Abstract
The problem is posed on the class-preserving extension of functions of Sobolev class
is a finite domain in the in-dimensional space
) onto the whole of
such that the supports of the extended functions would lie in a specified finite domain
and such that the so-called constant of extension would be minimal. The existence of such an extension is proved under constraints on
of the type of a certain minimal smoothness; an algorithm for the approximate computation of the minimal constant of extension is indicated forp=2. Ifp=2 andS=I while
are concentric balls, then the exact value of the constant of extension has been computed under the usual definition of a norm in
; it is expressed in terms of Bessel functions of an imaginary argument. The exact value mentioned permits the estimation from above and from below of the constant of extension in the case when the domain is diffeomorphic to a sphere.
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 150–185, 1979.
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Mikhlin, S.G. Smallest constant of extension of functions of Sobolev classes. J Math Sci 20, 2011–2036 (1982). https://doi.org/10.1007/BF01680567
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DOI: https://doi.org/10.1007/BF01680567