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)
Variational models for microstructure and phase transitions. (English) Zbl 0968.74050
Hildebrandt, S. (ed.) et al., Calculus of variations and geometric evolution problems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 15-22, 1996. Berlin: Springer. Lect. Notes Math. 1713, 85-210 (1999).

Summary: This article covers recent developments in the analysis of microstructures that arise from solid-solid phase transition, involving methods and models from the variational calculus. The material is presented in seven sections, starting with a comprehensive presentation of basic problems related to the formation of microstructures. Section 2 contains examples known as k-gradient problems, and collects some results concerning approximate and exact solutions for k4. In section 3 the reader becomes familiar with the notion of Young measures and with the way how the Young measures can be used to represent the limits of variational integrals. Section 4 is devoted to the study of gradient Young measures, i.e. Young measures which arise from sequences of gradients. Among other things, the author describes here the classification of gradient Young measures due to D. Kinderlehrer and P. Pedregal [J. Geom. Anal. 4, No. 1, 59–90 (1994; Zbl 0808.46046)] which relies on the concept of quasiconvexity of variational integrals. Moreover, Šveràk’s counterexample is reviewed, and it is indicated how to obtain J. Kristensen’s result [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 1, 1–13 (1999; Zbl 0932.49015)] saying that the quasiconvexity is not a local condition. In Section 5 the problem of exact solutions is discussed, i.e. the problem of finding all Lipschitz maps u satisfying DuK a.e. for a given compact set K in the space of matrices (for example, the author considers the case K=SO(2)ASO(2)B for special choices of A and B). Section 6 briefly describes some reasonable penalty terms which can be added to the pure elastic energy in order to exclude an infinitesimally fine mixture of phases. The final section 7 contains alternative descriptions of microstructures, gives comments on the dynamics and computation of microstructures, and finishes with some solved and unsolved problems.

The material is presented with great care, showing the author’s ability to explain the main methods which have been developed in this rapidly growing area of applied mathematics. Clearly, most of the proofs have not been written down, the interested reader who wants to go into details will find an exhaustive list of more than 230 references together with some comments on the history. The article is addressed to graduate students and researchers in applied analysis, applied mathematics, mechanics, material science and engineering.


MSC:
74N15Analysis of microstructure (solid mechanics)
74G65Energy minimization (equilibrium problems in solid mechanics)
74A60Micromechanical theories
49S05Variational principles of physics