My research interests cover several areas in
partial differential equations and their applications.
My main results are in
harmonic maps, elliptic-parabolic equations, mean curvature
equations in phase change systems and filtration flow.
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| Harmonic Maps
In harmonic maps my contribution is an interpolation lemma
for boundary values which can be used to prove partial regularity results for
minimizers of fairly general variational problems, say p-harmonic obstacle
problems.
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| Elliptic-Parabolic Equations
In elliptic parabolic equations my results
together with H. W. Alt on existence of weak solutions, later extended with A.
Visintin to Stefan type systems, have been used widely in situations where the
semigroup approach does not work.
Main applications are the equations of
filtration flow and the already mentioned phase change systems.
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| Surface Tension in Phase Change Systems
The investigations of phase
change systems with surface tension bring together my earlier interest in the
mean curvature equation and the work on Stefan problems. Mathematically the main
result is a capacity type estimate which allows to estimate oscillations in time
of the phase in systems which lack time derivatives for these functions.
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| Filtration Flow
The investigations of filtration flow have lead on to
work on periodic homogenization with singular scalings. In the work with
Bourgeat and Mikelic a blow up (or periodic modulation) technique was introduced
for such systems. This technique has been refined in a recent paper with Filo to
show the relationship between uniqueness results for the blow up equation in
the unbounded space and homogenization. |