Background Material on Asymptotic Analysis of Extremal Problems

  • Peter I. Kogut
  • Günter R. Leugering
Part of the Systems & Control: Foundations & Applications book series (SCFA)


This chapter is intended to provide various facts, notions, and concepts which play a fundamental role in modern asymptotic analysis of optimization problems. We recall some main concepts and basic results of measure theory, Sobolev spaces, and boundary value problems which are used later. We include proofs only if the line of arguments is of importance for the understanding of subsequent remarks. For a deeper insight in the subject, we refer to the books of Adams (1975), Bucur and Buttazzo (2005), Evans and Gariepy (1992), Kantorovich and Akilov (1976), Lions and Magenes (1972), Maz’ya (1986), Yosida (1965), Ziemer (1989), and so on.


Banach Space Sobolev Space Dirichlet Problem Weak Convergence Extremal Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 2.
    R. Adams. Sobolev Spaces. Academic Press, New York, 1975. Google Scholar
  2. 38.
    D. Bucur and G. Buttazzo. Variational Methodth in Shape Optimization Problems. Birkhäuser, Boston, 2005. Google Scholar
  3. 106.
    L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, New York, 1992. Google Scholar
  4. 128.
    L. V. Kantorovich and G. P. Akilov. Functional Analysis. Nauka, Moskow, 1976. (in Russian) Google Scholar
  5. 173.
    J.-L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin, 1972. Google Scholar
  6. 185.
    V. G. Maz’ya. Sobolev Spaces. Leningrad University Press, Leningrad, 1986. (English transl.: Springer-Verlag, Berlin, 1985) Google Scholar
  7. 251.
    K. Yosida. Functional Analysis. Springer, Berlin, 1965. Google Scholar
  8. 267.
    W. P. Ziemer. Weakly Differentiable Functions. Springer, Berlin, 1989. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and Mechanics, Department of Differential EquationsOles Honchar Dnipropetrovsk National UniversityDnipropetrovskUkraine
  2. 2.Department of Mathematics, Chair of Applied Mathematics IIFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations