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

Boivin, André ; Gauthier, Paul M. ; Paramonov, Petr V.

Canadian journal of mathematics, Tome 54 (2002) no. 5, pp. 945-969

Résumé

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.

Détail

DOI : 10.4153/CJM-2002-035-4

Mots-clés : 30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30, approximation on closed sets, elliptic operator, strongly elliptic operator, L-meromorphic and L-analytic functions, localization operator, Banach space of distributions, Dirichlet problem

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