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

  Published:2002-10-01
 Printed: Oct 2002
  • André Boivin
  • Paul M. Gauthier
  • Petr V. Paramonov
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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 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
 

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