14. Cegrell classes of plurisubharmonic functions
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Polish supervisor:
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S³awomir Ko³odziej
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Cooperating partners:
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Urban Cegrell (Umea University, Sweden)
Ahmed Zeriahi (Universite Paul Sabatier, Toulouse)
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In 1998 Cegrell introduced finite p-energy classes of
plurisubharmonic, possibly unbounded functions, and gave a
characterization of measures for which complex Monge-Ampère
equation has weak solutions within those classes. Further work
lead to a description of the maximal domain of definition of the
Monge-Ampère operator and a natural question arises about the
range of that operator. It is now known that for measures
vanishing on pluripolar sets the Dirichlet problem is solvable.
However if a measure lives on a pluripolar set then only in some
particular cases one can solve the problem. Guedj and Zeriahi
carried over the Cegrell's finite energy methods to the study of
the Monge-Ampère equation on compact Kähler manifolds. It was
successfully applied to some geometrical problems like a proof of
the existence of generalized Kähler-Einstein metrics.
In Cegrell's approach some ideas of classical potential theory,
energy methods, are implemented in the setting of non-linear
pluripotential theory. This non-linearity makes the theory much
more difficult as compared to the classical counterpart. Thus, for
instance, the maximal plurisubharmonic functions which correspond
to harmonic functions in the classical case are not in general
continuous, let alone smooth. Therefore pluripotential theory must
use a lot of other tools like positive currents.
The aims of this project are the following: find necessary and
sufficient conditions on a measure for which the Dirichlet problem
for the Monge-Ampère equation is solvable under classical or
generalized boundary conditions; describe the boundary values of
functions belonging to the maximal domain of definition of the
Monge-Ampère operator; solve the Monge-Ampère equation on
compact Kähler manifolds for measures possibly supported on
pluripolar sets. Here partial results are also interesting
provided the progress is significant.
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