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UBC Theses and Dissertations

An inequality in generalized sobolev spaces Kanigan, Lawrence Louis


In the study of the spaces (formula omitted) of functions for which the pth powers of all the derivatives up to order ℓ are summable in the domain Ω⊂R, it has been found that there are mutual relations between various spaces. These relations were developed under the name "embedding theorems". The first embedding theorem (for spaces (formula omitted) were proved by Sobolev [3]*. Subsequently these spaces became known as Sobolev Spaces. However, in the study of existence of solutions for well-posed boundary value problems, there arose the necessity to consider spaces of distributions: an example is the space dual to (formula omitted). For a thorough development of distributions see L. Schwarz's texts [4]. Furthermore, the classes of Sobolev spaces had to be widened to fractional values of ℓ, the latter spaces being particularly useful in the study of non-linear problems. This thesis follows the development of generalized Sobolev spaces as in Volevich and Panayakh [1]. In section I we prove the basic theorems in this formulation. In section II, the existence of a function is proved using the formulation of section II. The proof of the proposition in which a modification has been made was given by Agranovich and Vishik [2]. The proposition is essential to the applications of Sobolev spaces to differential operators. The result states that ll u ll µ ≤ constant ll u,Ω ll µ for (formula omitted) for the particular case when the weighting function is (formula omitted) and Ω is a half-line. (For definitions see section I). Section III is devoted to a brief comparison of this formulation of Sobolev spaces to other approaches.

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