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Our broad interest is the mathematical modeling and analysis of problems arising in the physical sciences. Of particular concern are the combination of analysis and computational methods to gain insight into the mathematical structure and physical predictions of the model. Further, we seek to combine methods from analysis and probability theory to learn about the behavior of complex physical systems with random input. The major areas of our research are the fields of calculus of variations with nonconvex energy functionals and free boundary evolution problems. In particular, we have been working on the following projects:

The main goal of this project is to derive macroscopic evolution laws for interfaces in heterogeneous, random environments, described on a small scale by nonlinear PDEs with random coefficients. In particular, we want to go beyond classical homogenisation in order to treat systems where a long-range collective behaviour emerges.

Such problems of interface evolution in heterogeneous media arise in a large number of physical models. Common to these models is a regularizing operator, for example line tension, and the competition between an external applied driving force $F$ and a force field $f(x,y)$ describing the inhomogeneities. A canonical example is $$ u_t(x,t) = \Delta u(x,t) + f(x,u(x,t)) +F, $$ with appropriate boundary conditions and initial condition $u(\cdot,0) = 0$. The evolving interface is given by the graph of the function $u\colon \mathbb{R}^n \to \mathbb{R}$, the heterogeneous force term $f(x,u(x))$, with $f\colon \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, is evaluated at the current location of the interface, which makes the problem non-linear.

We consider the problem of finding minimal-energy configurations (or at least stationary states) of confined thin elastic bodies. A good example are the inner membranes of mitochondria: they consist of an elastic lipid bilayer and are confined by an additional outer mitochondrial membrane of much smaller surface area. A very important consideration for mathematical models of these membranes is their topological side condition. Organelle membranes, for example of mitochondria, are elastic bilayers separating a connected inside of the organelle from a connected outside. This implies that the boundary between the inside and the outside must also be a connected surface without any self-interpenetration.

In this project we focus on phase-field, or diffuse-interface, descriptions for such physical models and their sharp-interface limits. While this adds an additional space dimension to the model (instead of considering the interface as an $n-1$-dimensional submanifold directly, we work with a function in $n$ dimensions whose zero-level set approximates the interface), such phase-field models nevertheless have many advantages. In particular numerical implementation and its rigorous analysis is often feasible only for diffuse-interface models. Furthermore, coupling of the surface equations to bulk processes in the cell always requires treating the full $n$-dimensional problem, regardless of how the free boundary is encoded.

The main goal of this project is to answer questions that arise in the transition of scales in continuum mechanical and atomistic problems. In particular, we want to rigorously derive the coarse scale behavior from microscopic models for plastic materials. One of the challenges is to understand the formation of the intricate sub-grain patterns that plastically deformed materials exhibit. These patterns are an observable effect of non-homogeneous plastic deformation which in turn has a great influence of the further plastic behavior of crystals. Some of the questions we want to answer are

- Which continuum plasticity models lead to the formation of microstructure?
- What is the morphology of the microstructure predicted by the models?
- Which experiments are suitable to verify the microstructure predictions?
- Which atomistic or mesoscopic models give rise to pattern forming continuum models?
- Can we derive coarse grained models that incorporate the effect of, but do not resolve the small scale microstructure (i.e., relaxed models)?