Analytic methods for partial differential equations

Project: Research area

Description

Classical methods for solving partial differential equations (=PDE) are mainly based on explicit representations of solutions such as the Poisson Integral Formula for the solution of the Dirichlet boundary value problem for the Laplace equation in balls. Present methods for solving (linear and non-linear) PDE make use of functional-analytic toole such as the reduction of problems for PDEs to fixed-point problems and their solution by suitable topological methods. Such reductions are possible, for instance, by the inversion of (elliptic) differential operators by singular integrla operators.

Main areas of the research project are boundary and initial value problems and their solution in suitable chosen function spaces (such as Banach, Hilbert and Sobolev spaces), where in many cases the above described functional-analytic methods can be combined with methods of complex and Clifford analysis. An important topic of complex analysis in the plane is the theory of generalized analytic functions making it possible to solve fully non-linear (elliptic) first-order systems in the plane (instead of the Cauchy-Riemann system). Clifford analysis, further, generalizes complex methods to higher-dimensional Euclidean spaces. Finally, the research programme includes the optimization of fixed-point methods.
StatusActive
Effective start/end date1/01/95 → …