# Achievements of Research Area B

## Variational problems in geometry and elasticity.

Thin elastic objects display a fascinating variety of shapes and microstructures and recently attracted a lot of attention in the (bio-) physics community. The rigidity estimates of Friesecke, James and S. Müller have led to a rigorous understanding of a family of lower dimensional theories, including the von Kármán theory [B:LM09], as $\Gamma$-limits of three dimensional nonlinear elasticity. Frequently, a complex interaction of nonconvex stretching energies favouring locally isometric deformations and bending energies penalising high curvature arises and requires tools beyond $\Gamma$-convergence. A prominent example is the crumpling of a thin sheet leading to the formation of complex folding patterns over several length scales. Based on a combination of explicit constructions and general results from differential geometry, Conti and Maggi derived in [B:CM08] a power law for an upper bound of the energy relative to the thickness in agreement with previous conjectures from physics. Related to this, Conti, Székelyhidi (BJF) with De Lellis (Lipschitz lecturer at Bonn in 2009), studied the h-principle and rigidity of (almost) isometric embeddings of Riemannian manifolds in low co-dimension. They proved that the classical approximation with $C^1$ isometric embeddings by Nash and Kuiper can be extended to local $C^{1,\alpha}$ embeddings. Spadaro (HCM-Postdoc) and De Lellis initiated a major research program on a new approach to Almgren’s 1000 page treatment of the regularity of mass-minimising currents [B:DLS11].
Bartels developed and analysed new predictor corrector type algorithms for the finite element approximation of harmonic maps between surfaces [B:Bar10]. The convergence analysis is based on a weak compactness result by Freire, S. Müller, and Struwe. This concept has also been applied to the problem of approximating wave maps.

## Pattern formation in materials science.

Inspired by the Ball–James theory of martensitic microstructure recently new shape memory alloys with very low hysteresis have been synthesised. Zhang, James, S. Müller [B:ZJM09] and Barbara Zwicknagl developed a theory for low hysteresis alloys based on a careful analysis of the energy stored in transition layers. S. Müller and Dondl together with Dmitrieva and Raabe (MPI für Eisenforschung) identified in [B:DDMR09] lamination parameters in shear deformed copper single crystals by applying the theory of kinematically compatible lamination. Using a twoscale analysis in the context of statistical physics in [B:CDM09] S. Müller together with Cotar and Deuschel showed that non-convex perturbations of a convex potential in a gradient interface model on the scale of an atomic lattice can lead to a strictly convex surface tension at moderate temperatures. Anja Schlömerkemper working at the HCM on G-equivalent theories for discrete-to-continuum limits recently accepted an offer for a W3 professorship at Würzburg.
Following work by Ortiz et al. a rigorous theory of plasticity based on pattern formation due to crystal dislocation is now evolving. Conti, S. Müller and Garroni [B:CGM11] derived a linetension model for dislocations. Observing that the microstructure must be approximately onedimensional on most length scales they could obtain a sharp-interface limit in the sense of $\Gamma$-convergence. With Celia Reina (HCM-Postdoc), who obtained her PhD at Caltech in the computational mechanics group, the interdisciplinary collaboration with Ortiz is deepened. Very recently, Caterina Zeppieri (HCM-Postdoc) and Lucia Scardia rigorously derived a strain-gradient theory for plasticity including line tension of dislocations. Conti jointly with Dolzmann and Kreisbeck demonstrated that the rigid-plastic limit of geometrically nonlinear elastoplasticity is realistic and non soft if in addition to dissipation linear hardening is included. Frehse and Löbach [B:FL09] investigated the regularity of elastic plastic deformation with  hardening. Bartels and Roubícek gave a constructive existence theory in non-isothermal elastoplasticity, where solutions are constructed via finite element approximations and careful limit passages.
Jointly with the experimentalist Schäfer, Otto (former principal investigator of Research Area B) studied wall patterns in thin-film ferromagnetic elements and with Jutta Steiner new insight was obtained on the concertina pattern based on analysis and numerics of a reduced variational model rigorously derived from 3D micromagnetics [B:OS10]. Otto and Gloria (HCM-Postdoc) established optimal and fully practical error estimates in stochastic homogenization [B:GO11] (cf. also Research Area G). There is a close link to Research Area A via the geometry of dispersive evolution problems, e.g. the nonlinear Schrödinger equation. Koch and Tataru obtained global in time control of certain norms below L2 for solutions to the nonlinear Schrödinger equation, which implies control of the flow of the energy, even if uniqueness below L2 remains an outstanding problem [B:KT07]. In 2012 the HIM will host a trimester program “Mathematical challenges from materials science and condensed matter physics”, co-organised by Conti, S. Müller and Schlein (Research Area C and Research Area J).

## Variational methods in imaging and vision.

In the last decade a Riemannian perspective to the infinite dimensional space of shapes attracted a lot of attention. Rumpf and BIGS scholar Wirth linked the geometry of shape space to concepts of viscous flow and nonlinear elasticity. In [B:WBRS11] geodesics paths in shape space are approximated via a minimisation of the sum of nonlinear deformation energies on a time discrete sequence of shapes. In an alternative elastic approach a dissimilarity measure between shapes is given by the minimal stored elastic energy of deformations matching the shapes [B:RW09]. Polyconvex energy functionals combined with phase field approximations lead to well-posed, numerically efficient models for shape averaging and an elastic principal component analysis of shapes. Substantial impulses came from a workshop of the Research Area B on “Geometry and Statistics of Shape” co-organised by Cremers, Mumford, Rumpf and Trouvé in 2008. Pock (HCM-Postdoc) and Cremers jointly with Chambolle (several times guest at the HCM) pioneered the development of convex relaxation techniques for shape optimisation in computer vision [B:PSG+08]. At the atomic scale thin elastic films also pose challenging problems in vision. Berkels, Rumpf and coworkers proposed a Mumford–Shah model to extract grain boundaries and mesoscopic elastic deformation from transmission electron microscopy images.