# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Partial differential equations. III: Nonlinear equations. 2nd ed. (English) Zbl 1206.35004
Applied Mathematical Sciences 117. New York, NY: Springer (ISBN 978-1-4419-7048-0/hbk; 978-1-4419-7049-7/ebook). xxii, 715 p. EUR 99.95/net; £ 86.50; SFR 148.00 (2011).

The third of three volumes on partial differential equations is devoted to nonlinear PDEs. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of ${L}^{p}$ Sobolev spaces, Hölder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderón-Zygmund theory and paradifferential operator calculus. In this third volume there are six chapters.

Chapter 13 is devoted to function space and operator theory for nonlinear analysis.

Chapter 14 concerns nonlinear elliptic equations (DeGiorgi-Nash-Moser theory for variational operators, Krylov-Safonov estimates for nonvariational operators, Monge-Ampère equation, Leray-Schauder fixed point theorem).

Chapter 15 analyses nonlinear parabolic equations (reaction diffusion equations, Trotter product formula and Moser iteration method).

Chapter 16 is focused on nonlinear hyperbolic equations (Cauchy-Kowalewsky theorem, conservation laws and Riemann problem).

Chapter 17 is devoted to incompressible fluids governed by Navier-Stokes equations.

Chapter 18 concerns Einstein’s gravitational equation and Schwarzschild solution.

The book is targeted at graduate students and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.

In this second edition, there are a new appendix in Chapter 13 and there are two new sections in Chapter 17. Moreover several other sections have been substantially rewritten, and numerous others polished to reflect insights obtained through the use of these books over time.

##### MSC:
 35-01 Textbooks (partial differential equations) 35-02 Research monographs (partial differential equations) 35J60 Nonlinear elliptic equations 35K55 Nonlinear parabolic equations 35L60 Nonlinear first-order hyperbolic equations 35L70 Nonlinear second-order hyperbolic equations 35J96 Elliptic Monge-Ampère equations 35Q30 Stokes and Navier-Stokes equations 35Q31 Euler equations 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
##### Keywords:
operator theory; nonlinear analysis; new sections