Partial Differential Equations

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

Partial differential equations form an adequate and powerful instrument to provide a mathematical model for nature. At the Weierstrass Institute this research has two essential focuses: (a) Regularity for the solutions of linear elliptic equations and (b) Existence, uniqueness and regularity for evolution equations. [>> more]

Thermomechanical 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]

At 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 and, finally, to mathematical problems of semiconductor models. [>> more]

Time-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]

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

The 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]

A 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]

Because of the ongoing miniaturization, modern devices in mechanics, electronics, or optics become smaller and smaller. Their working principles depend more and more on effects on different spatial scales. The aim is to optimize the efficiency by adapting the arrangement of interfaces or periodic microstructures. 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]

The mathematical description of many scientific and technological problems leads to systems of partial differential equations (PDEs). [>> more]

Many 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]