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

Author:
R. A. Adams

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

MSC:
Primary 46E35

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

MathSciNet review:
0322494

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**R. A. Adams and John Fournier,*Some imbedding theorems for Sobolev spaces*, Canad. J. Math.**23**(1971), 517–530. MR**0333705**, https://doi.org/10.4153/CJM-1971-055-3**[2]**Rolf Andersson,*Unbounded Soboleff regions*, Math. Scand.**13**(1963), 75–89. MR**0179600**, https://doi.org/10.7146/math.scand.a-10690**[3]**C. W. Clark,*Introduction to Sobolev spaces*, Seminar Notes, University of British Columbia, Vancouver, 1968.**[4]**Emilio Gagliardo,*Proprietà di alcune classi di funzioni in più variabili*, Ricerche Mat.**7**(1958), 102–137 (Italian). MR**0102740****[5]**I. G. Globenko,*Embedding theorems for a region with null angular points*, Soviet Math. Dokl.**1**(1960), 517–519. MR**0130467****[6]**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****[7]**Günter Hellwig,*Differential operators of mathematical physics. An introduction*, Translated from the German by Birgitta Hellwig, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR**0211292****[8]**V. G. Maz′ja,*Classes of domains and imbedding theorems for function spaces*, Soviet Math. Dokl.**1**(1960), 882–885. MR**0126152****[9]**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****[10]**Norman G. Meyers and James Serrin,*𝐻=𝑊*, Proc. Nat. Acad. Sci. U.S.A.**51**(1964), 1055–1056. MR**0164252****[11]**C. B. Morrey Jr.,*Functions of several variables and absolute continuity, II*, Duke Math. J.**6**(1940), 187–215. MR**0001279****[12]**S. L. Sobolev,*Applications of functional analysis in mathematical physics*, Translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. MR**0165337****[13]**Neil S. Trudinger,*On imbeddings into Orlicz spaces and some applications*, J. Math. Mech.**17**(1967), 473–483. MR**0216286****[14]**Antoni Zygmund,*Trigonometrical series*, Chelsea Publishing Co., New York, 1952. 2nd ed. MR**0076084**

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Additional Information

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

Keywords:
Sobolev space,
imbedding theorem,
integral inequality

Article copyright:
© Copyright 1973
American Mathematical Society