Distributed Parameter Systems, Eugenio Aulisa and David S. Gilliam Reconstructions from the Data of Integrals, Victor Palamodov Special Integrals of Gradshetyn and Ryzhik: the Proofs – Volume ll, Victor H. Moll Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions, Irina V. Melnikova and Alexei Filinkov Symmetry and Quantum Mechanics, Scott Corry MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Partial Differential Equations with Variable Exponents Variational Methods and Qualitative Analysis Vicenţiu D. Rădulescu King Abdulaziz University Jeddah, Saudi Arabia Dušan D. Repovš University of Ljubljana Ljubljana, Slovenia CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150526 International Standard Book Number-13: 978-1-4987-0344-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Vicen¸tiu D. R˘adulescu dedicates this book with deep gratitude to the memory of his beloved parents, Professor Dumitru R˘adulescu (1914–1982) and Ana R˘adulescu (1923–2011) Contents Preface xiii List of Figures xv List of Tables xvii Symbol Description xix About the Authors xxi I Isotropic and Anisotropic Function Spaces 1 1 Lebesgue and Sobolev Spaces with Variable Exponents 3 1.1 History of function spaces with variable exponents . . . . . . 4 1.2 Lebesgue spaces with variable exponents . . . . . . . . . . . 7 1.3 Sobolev spaces with variable exponents . . . . . . . . . . . . 10 1.4 Dirichlet energies and Euler-Lagrangeequations . . . . . . . 11 1.5 Lavrentiev phenomenon . . . . . . . . . . . . . . . . . . . . . 12 1.6 Anisotropic function spaces . . . . . . . . . . . . . . . . . . . 14 1.7 Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 II Variational Analysis of Problems with Variable Exponents 23 2 Nonlinear Degenerate Problems in Non-Newtonian Fluids 25 2.1 Physical motivation . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 A boundary value problem with nonhomogeneous differential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Properties of the energy functional . . . . . . . . . . . 30 2.2.2 A mountain pass-type critical point . . . . . . . . . . 37 2.3 Nonlinear eigenvalue problems with two variable exponents . 42 2.3.1 Ekeland variational principle versus the mountain pass geometry . . . . . . . . . . . . . . . . . . . . . . . . . 43 ix
The list of books you might like
