Abstract view
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications


Published:20021001
Printed: Oct 2002
AndrĂ© 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 (jetdistributions) 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
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, 30E10, 31B35, 35Jxx, 35J67, 41A30 show english descriptions
Cluster sets, prime ends, boundary behavior Approximation in the complex domain Connections with differential equations Elliptic equations and systems [See also 58J10, 58J20] Boundary values of solutions to elliptic equations Approximation by other special function classes
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
