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Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
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- Journal:
- Canadian Journal of Mathematics / Volume 54 / Issue 5 / 01 October 2002
- Published online by Cambridge University Press:
- 20 November 2018, pp. 945-969
- Print publication:
- 01 October 2002
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- Article
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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.
