Prof. Dr. (em.) Jens Frehse

Biography

1966 - 1969 Assistant Professor, University of Frankfurt 1969 - 1970 Postdoc, National Research Council (CNR), Rome, Italy 1970 Habilitation, University of Frankfurt 1972 DFG research, Pisa, Italy 1972 - 1973 Substitution Professor (C4), University of Heidelberg 1973 Visiting Associate Professor, University of California, Berkeley, CA, USA 1973 - 2010 Professor (C4), University of Bonn Since 2010 Professor Emeritus, University of BonnMainly I consider myself as a specialist in regularity theory in the field of nonlinear elliptic and parabolic equations and variational inequalities. Besides the classical meaning, “regularity” may also mean “improved p-integrability” or “improved fractional differentiability” in situations where more regularity cannot be obtained (e.g. see my work on compressible fluids and elastic-plastic deformation with hardening). The equations considered come from continuum mechanics, fluid mechanics, Bellmann system to stochastic games. Euler equations to variational problems motivated by differential geometry. In the last two years, I developed, in collaboration with Miroslav Bulicek, several new weighted norm techniques which allow to treat a considerable broader class of variational problems with p-growth, and also stochastic differential games like Stackelberg games rather than Nash games. These techniques will be applied in the framework of mean field games in the sense of Lasry-Lions, furthermore for proving the long time existence of certain variational flows. A recent manuscript on the Prandtl Reuss problem (submitted) yields a new technique to obtain fractional derivatives of stress velocities. This improves the regularity theory of related problems in several directions.

Research Interest

Research Area A (until 10/2012) Regularity analysis for elliptic and parabolic systems:  In [1] and [2], we constructed an irregular complex valued solution of an elliptic or parabolic scalar equation on a domain with dimension 2. This shows that the DeGiorgi-Nash theorem does not hold in the complex case. It gives a simple counter example of a real valued system with two equations.  Regularity to solutions of Euler systems to variational problems with p-growth (with Bulicek):  In [3], we obtained everywhere-Hoelder-continuity for solutions of Euler systems which are, concerning their structure, far away from Uhlenbeck-type systems. The  estimate even works for a class of nonconvex coercive systems. Research Area B Stationary compressible fluids (with Steinhauer and Weigant):  We focused on the Navier-Stokes-equations with pressure dependent density . We succeeded to treat physical relevant pressure laws with . See [4], [5], [6].  p-Fluids (with Ruzicka and Malek):  In [7], we obtained long time solutions (overcoming a certain monotonicity problem). Further results (together with Ruzicka and Malek) cover the temperature dependent case.  Regularity for elastic-perfect-plastic deformations (with Loebach):  In [8], we obtained fractional boundary differentiability for the stresses which lead to a differentiability order greater 1/2. Further boundary differentiability results were obtained with Bulicek and Malek. Research Area I (until 10/2012) We consider Bellmann systems to stochastic differential games with Hamiltonians growing quadratically. The analytical difficulty consists in this behaviour of the Hamiltonian which leads to the problem that a lot of game problems have not been solved yet due to the unorthodox structure of the Hamiltonian. The group (Bensoussan, Bulicek, Frehse, Vogelgesang) developed new analytical tools which allowed the solution of Stackelberg games and cyclic Nash games, see [9] and “Nash and Stackelberg Differential Games”, Chinese Annales of Mathematics Series B (accepted). Furthermore we mention our work on stochastic differential games with discount control which lead to interesting new PDE aspects.