# Calculus of variations

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

- the crystallization problem: under which conditions is it optimal for atoms to assemble into crystalline order?
- atomistic-to-continuum limits (including effective theories for thin films, shape models)
- 3D-to-2D reduction of nonlinear elasticity theory to membrane-, plate- and shell theories
- phase transitions and fracture in 3D elasticity.

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.

## Publications

- Y.Au Yeung, G.Friesecke, B.Schmidt, Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape, Calc. Var. PDE 44, 81-100, 2012 Article
^{} Preprint ^{}
- S.Capet, G.Friesecke, Minimum energy configurations of classical charges: Large N asymptotics, Appl. Math. Research Express, doi:10.1093/amrx/abp002, 2009 Article
^{} Preprint ^{}
- B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy,
*J. Math. Pures Appl.* 88, 107-122, 2007
- B. Schmidt, Minimal energy configurations of strained multi-layers,
*Calc. Var. Partial Diff. Eq.* 30, 477 - 497, 2007
- G.Friesecke, R.D.James, S.Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence,
*Arch. Rat. Mech. Analysis* 180 No. 2, 183-236, 2006 Article ^{} Preprint ^{}
- G.Friesecke, R.D.James, S.Müller, The Föppl-von Karman plate theory as a low energy Gamma limit of nonlinear elasticity,
*C. R. Acad. Sci. Paris*, 2004
- G.Friesecke, R.D.James, S.Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity,
*C. R. Acad. Sci. Paris* Ser. I 334, 173-178, 2002
- G.Friesecke, R.D.James, S.Müller, A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity,
*Commun. Pure Appl. Math.* Vol LV, 1461-1506, 2002 Preprint ^{}
- G.Friesecke, F.Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice,
*J. Nonl. Sci.* 12 No. 5, 445-478, 2002
- G.Friesecke, R.D.James, A Scheme for the Passage from Atomic to Continuum Theory for Thin Films, Nanotubes and Nanorods,
*J. Mech. Phys. Solids* 48, 1519-1540, 2000