## The group contributes to the following mathematical research topics of WIAS:

Analysis of Partial Differential Equations and Evolutionary Equations

Partial differential equations form an adequate and powerful instrument to provide a mathematical model for nature. At the Weierstrass Institute this research has three essential focuses: (a) Rigorous mathematical analysis of general evolution equations in terms of existence, uniqueness and regularity of different types of solutions, (b) Development of variational approaches using the toolbox of the calculus of variations, (c) Regularity results for solutions of elliptic and parabolic partial differential equations. [>> more]

Direct and inverse problems in thermomechanicsThermomechanical models are the basis for the description of numerous technological processes. The consideration of phase transitions and of inelastic constitutive laws raise exciting new questions regarding the analysis of direct problems as well as the identification of materials parameters. [>> more]

Functional analysis and operator theoryAt WIAS, functional analysis and operator theory are related, in particular, to problems of partial differential equations and evolutions equations, to analysis of multiscale, hybrid and rate-independent models. [>> more]

Hysteresis operators and rate-independent systemsTime-dependent processes in physics, biology, and economics often exhibit a rate-independent input-output behavior. Quite often, such processes are accompanied by the occurrence of hysteresis phenomena induced by inherent memory effects. There are two methods to describe such processes. 1. In order to describe the current state, one can add internal variables to the observable variables, and describes the evolution of internal variables. 2. For many cases one can find hysteresis operators directly describing the input-output-behavior, such that the state depends on the history of the considered process. [>> more]

Interacting stochastic particle systemsIn the mathematical modeling of many processes and phenomena in the Sciences and Technology one employs systems with many random particles and interactions. [>> more]

Large deviationsThe theory of large deviations, a branch of probability theory, provides tools for the description of the asymptotic decay rate of a small probability, as a certain parameter diverges or shrinks to zero. Examples are large times, low temperatures, large numbers of stochastic quantities, or an approximation parameter. [>> more]

Modeling, analysis and numerics of phase field modelsA diffuse phase field model is a mathematical model for describing microstructural phenomena and for predicting morphological evolution on the mesoscale. It is applied to a wide variety of material processes such as solidification, coarsening in alloys, crack propagation and martensitic transformations. [>> more]

Multi-scale modeling, asymptotic analysis, and hybrid modelsTo understand the interplay between different physical effects one often needs to consider models involving several length scales. The aim in this mathematical topic is the derivation of effective models for the efficient description of the processes. The understanding of the transfer between different scales relies on mathematical methods such as homogenization, asymptotic analysis, or Gamma convergence. The generated effective models are coupled partial differential equations combining volume and interfacial effects. [>> more]

Systems of partial differential equations: modeling, numerical analysis and simulationThe mathematical modelling of many scientific and technological problems leads to (initial) boundary value problems with systems of partial differential equations (PDEs). [>> more]

Variational methodsMany physical phenomena can be described by suitable functionals, whose critical points play the role of equilibrium solutions. Of particular interest are local and global minimizers: a soap bubble minimizes the surface area subject to a given volume and an elastic body minimizes the stored elastic energy subject to given boundary conditions. [>> more]

## Research Groups

- Partial Differential Equations
- Laser Dynamics
- Numerical Mathematics and Scientific Computing
- Nonlinear Optimization and Inverse Problems
- Interacting Random Systems
- Stochastic Algorithms and Nonparametric Statistics
- Thermodynamic Modeling and Analysis of Phase Transitions
- Nonsmooth Variational Problems and Operator Equations