| 概要 |
In this talk, we characterize the space of harmonic vector fields in
$L^r$ on the 3D exterior domain with smooth boundary.
There are two kinds of boundary conditions.
One is such a condition as the vector fields are tangential to the
boundary,
and another is such one as those are perpendicular to the boundary.
In bounded domains both harmonic vector spaces are of finite dimensions
and characterized in terms of
topologically invariant quantities which we call the first and the
second Betti numbers.
These properties are closely related to characterization the null spaces
of solutions
to the elliptic boundary value problems associated with the operators
div and rot.
We shall show that, in spite of lack of compactness, spaces of harmonic
vector fields in $L^r$
on the 3D exterior domain are of finite dimensions and characterized
similarly to those in bounded domains.
It will be also clarified the difference between interior and exterior
domains in accordance with the
integral exponent $1 < r < \infty$.
This is based on the joint work with Profs. M.Hieber, A.Seyferd,
S.Shimizu and T.Yanagisawa.
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