Applied analysis links mathematics and scientific computing to engineering and natural sciences. Modeling in the applied sciences typically leads to systems of nonlinear ordinary or partial differential equations or variational problems. The nonlinearities reflect complex phenomena which result in oscillations, concentrations, or singularities in the mathematical model. Most problems involve the interaction of processes on different time and length scales, leading to so-called multiscale systems. Resolution on all scales is usually impossible, which requires deriving hierarchies of models to describe the problem on the various scales.

Berlin applied analysis is highly visible internationally through the groups at FU, HU, and WIAS. Their very active cooperation is shown through several joint research activities as well as the seminar on Nonlinear Dynamical Systems (FU and WIAS) and the traditional Langenbach Seminar on nonlinear PDEs (WIAS and HU).

Several research projects of the applied analysis groups in Berlin are motivated by phase transformation phenomena (Dreyer, Knees, Kraus, Mielke, Sprekels). For example, the growth of crystals from a melt involves free boundaries, and damage processes in elastic materials have to be understood in their interaction with phase separation in solder alloys. Techniques from the calculus of variations are relevant in the study of local or global minimizers of certain energies. Often small parameters are involved, representing a specific regime of interest for the corresponding applications. The framework of Γ - convergence allows one to study the corresponding family of minimization problems and to identify the limit problem. Special interest lies in developing a theory of Γ - convergence for evolutionary problems. Since sharp-interface limits lead to lower-dimensional singular structures, there is a strong link to differential geometry.

Nonlinear dynamical systems (Fiedler, Mielke, Recke, Yanchuk) are studied in Berlin, for example, in the context of nonlinear processes in opto-electronic devices, in spatiotemporal patterns in nonlinear excitable systems, or in synchronization and cluster formation in coupled neuron systems. Bifurcation and center manifold theory are the key analytical tools for the understanding of these processes. Due to the inherent multiple scales, singular perturbation theory is also relevant.

Asymptotic analysis and multiscale modeling play an important role in applications and form a prominent source of mathematical challenges. Homogenization, averaging techniques, and asymptotic expansions are used to extract reduced models for the statistics of large systems. A few examples are the following:

• Solid-liquid phase transitions occur in metal soldering or in crystals used in wafer fabrication. The early stage of such a phase transformation requires a microscopic description on the atomic scale. Here there is a strong link to stochastic analysis, where the corresponding particle models are studied. The second stage, growth, is often modeled by nonlinear parabolic systems for the respective fields (temperature, concentration, potentials), whereas the late stage, coarsening, is best described as a free boundary problem.
• In the context of thin liquid films (Wagner), the separation of length scales allows for the derivation of simplified models derived from the basic equations of fluid mechanics.Corresponding reduced theories result in degenerate higher-order equations for which a general mathematical theory is still lacking.
• In semiconductor physics, the importance of active interfaces leads to new mathematical questions concerning stable and thermodynamically correct bulk-interface couplings (Glitzky, Mielke). Moreover, the usage of organic semiconductor materials requires new stochasic models for solar cells (Wagner).
• Multiple scale asymptotics has recently opened highly innovative routes of theoretical research in meteorology and climate science (Klein). It provides a unifying framework for the large variety of simplified models known in theoretical meteorology, and it paves the way for the systematic investigation and robust numerical simulation of the ubiquitous scale interactions in the atmosphere and the oceans.

All of the research topics described above require at some level efficient simulations of the different mathematical models. The corresponding links to numerical analysis and scientific computing are also reflected in the Application Areas C and D of MATHEON. The evolution of microstructures in elastoplastic materials is studied within the DFG Research Unit 787 MicroPlast. Additional interdisciplinary contacts are established in the SFB 910 - "Control of self-organizing nonlinear systems" and in the competence center PVcomB on nanotechnology for photovoltaics. The links to differential geometry mentioned above are represented in the SFB Space, Time, Matter. The combination of analytical and stochastic tools to understand complex phenomena is currently a very active research area.