| Summary: | 碩士 === 國立中正大學 === 應用數學研究所 === 91 === Sobolev spaces play an important role in partial differentiable equations. In this paper we will discuss some properties and useful inequalities about Sobolev spaces.
First we introduce the concept of weak derivatives and then define the function spaces W^{k,p}(U), whose members have weak derivatives of various orders lying in various L^{p} spaces. Base on the definition of weak derivatives, we have some element properties of Sobolev spaces in §2.3. In order to study the deeper properties of Sobolev spaces, in §2.4 we develop some procedures for approximation a function in a Sobolev space by smooth functions. Next we extend functions in the Sobolev space W^{1,p}(U) to become functions in the Sobolev space W^{1,p}(ℝⁿ). In §2.7 we consider the question: If a function u belongs to W^{1,p}(U), does u automatically belong to certain other spaces? The answer is "YES", and the Sobolev-type inequalities ( Gagliardo-Nirenberg-Sobolev inequality and Morrey's inequality) will help us to answer the question. We also demonstrate that W^{1,p}(U) is in fact compactly embedded in L^{q}(U) for 1≤q<p^{∗}.
In the second part of this paper, we use the compactness assertion in §2.8 to generate Poincaré's inequalities. Then we study difference quotient approximations to weak derivatives and discuss the connections between weak partial derivatives and partial derivatives in the usual calculus
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