Calculus of Variations and Nonlinear PDE-Systems
Many problems in analysis, geometry, physics, engineering, and economics can be cast into the form of minimizing a functional I(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 systems of equations.
In particular my interest is about existence and regularity of nonlinear PDE-Systems deriving from engineering applications and structural mechanics, as e.g., plasticity problems, extended continuum mechanics, shells and plates etc. Useful tools are newly derived coercive inequalities, like a generalized Korn‘s inequality or an inequality connecting the curl of a rotation to all partial derivatives. The precise amount of regularity of the (weak) solution is also a decisive information for subsequent FEM-implementations.
I have first applied successfully the fundamental notion of John Balls polyconvexity to materials undergoing large deformations with preferred directions (anisotropy). A specific application is e.g. biological tissue. Thus I have partly answered problem 2/18 of John Ball‘s „Some open problems in elasticity“: „Are there ways to verifying polyconvexity and quasiconvexity for a useful class of anisotropic stored energy functions?“ This topic is part of ongoing research, fully funded by the DFG.
I am also interested in 3D-to-2D reduction of nonlinear elasticity theory to membrane-, plate- and shell theories. A main mathematical tool here 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.
Plasticity, either small strain or large strain, takes a prominent role in my investigations. I have developed a novel model for a geometrically exact description based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts with viscoelastic effects for which well-posedness is shown by the above mentioned methods. In this field this is one of the view results establishing the existence of classical, smooth solutions. In small strain plasticity, I have extended the Cosserat model to incorporate plastic effects and more recently I have used variational inequalities to show existence for a weak reformulation of gradient plasticity.
The mathematics of extended continuum models is another field of research. Here, I treat e.g. Cosserat and micromorphic models which may arise through certain homogenization schemes from small scale to large scale, think e.g. of a continuum description of a metallic foams. I have set up a new finite strain Cosserat model and I have shown existence of minimizers. A major challenge here is given by the nonlinear structure of the group of rotations SO(3). Recently, I have also re-examined the small strain Cosserat model with very weak curvature conditions, thus eliminating some troubling physical inconsistency problems of the model.