Abstract
In the modern approach to partial differential equations a pivotal role is played by various function spaces which are defined in terms of the existence of derivatives (either in the classical or in a generalized, weaker sense). In this chapter we develop the study of such spaces to the extent required for the investigation of second-order elliptic problems.
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Bibliographical Notes
The contents of Section 1.2 are certainly familiar to most readers. We only mention that the example in Section 1.2.1 is taken from D. Gilbarg and N. S. Trudinger [67], and the proof of Theorem 1.2 from A. Kufner, O. John, and S. Fucik [92].
In Section 1.3 the only nonstandard topics are those of the last subsection. Their presentation is largely based on A. Kufner, O. John, and S. Fučik [92], with some modifications in the proof of Theorem 1.12.
Almost all results of Section 1.4 are the contribution of S. Campanato [27, 28] (see also N. Meyers [110] for what concerns Theorem 1.17); functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [81].
The theory of Sobolev spaces stems from the works of several authors: S. L. Sobolev [137, 138], of course, but also, e.g., B. Levi [100], L. Tonelli [145], C. B. Morrey, Jr. [116], J. Deny and J. L. Lions [45]. The equivalence of Levi’s and Sobolev’s definitions can be obtained as a consequence of Theorem 1.20 (whose proof as adopted here follows J. Necas [127]). For what concerns density results we mention N. Meyers and J. Serrin [111] (Theorem 1.26) and S. Agmon [2] (Theorem 1.27).
Sobolev inequalities (Sections 1.6.1 and 1.6.3) are due to S. L. Sobolev [138], L. Nirenberg [129], and E. Gagliardo [58] for kp < N, to C. B. Morrey, Jr. [117] for kp > N. The proof of Theorem 1.34 (see F. Rellich [132] and V. I. Kondrachov [89]) is taken from H. Brézis [19]; the other results of Section 1.6.2 are based on C. B. Morrey, Jr. [118].
Both Sections 1.7 and 1.8 utilize a more or less standard approach to their arguments (although in the literature the intrinsic definition of H 1/p′, p (Γ) is usually preferred to its more rapid construction via the quotient space technique adopted here: e.g., see A. Kufner, O. John, and S. Fučik [92]). Section 1.8.1 is based on H. H. Schaefer [134]; Theorem 1.55 is due to R. Klee [88]. The core of Section 1.8.2 can be considered to be Theorem 1.56 (G. Stampacchia [143]), whose proof here is taken, for its first part, from D. Gilbarg and N. S. Trudinger [67]. (The proof of the last statement, in our Step 2, is simpler than the one suggested by D. Kinderlehrer and G. Stampacchia [87].)
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© 1987 Springer Science+Business Media New York
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Troianiello, G.M. (1987). Function Spaces. In: Elliptic Differential Equations and Obstacle Problems. University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-3614-1_1
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DOI: https://doi.org/10.1007/978-1-4899-3614-1_1
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