Partial Differential Equations with Variable Exponents

Partial Differential Equations with Variable Exponents PDF

Author: Vicentiu D. Radulescu

Publisher: CRC Press

Published: 2015-06-24

Total Pages: 321

ISBN-13: 1498703445

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Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational

Ordinary and Partial Differential Equations for the Beginner

Ordinary and Partial Differential Equations for the Beginner PDF

Author: László Székelyhidi

Publisher: World Scientific Publishing Company

Published: 2016-05-24

Total Pages: 256

ISBN-13: 9814725013

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This textbook is intended for college, undergraduate and graduate students, emphasizing mainly on ordinary differential equations. However, the theory of characteristics for first order partial differential equations and the classification of second order linear partial differential operators are also included. It contains the basic material starting from elementary solution methods for ordinary differential equations to advanced methods for first order partial differential equations. In addition to the theoretical background, solution methods are strongly emphasized. Each section is completed with problems and exercises, and the solutions are also provided. There are special sections devoted to more applied tools such as implicit equations, Laplace transform, Fourier method, etc. As a novelty, a method for finding exponential polynomial solutions is presented which is based on the author's work in spectral synthesis. The presentation is self-contained, provided the reader has general undergraduate knowledge.

Lebesgue and Sobolev Spaces with Variable Exponents

Lebesgue and Sobolev Spaces with Variable Exponents PDF

Author: Lars Diening

Publisher: Springer

Published: 2011-03-29

Total Pages: 516

ISBN-13: 3642183638

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The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.

Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents

Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents PDF

Author: Alex Kaltenbach

Publisher: Springer Nature

Published: 2023-09-12

Total Pages: 364

ISBN-13: 3031296702

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This book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier–Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner–Lebesgue spaces is not applicable. As a substitute for Bochner–Lebesgue spaces, variable Bochner–Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier–Stokes equations under general assumptions. Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory and non-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject.

Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces

Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces PDF

Author: Iwona Chlebicka

Publisher: Springer Nature

Published: 2021-11-01

Total Pages: 389

ISBN-13: 3030888568

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This book provides a detailed study of nonlinear partial differential equations satisfying certain nonstandard growth conditions which simultaneously extend polynomial, inhomogeneous and fully anisotropic growth. The common property of the many different kinds of equations considered is that the growth conditions of the highest order operators lead to a formulation of the equations in Musielak–Orlicz spaces. This high level of generality, understood as full anisotropy and inhomogeneity, requires new proof concepts and a generalization of the formalism, calling for an extended functional analytic framework. This theory is established in the first part of the book, which serves as an introduction to the subject, but is also an important ingredient of the whole story. The second part uses these theoretical tools for various types of PDEs, including abstract and parabolic equations but also PDEs arising from fluid and solid mechanics. For connoisseurs, there is a short chapter on homogenization of elliptic PDEs. The book will be of interest to researchers working in PDEs and in functional analysis.

Evolution PDEs with Nonstandard Growth Conditions

Evolution PDEs with Nonstandard Growth Conditions PDF

Author: Stanislav Antontsev

Publisher: Springer

Published: 2015-04-01

Total Pages: 417

ISBN-13: 9462391122

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This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.

Analysis, Partial Differential Equations and Applications

Analysis, Partial Differential Equations and Applications PDF

Author: Alberto Cialdea

Publisher: Springer Science & Business Media

Published: 2010-01-14

Total Pages: 342

ISBN-13: 3764398981

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This volume includes several invited lectures given at the International Workshop "Analysis, Partial Differential Equations and Applications", held at the Mathematical Department of Sapienza University of Rome, on the occasion of the 70th birthday of Vladimir G. Maz'ya, a renowned mathematician and one of the main experts in the field of pure and applied analysis. The book aims at spreading the seminal ideas of Maz'ya to a larger audience in faculties of sciences and engineering. In fact, all articles were inspired by previous works of Maz'ya in several frameworks, including classical and contemporary problems connected with boundary and initial value problems for elliptic, hyperbolic and parabolic operators, Schrödinger-type equations, mathematical theory of elasticity, potential theory, capacity, singular integral operators, p-Laplacians, functional analysis, and approximation theory. Maz'ya is author of more than 450 papers and 20 books. In his long career he obtained many astonishing and frequently cited results in the theory of harmonic potentials on non-smooth domains, potential and capacity theories, spaces of functions with bounded variation, maximum principle for higher-order elliptic equations, Sobolev multipliers, approximate approximations, etc. The topics included in this volume will be particularly useful to all researchers who are interested in achieving a deeper understanding of the large expertise of Vladimir Maz'ya.

A First Course in Partial Differential Equations

A First Course in Partial Differential Equations PDF

Author: H. F. Weinberger

Publisher: Courier Corporation

Published: 2012-04-20

Total Pages: 482

ISBN-13: 0486132048

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Suitable for advanced undergraduate and graduate students, this text presents the general properties of partial differential equations, including the elementary theory of complex variables. Solutions. 1965 edition.

Lebesgue and Sobolev Spaces with Variable Exponents

Lebesgue and Sobolev Spaces with Variable Exponents PDF

Author: Lars Diening

Publisher: Springer Science & Business Media

Published: 2011-03-31

Total Pages: 516

ISBN-13: 364218362X

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The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.

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