The working group develops functional analytic methods for the treatment of partial differential equations, and it applies these techniques in particular to parabolic evolution equations. In the focus of our interests are the spectral theory of partial differential operators and the properties of their resolvents.
1. Functional calculus
We investigate -calculi for (bi-)sectorial operators. Here we combine methods of harmonic analysis and stochastic analysis (e.g., random sums in Banach spaces) with methods from functional analysis, in particular of the geometry of Banach spaces. Together with items 2 and 4, this research area was part of the project " functional calculus and its applications to partial differential equations" sponsored by the DFG (until 2007).
2. Vector valued harmonic analysis and wavelets
The boundedness of Fourier multipliers and Calderon-Zygmund operators with operator valued kernels on Bochner and Besov spaces is established. These operators are used as tools in the research areas 1,3 and 4, and they allow to study e.g. the convergence of wavelet decompositions in vector valued function spaces.
3. Maximal regularity and nonlinear problems
Using methods from vector valued harmonic analysis the maximal regularity of linear parabolic evolution equations was characterized and a comprehensive theory of maximal regularity was developed. Such regularity properties allow the treatment of nonlinear parabolic problems by means of linearization and fixed point methods. For instance, for equations with fully nonlinear boundary conditions we constructed invariant manifolds and used them to show the attractivity of equilibria under suitable spectral conditions on the linearization. These topics are investigated in the framework of the DFG project "Qualitative behavior of parabolic problems with nonlinear dynamical and static boundary conditions" (04/09 - 03/11).
4. Stochastic differential equations
The techniques from the research areas 1-3 have allowed to develop a powerful theory of stochastic integration on Banach spaces. Based on this theory, we have solved stochastic differential equations for systems with infinite dimensional state space, and we have investigated the qualitative behaviour of the solutions.
5. Elliptic differential operators and Gaussian estimates
We study regularity properties of elliptic differential operators with irregular or unbounded coefficients and of the semigroups generated by these operators. In particular, we are interested in estimating the kernels of the semigroups (e.g., by the Gaussian kernel). Such estimates play a crucial role in the qualitative behaviour of the underlying parabolic problem. Generalizations of Gaussian estimates have led to a Calderon Zygmund theory for non-integral operators which can be applied to Riesz transforms of functional calculi.
6. Control theory
Problems from control theory (such as wellposedness, feedback theory, or controllability) are investigated for parabolic and hyperbolic equations by means of the theory of operator semigroups and the methods from areas 1 and 3.
7. Asymptotics of linear evolution equations
The members of the working group have made numerous contributions to the theory of the long term behaviour of linear evolution equations. The main tools were spectral theory and the Laplace transform.