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Partial Differential Equations de Jürgen Jost

Partial Differential Equations: Graduate Texts in Mathematics, cartea 214

Autor Jürgen Jost
en Limba Engleză Paperback – 25 noi 2010

Prezentat sub formă de manual de nivel universitar avansat, Partial Differential Equations reprezintă o resursă fundamentală pentru studenții la masterat și doctorat în matematică. Această a treia ediție, publicată de Springer în prestigioasa serie Graduate Texts in Mathematics, a fost substanțial extinsă pentru a include cercetări recente și noi perspective pedagogice. Subliniem modul în care Jürgen Jost reușește să creeze o punte între structurile matematice clasice și fenomenele complexe, precum mișcarea browniană sau procesele de formare a structurilor.

Considerăm că forța acestui volum rezidă în organizarea sa riguroasă în două părți distincte. Prima parte este dedicată tipologiei ecuațiilor — eliptice, parabolice și hiperbolice —, în timp ce a doua parte se concentrează pe scheme de existență și estimări de regularitate. Noua ediție aduce capitole esențiale despre sistemele de ordinul întâi și ecuațiile Fokker-Planck, oferind o claritate sporită utilizării inegalității lui Harnack. Comparativ cu A Short Introduction to Partial Differential Equations de Arian Novruzi, care oferă o perspectivă sintetică axată pe soluții analitice și slabe, lucrarea lui Jost este mult mai detaliată în ceea ce privește teoria regularității (de Giorgi-Nash) și tehnicile de iterație.

Remarcăm continuitatea tematică în opera autorului; dacă în Calculus of Variations el explora bazele geometrice și topologice, aici aplică acea rigoare analitică asupra spațiilor Sobolev și a metodelor variaționale. Deși acoperă aceeași arie tematică precum volumul lui Hervé Le Dret, lucrarea de față se distinge printr-o abordare pur teoretică și analitică, lăsând deoparte aproximările numerice în favoarea unei profunzimi conceptuale superioare.

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Specificații

ISBN-13: 9781441923806
ISBN-10: 1441923802
Pagini: 376
Ilustrații: XIV, 356 p. 10 illus.
Dimensiuni: 155 x 235 x 21 mm
Greutate: 0.57 kg
Ediția:Softcover reprint of hardcover 2nd edition 2007
Editura: Springer
Colecția Graduate Texts in Mathematics
Seria Graduate Texts in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Recomandăm acest manual cercetătorilor la început de drum care doresc să stăpânească tehnicile moderne de analiză pentru ecuațiile eliptice. Cititorul câștigă o înțelegere profundă a spațiilor Sobolev și a metodelor de regularitate Schauder, esențiale în cercetarea matematică actuală. Este o investiție pe termen lung în biblioteca oricărui analist, oferind fundamentul teoretic necesar pentru abordarea literaturii de specialitate.

Despre autor

Jürgen Jost este un matematician german de renume, afiliat Institutului Max Planck pentru Matematică în Științe. Opera sa este impresionantă prin diversitate, publicând lucrări fundamentale de la geometrie algebrică, precum Spektren, Garben, Schemata, până la analize complexe în Calculus of Variations. Această polivalență îi permite să abordeze ecuațiile cu derivate parțiale nu doar ca pe un exercițiu de analiză, ci ca pe un instrument vital în înțelegerea structurilor geometrice și fizice. Expertiza sa în sisteme dinamice și analiză geometrică transpare în rigoarea cu care tratează regularitatea soluțiilor în prezentul volum.

Descriere scurtă

This textbook is intended for students who wish to obtain an introduction to the theory of partial di?erential equations (PDEs, for short), in particular, those of elliptic type. Thus, it does not o?er a comprehensive overview of the whole ?eld of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding qu- tion is how one can ?nd a solution of such a PDE. Such a solution will, of course, depend on given constraints and, in turn, if the constraints are of the appropriate type, be uniquely determined by them. We shall pursue a number of strategies for ?nding a solution of a PDE; they can be informally characterized as follows: (0) Write down an explicit formula for the solution in terms of the given data (constraints). This may seem like the best and most natural approach, but this is possible only in rather particular and special cases. Also, such a formula may be rather complicated, so that it is not very helpful for detecting qualitative properties of a solution. Therefore, mathematical analysis has developed other, more powerful, approaches. (1) Solve a sequence of auxiliary problems that approximate the given one, and show that their solutions converge to a solution of that original pr- lem. Di?erential equations are posed in spaces of functions, and those spaces are of in?nite dimension.

Cuprins

Introduction: What Are Partial Differential Equations?.- The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order.- The Maximum Principle.- Existence Techniques I: Methods Based on the Maximum Principle.- Existence Techniques II: Parabolic Methods. The Heat Equation.- Reaction-Diffusion Equations and Systems.- The Wave Equation and its Connections with the Laplace and Heat Equations.- The Heat Equation, Semigroups, and Brownian Motion.- The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III).- Sobolev Spaces and L2 Regularity Theory.- Strong Solutions.- The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV).- The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash.

Recenzii

From the reviews of the second edition:
"Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations."
- Alain Brillard, Mathematical Reviews
"Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics."
- Nick Lord, The Mathematical Gazette
“It is an expanded translation by the author of the German original. … The range of methods is wide, covering integral kernels, maximum principles, variational principles, gradient descents, weak derivatives and Sobolev spaces. … the proof are clear and pleasant, provided the reader has a good command in integration theory. … This book is an interesting introduction to the multiple facets of partial differential equations –– especially to regularity theory –– for the reader who has already a good background in analysis.” (Jean Van Schaftingen, Bulletin of the Belgian Mathematical Society, 2007)

Textul de pe ultima copertă

This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. This book can be utilized for a one-year course on partial differential equations.
For the new edition the author has added a new chapter on reaction-diffusion equations and systems. There is also new material on Neumann boundary value problems, Poincaré inequalities, expansions, as well as a new proof of the Hölder regularity of solutions of the Poisson equation.
Jürgen Jost is Co-Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Dynamical Systems (2005), Postmodern Analysis (3rd ed. 2005, also translated into Japanese), Compact Riemann Surfaces (3rd ed. 2006) and Riemannian Geometry and Geometric Analysis (4th ed., 2005). The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998).
 
About the first edition:
Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-orderpartial differential equations.
- Alain Brillard, Mathematical Reviews
Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics.
- Nick Lord, The Mathematical Gazette

Caracteristici

Modern and systematic treatment of main approaches Several additions have been made to the 2nd edition, including a new chapter on Reacition-diffusion equations Emphasis on methods relevant for both linear and nonlinear equations Contains chapter summaries, detailed illustrations and numerous exercises