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1. CJM 2002 (vol 54 pp. 945)
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications |
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications 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 Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 |
2. CJM 1997 (vol 49 pp. 568)
A counterexample in $L^p$ approximation by harmonic functions For ${n \over {n-2}}\leq p<\infty$ we show that the
conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus
X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not
sufficient to characterize the compact
sets $X$ with the property that each function harmonic on $\dox$
and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence
of functions which are harmonic on neighbourhoods of $X$.
Categories:41A30, 31C15 |