Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Mario Kaip
Universität Konstanz
Mixed order systems and application to parabolic boundary value problems
In this talk we want to present some results on mixed order systems on L p -spaces and applications
to free boundary value problems. Using the H ∞ -calculus concept we introduce an functional calculus
for the time and space derivatives. After an analysis of the order structure of each component of the
mixed order system we can determine mapping properties in a comfortable way. One major aspect of
this work is the possibility to consider mixed order systems on spaces of mixed Besov and Bessel scales
as 0B 1−1/(2p)
p
(R+,Lp(R n )) ∩ Lp(R+,B 2−1/p
p
(R n )). Therefore we are able to handle mixed order systems
realized on parabolic trace spaces for example.
In L p theory free boundary value problems are usually reduced to the boundary where the natural spaces
are of mixed Besov and Bessel type. With our method we can interpret the reduced problem as a mixed
order system and solve this at one stroke. With this approach we can avoid many auxiliary problems
which occurs when the problem is solved ’manually’. The method can be applied to the generalized Lp-Lq
Stokes problem on R n , the generalized thermo-elastic plate equations on R n , the spin-coating process,
the two-phase Navier-Stokes with Boussinesq-Scriven surface and gravity, the two phase Stefan problem
with Gibbs-Thomson correction, etc.
Literatur
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Denk, R. and Saal, J. and Seiler, J. (2008). Inhomogeneous symbols, the Newton polygon, and maximal
L p -regularity. Russian J. Math. Phys, 8, 523-556.
Denk, R. and Volevich, L. R. (2008). Parabolic boundary value problems connected with the Newtons’s
polygon and some problems of crystallization. J. Evol. Equ., 8, 523-556.
Dore, G. and Venni, A. (2005). H ∞ functional calculus for sectorial and bisectorial operators. Studia Math.,
166, 3, 221-241.
Gindikin, S. and Volevich, L. R. (1992). The method of Newton’s polyhedron in the theory of partial differential
equations. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group,
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Prüss, J. and Simonett, G. (2010). On the two-phase Navier-Stokes equations with surface tension. Interfaces
and Free Boundaries. Mathematical Modelling, Analysis and Computation, 12, 3, 311-345.
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