Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Sascha Trostorff
Technische Universität Dresden
Well-posedness and causality of a class of evolutionary inclusions
In 2009 R. Picard has shown that most of the problems occurring in classical mathematical physics posses
the form
� ∂0M(∂ −1
0 ) + A� U = F,
where ∂0 denotes the time derivative established as a normal operator in suitable weighted Hilbert space,
M(∂ −1
−1
0 ) is an analytic, operator valued function of ∂0 , describing the behaviour of the underlying material
and A is a skew-selfadjoint operator. The well-posedness and causality of this problem class has
been shown. We extend these results to differential inclusions, by replacing the operator A by a maximal
monotone relation. Thus we arrive at a class of differential inclusions which are encountered in many
applications, for example in hysteresis models or for switched dynamical systems. We apply our solution
theory to a system describing the diffusion of a fluid through a saturated poro-plastic media, including
hysteresis effects (see Showalter & Stefanelli 2004).
Literatur
Picard, R. (2009). A structural observation for linear material laws in classical mathematical physics, Math.
Methods Appl. Sci., 32 (14), 1768 - 1803.
Showalter, R. E., Stefanelli, U. (2004). Diffusion in poro-plastic media. Math. Methods Appl. Sci., 27 (18),
2131 - 2151.
Carsten Trunk
Technische Universität Ilmenau
Variational Principles for Eigenvalues of a Class of Block Operator Matrices
We consider eigenvalues of block operator matrices
�
�
0 I
A =
−A0 −D
which can be written as a quadratic eigenvalue problem. Under certain assumptions (A0 uniformly positive
with a compact resolvent, A −1/2
0
DA−1/2
0 bounded and nonnegative) the essential spectrum of A is located
on the negative half-axis. We characterise eigenvalues above and below the essential spectrum, even in
the presence of complex eigenvalues. As a consequence we can compare these eigenvalues if the entries
A0 and D are increased/decreased.
Operators of the form (4) arise, e.g., in the investigation of small transverse oscillations of a pipe carrying
steady-state fluid of ideal incompressible fluid. This talk is based on a joint work with Birgit Jacob
(Bergische Universität Wuppertal, Germany) and Matthias Langer (University of Strathclyde, Glasgow).
(4)
189