Variational and Topological Methods

  • Leszek Gasiński
  • Nikolaos S. Papageorgiou
Part of the Problem Books in Mathematics book series (PBM)


Let X be a set and let \(\varphi: X\longrightarrow \overline{\mathbb{R}} = \mathbb{R} \cup \{ +\infty \}\) be a map. One of the main problems of the Calculus of Variations is to find the minimum value
$$\displaystyle{m(\varphi )\ =\ \inf _{u\in X}\varphi (u)}$$
and also investigate the set of minimizers, that is,
$$\displaystyle{S(\varphi )\ =\ \left \{u \in X:\ m(\varphi ) =\varphi (u)\right \}.}$$
The classical way to approach this problem is the so-called direct method of Tonelli. This method consists of determining a suitable topology on X for which the function \(\varphi\) is lower semicontinuous and coercive.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Leszek Gasiński
    • 1
  • Nikolaos S. Papageorgiou
    • 2
  1. 1.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Department of MathematicsNational Technical UniversityAthensGreece

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