Partial differential equations with variable exponents : variational methods and qualitative analysis

321 Pages·2015·3.228 MB·English

by  Rădulescu, Vicenţiu D., Repovš, Dušan

Mathematics
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
P Partial Differential
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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

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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
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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
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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
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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

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Features
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• Provides a modern, unified approach to analyzing PDEs p
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• Presents elliptic equations with variable exponents from different
n
viewpoints e
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• Demonstrates the power of Sobolev spaces in analysis t
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• Reveals a number of surprising interactions among various topics
• Covers applications in elasticity, heat diffusion, and other physical
areas RR
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še Vicenţiu D. Rădulescu
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Dušan D. Repovš
K24661
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Partial Differential
Equations with
Variable Exponents
Variational Methods
and Qualitative Analysis
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
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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)
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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