Abstract
An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ1,..., λn) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pi, where pi are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A0 determined by theID-parameters. A0 turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pi in terms of certain specific properties of A0. Let A0 be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ *G is determined by Λ(G) and A0, and may be found by a linear optimization procedure. The following cases are possible: a)\(\lambda _G^* = \lambda _{E^n }^* \), b)\(\lambda _G^* \ne \lambda _{E^n }^* \), c) λ *G does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pi exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to En. Bibliography: 20 titles.
Similar content being viewed by others
Literature Cited
V. P. Il'in, “On existence and optimal choice of parameter values in inequalities assuring the validity of embedding theorems,”Zap. Nauchn. Semin. LOMI,111, 63–87 (1982); English translation in J. Soviet Math.
V. P. Il'in, “To the question on embedding theorems on domains of the spaceE n,”Zap. Nauchn. Semin. LOMI,124, 164–196 (1983); English translation in J. Soviet Math.
V. P. Il'in, “To the embedding theory of function spaces with power smoothness on domains,”Tr. MIAN SSSR,187, 78–97 (1989); English translation in Proc. Steklov Inst.
V. P. Il'in, “To the embedding theory of anisotropic classes of functions with power smoothness and metric ofL p type,”Tr. MIAN SSSR,192, 68–84 (1990); English translation in Proc. Steklov Inst.
O. V. Besov, V. P. Il'in, and S. M. Nikolskii,Integral Representations of Functions and Embedding Theorems, English translation: Scripta Ser. in Math.,1 (1978),2 (1979); John Wiley & Sons, Halsted Press, New York, etc.1 MR80f:46030a;2 MR80f:46030b.
A. Berman and R. J. Plemmons,Nonnegative Matrices in the Mathematical Sciences, New York, etc. Academic Press, (1979).
S. N. Kruzhkov and I. M. Kolodyi, “To the embedding theory of Sobolev spaces,”Usp. Mat. Nauk,38, No. 2, 207–208 (1983); English translation in Russian Math. Surveys.
O. V. Besov, “Integral representations of functions on domains satisfying the flexible horn condition and embedding theorems,”DAN SSSR,273, No. 6, 1294–1297 (1983); English translation in Math. USSR — Doklady.
I. G. Globenko, “Some topics of the embedding theory for domains having singularities on the boundary,”Mat. Sb.,57(99), No. 2, 201–224 (1962); English translation in Math. USSR — Sbornik.
S. Campanato, “Il teorema di immersione di Sobolev per una classe di sparti non dotati della proprieta di cono,”Ric. di Mat.,11, No. 1, 103–122 (1962).
V. G. Maz'ya, “The classes of domains and the embedding theorems of function spaces,”DAN SSSR,133, No. 3, 527–530 (1960); English translation in Math. USSR — Doklady.
V. G. Maz'ya, “‘p-conductivity’ and embedding theorems of some function spaces into the spaceC,”DAN SSSR,140, No. 2, 299–302 (1961); English translation in Math. USSR — Sbornik.
V. G. Maz'ya,Sobolev Spaces, Springer Verlag, Berlin etc. (1985) MR87g:46056.
A. Ostrowski, “Über die Determinanten mit überwiegender Hauptdiagonale,”Comment. Math. Helv.,10, 69–96 (1937).
S. M. Nikol'skii,Approximation of functions of several variables and embedding theorems, Springer Verlag, Berlin etc. (1974).
S. M. Nikol'skii, “Embedding theorems for functions with partially derivatives, considered in various metrics,”Izv. AN SSSR,22, No. 3, 321–336 (1958); English translation in Math. USSR — Izvestiya.
S. N. Kruzhkov and A. G. Korolev, “To embedding theory of anisotropic function spaces,”DAN SSSR,285, No. 5, 1054–1057 (1985); English translation in Math. USSR — Doklady.
O. V. Besov, “Embeddings of an anisotropic Sobolev space for the domain under flexible horn condition,” in:Functional Theory and Related Problems of Analysis [in Russian], Proceeding of All-Union Conf. Dnepropetrovsk,180, 102–106 (1985).
O. V. Besov, “Embeddings of an anisotropic Sobolev space for the domain under flexible horn condition,”Tr. MIAN SSSR,181, 3–14 (1988); English translation in Proc. Steklov Inst.
Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94.
Translated by A. A. Mekler.
Rights and permissions
About this article
Cite this article
Il'in, V.P. On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems. J Math Sci 78, 142–180 (1996). https://doi.org/10.1007/BF02366032
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02366032