Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Bernd Kawohl
Universität zu Köln
On an overdetermined boundary value problem
Serrin proved in a well-known paper from 1971 that the overdetermined bvp −∆u = 1 in Ω, u = 0 AND
∂u/∂ν = const. on ∂Ω has a solution u in a bounded connected domain Ω if and only if Ω is a ball. While
Serrin’s Theorem is valid for fairly general and benign elliptic equations, Weinberger found a much simpler
proof for the special case mentioned above. In my lecture I report on related questions for degenerate
nonlinear equations, e.g. for −∆pu = 1 with p ∈ (1,∞]. Forp < ∞ Pohozaev-identities are one ingredient for
the proof. For p = ∞ it is remarkable that there are other domains than balls, on which (viscosity)-solutions
exist. These results were obtained in cooperation with the coauthors listed in the references.
Literatur
Kawohl, B. (2007) Overdetermined problems and the p-Laplacian, Acta Math. Univ. Comenianae 76, 77–
83.
Buttazzo, G. & Kawohl, B. (2011) Overdetermined boundary value problems for the ∞-Laplacian, Intern.Mathem.
Res. Notices 2011 237–247.
Kawohl, B. (2011) Fragalà, I., Gazzola, F. & Kawohl, B. Overdetermined problems for the ∞-Laplacian and
web functions. submitted
Hans Knüpfer
Universität Bonn
Propagation of three-phase contact lines - well-posedness and regularity
The propagation of a liquid drop on a plate is characterized by the evolution of the three-phase contact
line where air, liquid and solid meet. For thin viscous droplets, the evolution can be decribed by a class
of scalar fourth order degenerate parabolic equations, the so called thin-film equations. We consider
well-posedness and regularity for these thin-film equations and other fluid models in the presence of a
three-phase contact line. Since the considered do not satisfy a maximum principle, the analysis has to
be based on the underlying dissipative structure. One important point in the analysis is to find suitable
function spaces such that the model is well-posed.
Stefan Krömer
Universität zu Köln
Weak lower semicontinuity of multiple integrals revisited: the role of lower bounds
For integral functionals on W 1,p , the standard result due to Acerbi and Fusco shows equivalence of
quasiconvexity of the integrand and weak lower semincontinuity of the functional. This result relies on an
additional assumption, a lower bound for the integrand, which is not purely technical (as a well-known
example involving the determinant illustrates), but not optimal either. I will present a less restrictive
condition that is also necessary. Interestingly, this optimal condition is weaker than boundary quasiconvexity
as defined by Ball and Marsden, although for p-homogeneous integrands, it reduces to the latter.
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