һ�Sݗ�����w���̽M��Һ�����̽M������w�����ԡ��u���Ժ����t��(Ӣ��) ���x

Chapter 1 Preliminary ����This chapter will introduce some basic results�� most of which will be used in the following chapters. First we shall recall some basic inequalities whose detailed proofs can be found in the related literature�� see�� e.g.�� Adams [1��2]�� Friedman [37�� 38]�� Gagliardo [40��41]�� Nirenberg [95�� 96]�� Yosida [148]�� etc. ����1.1 Some Basic Inequalities ����1.1.1 The Sobolev Inequalities ����We shall first introduce some basic concepts of Sobolev spaces. ����Definition 1.1.1. Assume is a bounded or an unbounded domain with a smooth boundary r. For 1 n and Q is bounded�� and u G then u G C(Q) and ����(1.1.5) ����While�� then ����(1.1.6) ����where measure the n-dimensional unit ball�� T is the Euler gamma function and. ����Remark 1.1.1. The Sobolev inequality (1.1.4) does not hold for p = n�� p* = +oo. ����(1.1.4) was first proved by Sobolev [138] in 1938. Sobolev [138] stated that the Lp* norm of u can be estimated by�� the Sobolev norm of u. However�� we can bound a higher Lp norm of u by exploiting higher order derivatives of u as shown in the next theorem which generalizes theorem 1.1.1 from m = 1�� p > n to an integer. ����Theorem 1.1.2. Assume QCMn is an open domain. There exists a constant C = C(n��p) > 0 such that ����(1) if�� and u G then and ����(1.1.7) ����(2) if mp > n�� and u S VF0m��p(n)�� then and ����(1.1.8) ����where and diamK is the diameter of K. ����(1.1.8) ����Remark 1.1.2. An important case considered in theorems 1.1.1 and 1.1.2 is Q = Mw. In this situation�� and therefore the results of theorems 1.1.1 and 1.1.2 apply to. ����For p > n�� the results of theorems 1.1.1 and 1.1.2 imply the fact that u is bounded. Indeed�� u is Hoder continuous�� which we shall state as follows. ����Theorem 1.1.3. If ue�� then where. ����Generally�� the embedding theorems are closely related to the smoothness of the domain considered�� which means that when we study the embedding theorems�� we need some smoothness conditions for the domain. These conditions include that the domain Q possesses the cone property�� and it is a uniformly regular open set in Mra�� etc. For example�� when or Lip�� Q has the cone property. Mathematically�� we need to define the special meaning of the word ��embedding�� or ��compact embedding��. ����Definition 1.1.2. Assume A and B are two subsets of some function space. Set A is said to be embedded into B if and only if ����(1)A C B; ����(2) the identity mapping I: A B is continuous��i.e.�� there exists a constant C > 0 such that for any x ^ A�� there holds that. ����If A is embedded into B�� then we simply denote by. ����A is said to be compactly embedded into B if and only if ����(1) A is embedded into B; ����(2) the identity mapping I: is a compact operator. ����If A is compactly embedded into B��then we simply denote by A B. ����Now we draw some consequences from theorem 1.1.1. In fact�� exploiting theorem 1.1.1�� we have the following result which is an embedding theorem. ����Corollary 1.1.1. If then u G Lq(Q) with�� and. Moreover��if p > n��u coincides. in Q with a (uniquely determined) function of C(Q). Finally��there holds that�� ����(1.1.9) ����(1.1.10) ����(1.1.11) ����where C = C(n��p��q) > 0 is a constant. ����We can generalize corollary 1.1.1 to functions from VF0m��p(O) which can be stated as the following embedding theorem. ����Theorem 1.1.4. Let. Then ����(1) ����(1.1.12) ����and there is a constant C�� > 0 depending only on m��p��q and n such that for all ����P�� ����(1.1.13) ����(2) if mp = n�� then we have��for all�� ����(1.1.14) ����and there is a constant C2 > 0 depending only on m�� p�� q and n such that for all�� ����(1.1.15) ����(3) if��each is equal a.e. in D. to a unique function in Ck(Q)�� for all and there is a constant C3 > 0 depending only on m�� p�� q and n such that ����(1.1.16) ����Remark 1.1.3. In case (2) of theorem 1.1.4�� the following exception case holds for ����(1.1.17) ����Now we give the following compact embedding theorem. ����Theorem 1.1.5 (Embedding and Compact Embedding Theorem). Assume that Q is a bounded domain