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Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications |
Canad. J. Math. 54(2002), 945-969
Published:2002-10-01
Printed: Oct 2002
Printed: Oct 2002
- André Boivin
- Paul M. Gauthier
- Petr V. Paramonov
Abstract
Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jet-distributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
| Keywords: | approximation on closed sets, elliptic operator, strongly elliptic operator, $L$-meromorphic and $L$-analytic functions, localization operator, Banach space of distributions, Dirichlet problem |
| MSC Classifications: |
30D40 - Cluster sets, prime ends, boundary behavior 30E10 - Approximation in the complex domain 31B35 - Connections with differential equations 35Jxx - Elliptic equations and systems [See also 58J10, 58J20] 35J67 - Boundary values of solutions to elliptic equations 41A30 - Approximation by other special function classes |