Some integral inequalities with applications to the imbedding of Sobolev spaces defined over irregular domains

Adams, R. A.

doi:10.1090/S0002-9947-1973-0322494-0

Some integral inequalities with applications to the imbedding of Sobolev spaces defined over irregular domains
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by R. A. Adams

Trans. Amer. Math. Soc. 178 (1973), 401-429

DOI: https://doi.org/10.1090/S0002-9947-1973-0322494-0

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Abstract:

This paper examines the possibility of extending the Sobolev Imbedding Theorem to certain classes of domains which fail to have the “cone property” normally required for that theorem. It is shown that no extension is possible for certain types of domains (e.g. those with exponentially sharp cusps or which are unbounded and have finite volume), while extensions are obtained for other types (domains with less sharp cusps). These results are developed via certain integral inequalities which generalize inequalities due to Hardy and to Sobolev, and are of some interest in their own right. The paper is divided into two parts. Part I establishes the integral inequalities; Part II deals with extensions of the imbedding theorem. Further introductory information may be found in the first section of each part.

References

R. A. Adams and John Fournier, Some imbedding theorems for Sobolev spaces, Canadian J. Math. 23 (1971), 517–530. MR 333705, DOI 10.4153/CJM-1971-055-3
Rolf Andersson, Unbounded Soboleff regions, Math. Scand. 13 (1963), 75–89. MR 179600, DOI 10.7146/math.scand.a-10690

Introduction to Sobolev spaces

Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137 (Italian). MR 102740
I. G. Globenko, Embedding theorems for a region with null angular points, Soviet Math. Dokl. 1 (1960), 517–519. MR 0130467
I. G. Globenko, Some questions in the theory of imbedding for domains with singularities on the boundary, Mat. Sb. (N.S.) 57 (1962), 201–224 (Russian). MR 0143022
Günter Hellwig, Differential operators of mathematical physics. An introduction, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. Translated from the German by Birgitta Hellwig. MR 0211292
V. G. Maz′ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882–885. MR 0126152
V. G. Maz′ja, $p$-conductivity and theorems on imbedding certain functional spaces into a $C$-space, Dokl. Akad. Nauk SSSR 140 (1961), 299–302 (Russian). MR 0157224
Norman G. Meyers and James Serrin, $H=W$, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055–1056. MR 164252, DOI 10.1073/pnas.51.6.1055
C. B. Morrey Jr., Functions of several variables and absolute continuity, II, Duke Math. J. 6 (1940), 187–215. MR 1279
S. L. Sobolev, Applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR 0165337, DOI 10.1090/mmono/007
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028
Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co., New York, 1952. 2nd ed. MR 0076084