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Calculus of variations

People:  Yuen Au Yeung
  Marco Cicalese
  Gero Friesecke

Many problems in analysis, geometry, physics, engineering, and economics can be cast into the form of minimizing a functional F(u) among a class of admissible functions u. Important early examples of such functionals minimized in nature are: time of travel of a light ray (Fermat's principle in optics, 1662), action of a trajectory of a mechanical system (Hamilton's principle, 1834), and energy of the electrostatic field outside a charged body (Dirichlet's principle, Dirichlet, Kelvin, Gauss, 1840s). Minimizers of the latter problem solve Laplace's equation Δu=0, linking the calculus of variations to the theory of partial differential equations. Many fascinating modern day minimization problems can be viewed as far-reaching nonlinear extensions of Dirichlet's principle, in that minimizers solve nonlinear partial differential equations. Another important contemporary research area are discrete problems, a paradigm being the travelling salesman problem of finding the shortest connection between a large number of cities.

Much of our own work is motivated by questions in continuum mechanics and atomistic mechanics. We are especially interested in

A main mathematical tool in our work is Gamma convergence, introduced by De Giorgi and developed notably by Dal Maso, Braides and coworkers. It provides a powerful and rigorous mathematical framework to pass from a finer-scale (or higher-dimensional) variational principle to a coarser-scale (or lower-dimensional) effective variational principle.

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