Research Interests
These lectures gives an impression of my present research interests: Marburg, Perugia, Nizza .
Boundary value problems for Willmore surfaces
The Willmore equation, i.e. the Euler-Lagrange equation of the Willmore functional, is a particularly challenging and important problem in nonlinear analysis. It is quasilinear and of fourth order. Many methods which are well established in second order problems do not apply any longer. Nevertheless, significant progress could be achieved in the past years e.g. by L. Simon, E. Kuwert, R. Schätzle, T. Riviere and many others. Most of these results, however, concern closed surfaces. As for boundary value problems only little is known so far since very hard compactness difficulties have to be overcome. By means of numerical studies and analytical investigations we want to study first quite symmetric prototype situations in order to outline in which direction one may expect to obtain a-priorily bounded minimising sequences and classical solutions. Our student Stephan Lenor has produced a number of pictures. This is a joint project with my colleagues Klaus Deckelnick und Friedhelm Schieweck who intend to develop numerical algorithms and convergence results also in more general situations, e.g. for graphs over two dimensional domains. Analysis, numerical analysis and numerics will intensively interact.
Supported by Deutsche Forschungsgemeinschaft, 1.10.2008-30.09.2010. Anna Dall'Acqua was working on this project.
Qualitative properties of solutions of elliptic boundary value problems of higher order
Here one may think of the clamped plate equation (biharmonic operator under Dirichlet boundary
conditions) as the most simple prototype for higher order elliptic boundary value problems. Of particular interest are positivity preserving properties of the corresponding solution operators:
Is it possible to find conditions on the domain and the differential operator such that positive data always yield positive solutions? Even for the "simple" example of the clamped plate equation it is up to now not possible to answer this question completely. There are as well examples of domains with positivity preserving as of domains, where change of sign occurs. I am mainly interested in the first class of domains: In two dimensions, e.g., in domains close to the disk, positive right hand sides always give rise to positive solutions. "Upwards pushing yields upwards bending."
Positivity properties of linear boundary value problems are not only of interest on their own, but one may also try to apply them to nonlinear problems. Although first results could be achieved it is still by far not obvious to what extent the (relatively restricted) positivity results will be of use for general nonlinear elliptic boundary value problems of higher order. I hope for some progress in problems from physics (mechanics, hydrodynamics) and differential geometry.
In this field I collaborate with Guido Sweers (Cologne, TU Delft) and Frederic Robert.
Semilinear eigenvalue problems involving critical Sobolev exponents
These equations are closely related to problems in conformal geometry. Here we are basing upon two widely known papers of Brezis-Nirenberg and Pucci-Serrin and try to find out in how far the results kown for second order problems extend to boundary value problems of higher order. Emphasis is laid on the interplay with qualitative properties of solutions. In this connection a modified version of a conjecture of Pucci and Serrin concerning the "critical dimension phenomenon" for semilinear polyharmonic problems could be proved. A full proof of the original conjecture, however, seems up to now out of reach.
Recently, the connections with the above mentioned positivity properties could be more intensively exploited. Here a decomposition method with respect to pairs of dual cones in higher order Sobolev spaces has proved to be particularly useful. This method replaces the decomposition in positive and negative part, which is no longer admissible in these spaces. In this way, compactness properties of the corresponding variational functionals could be described efficiently.
In this field I am particularly working with Filippo Gazzola.
Biharmonic equations with supercritical growth
Variational techniques are no longer available. Instead, comparison principles, the sub-/supersolution method or in the case of radially symmetric solutions methods from dynamical systems have to be applied. Combining supercritical growth with differential operators of higher order gives rise to subtle technical problems. Also here, I collaborate with Filippo Gazzola, which was 2005 and 2006 supported by the Vigoni-programme of DAAD (Bonn) and CRUI (ROM).
Parabolic systems with critical growth
Under consideration are semilinear systems, whose nonlinear terms grow critically with respect to the canonical energy norm of weak solutions, i.e. we are dealing with controllable growth conditions.
Without further assumptions (like e.g. sign conditions on the nonlinear term) it was shown in a joint paper with Wolf von Wahl that every weak solution is regular. For this result we crucially employ a continuity method with the time as parameter of continuity.
Further, we consider geometric evolution equations in order to construct Hermitian harmonic mappings between noncompact complete manifolds. This semilinear system has quadratic growth with respect to the gradient and is not in divergence form.
Navier-Stokes equations
Here I am interested in instationary problems in exterior domains with nonzero prescribed velocity at infinity. I have been investigating regularity of suitable weak solutions and the asymptotic behaviour, as the time tends to infinity, of perturbations of physically reasonable stationary solutions with small energetic Reynolds number.