Abstract
We consider the problem of the possibility of approximating solenoidal vectors from the Sobolev spaces\(\mathop W\limits^o{_\rho}{^1} (\Omega )\) by finite solenoidal vectors. The answer is positive if ΩcRn, n=2,3, is a strictly Lipschitz domain. We give examples of domains with noncompact boundaries for which such an approximation is not possible. We consider the auxiliary problem\(div \vec u = \varphi ,\vec u \in \mathop W\limits^o{_\rho}{^1} (\Omega )\) if the function τ ε Lp(Ω) is given.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 237–239, 1980.
In conclusion the author expresses his deep gratitude to V. A. Solonnikov for the formulation of the problem and for his guidance in the preparation of this paper.
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Piletskas, K.I. Three-dimensional solenoidal vectors. J Math Sci 21, 821–823 (1983). https://doi.org/10.1007/BF01094444
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DOI: https://doi.org/10.1007/BF01094444