Chapter 2 Sobolev spaces In this chapter, we give a brief overview on basic results of the theory of Sobolev spaces and their associated trace and dual spaces. 2.1 Preliminaries Let ­ be a bounded domain in Euclidean space lRd. We denote by ­ its closure and refer to ¡ = @­ := ­ n ­ as its boundary. Moreover, we d denote by ­e := lR n ­ the associated exterior domain. We consider functions u : ­ ! lR and denote by j®j ® @ u D u : = ®1 ®2 ®d @x1 @x2 :::@xd T d its partial derivatives of order j®j. Here, ® = (®1; :::; ®d) 2 lN0 is a Pd multiindex of modulus j®j = i=1 ®i. m We de¯ne by C (­); m 2 lN0; the linear space of continuous functions on ­ whose partial derivatives D®u; j®j · m; exist and are continuous. Cm(­) is a Banach space with respect to the norm ® kukCm(­) := max sup jD u(x)j : 0·j®j·m x2­ m;® We refer to C (­); m 2 lN0; 0 < ® < 1; as the linear space of func- tions in Cm(­) whose m-th order partial derivatives are HÄoldercon- d tinuous, i.e., for all ¯ 2 lN0 with j¯j = m there exist constants ¡¯ > 0 such that for all x; y 2 ­ ¯ ¯ ® jD u(x) ¡ D u(y)j · ¡¯ jx ¡ yj : We note that Cm;®(­) is a Banach space with respect to the norm jD¯u(x) ¡ D¯u(y)j m;® m kukC (­) := kukC (­) + max sup ® : j¯j=m x;y2­ jx ¡ yj m m;® Moreover, C0 (­) and C0 (­) are the subspaces of functions with compact support in ­). Finally, C1(­) stands for the set of functions with continuous partial 1 1 derivatives of any order and C0 (­) denotes the set of C (­) functions with compact support in ­. In the sequel, we will mainly deal with Lipschitz domains which are de¯ned as follows. 12 Finite Element Methods; Chapter 2 13 De¯nition 2.1 Lipschitz domain A bounded domain ­ ½ lRd with boundary ¡ is said to be a Lipschitz domain, if there exist constants ® > 0; ¯ > 0; and a ¯nite number r r r of local coordinate systems (x1; x2; :::; xd); 1 · r · R; and local Lipschitz continuous mappings r r r d¡1 r ar : fx^ = (x2; :::; xd) 2 lR j jxi j · ® ; 2 · i · dg ! lR such that [R r r r r r ¡ = f(x1; x^ ) j x1 = ar(^x ) ; jx^ j < ®g ; r=1 r r r r r r f(x1; x^ ) j ar(^x ) < x1 < ar(^x ) + ¯ ; jx^ j < ®g ½ ­ ; 1 · r · R; r r r r r r f(x1; x^ ) j ar(^x ) ¡ ¯ < x1 < ar(^x ) ; jx^ j < ®g ½ ­e ; 1 · r · R;: In particular, the geometrical interpretation of the conditions is that both ­ and ­e are locally situated on exactly one side of the boundary ¡. De¯nition 2.2 Cm-domain and Cm;®-domain A Lipschitz domain ­ ½ lRd is a Cm-domain (Cm;®-domain), if the m m;® functions ar; 1 · r · R; in De¯nition 2.1 are C -functions (C - functions). 2.2 Sobolev spaces We refer to Lp(­); p 2 [1; 1); as the linear space of p ¡ th order inte- grable functions on ­ and to L1(­) as the linear space of essentially bounded functions which are Banach spaces with respect to the norms Z ³ ´1=p p kvkp;­ := jv(x)j dx ; ­ kvk1;­ := ess supx2­ jv(x)j : Note that for p = 2, the space L2(­) is a Hilbert space with respect to the inner product Z (v; w)0;­ := vw dx : ­ Sobolev spaces are based on the concept of weak (distributional) deriv- atives: 14 Ronald H.W. Hoppe De¯nition 2.3 Weak derivatives 1 d Let u 2 L (­) and ® 2 lN0. The function u is said to have a weak ® 1 derivative Dwu, if there exists a function v 2 L (­) such that Z Z ® j®j 1 uD ' dx = (¡1) v' dx ; ' 2 C0 (­) : ­ ­ ® We then set Dwu := v. The notion 'weak derivative' suggests that it is a generalization of the classical concept of di®erentiability and that there are functions which are weakly di®erentiable, but not di®erentiable in the classical sense. We give an example. Example 2.1 Example of a weakly di®erentiable function Let d = 1 and ­ := (¡1; +1). The function u(x) := jxj; x 2 ­; is not di®erentiable in the classical sense. However, it admits a weak 1 derivative Dwu given by ½ ¡1 ; x < 0 D1 u = : w +1 ; x > 0 1 Indeed, for ' 2 C0 (­) we obtain by partial integration Z+1 Z0 Z+1 u(x)D1'(x) dx = u(x)D1'(x) dx + u(x)D1'(x) dx = ¡1 ¡1 0 Z0 Z+1 1 0 1 1 = ¡ Dwu(x)'(x) dx + (u')j¡1 ¡ Dwu(x)'(x) dx + (u')j0 = ¡1 0 Z+1 1 = Dwu(x)'(x) dx ¡ [u(0)]'(0) ; ¡1 where [u(0)] := u(0+) ¡ u(0¡) is the jump of u in x = 0. But u is continuous and hence, [u(0)] = 0 which allows to conclude. De¯nition 2.4 The Sobolev spaces W m;p(­); p 2 [1; 1] The linear space W m;p(­) given by m;p p ® p (2.1) W (­) := f u 2 L (­) j Dwu 2 L (­) ; j®j · m g Finite Element Methods; Chapter 2 15 is called a Sobolev space. It is a Banach space with respect to the norm ³ X ´1=p ® p (2.2) kvkm;p;­ := kDwvkp;­ ; p 2 [1; 1) ; j®j·m ® (2.3) kvkm;1;­ := max kDwvk1;­ : j®j·m Note that W m;2(­) is a Hilbert space with respect to the inner product X Z ® ® (2.4) (u; v)m;2;­ := DwuDwv dx : j®j·m ­ The associated norm will be denoted by k¢km;2;­. We will simply write m;2 (¢; ¢)m;­ and k ¢ km;­, if the context clearly indicates that the W (­)- inner product and the W m;2(­)-norm are meant. A natural question is whether Cm(­) is dense in W m;p(­). To this end, we consider the linear space m;¤ m C (­) := f u 2 C (­) j kukm;p;­ < 1 g : It is a normed linear space with respect to j ¢ jm;p;­, but not complete, i.e., a pre-Banach space. We denote its completion with respect to the m;p k ¢ km;p;­-norm by H (­), i.e., k¢k (2.5) Hm;p(­) := Cm;¤(­) m;p;­ : We note that for p = 1 we have Hm;1(­) = Cm(­) and hence, W m;1(­) 6= Hm;1(­). However, a famous result by Meyers/Serrin (cf., e.g., [1]) states that for a Lipschitz domain and p 2 [1; 1) the spaces W m;p(­) and Hm;p(­) are equivalent. Theorem 2.1 Characterization of Sobolev spaces d Let ­ ½ lR be a Lipschitz domain and m 2 lN0. Then, for p 2 [1; 1) there holds (2.6) W m;p(­) = Hm;p(­) : Remark 2.1 Remark on Meyers/Serrin The result (2.6) actually holds true for any open subset ­ ½ lRd. How- ever, Cm(­) is dense in W m;p(­), if ­ satis¯es the so-called segment property (cf., e.g., [1]). In particular, Lipschitz domains have this property. Theorem 2.1 asserts that for Lipschitz domains and p 2 [1; 1) the space Cm(­) \ W m;p(­) is dense in W m;p(­). A natural question to ask is whether or not functions in W m;p(­); m ¸ 1; belong to the Banach 16 Ronald H.W. Hoppe space Lq(­), or even are continuous. As we shall see, the latter only holds true, if m is su±ciently large (Sobolev imbedding theorem). However, let us ¯rst give an example which shows that in general we can not expect such a result: Example 2.2 Example of a weakly di®erentiable, but not es- sentially bounded function d 1 Let d ¸ 2 ; ­ := fx 2 lR j jxj < 2 g and consider the function u(x) := ln(jln(jxj)) : The function u has square-integrable ¯rst-order weak derivatives x® D®u(x) = ; j®j = 1 ; jxj2 ln(jxj) since in view of 1 jD®u(x)jd · ½(jxj) := ; j®j = 1 ; jxjd jln(jxj)jd it possesses a square-integrable majorant. On the other hand, u obviously is not essentially bounded. Theorem 2.2 Sobolev imbedding theorems d Let ­ ½ lR be a Lipschitz domain, m 2 lN0 and p 2 [1; 1]. Then, the following mappings represent continuous imbeddings ¤ 1 1 m d (2.7) W m;p(­) ! Lp (­) ; = ¡ ; if m < ; p¤ p d p d (2.8) W m;p(­) ! Lq(­) ; q 2 [1; 1) ; if m = ; p m;p 0;m¡ d d d (2.9) W (­) ! C p (­) ; if < m < + 1 ; p p d (2.10) W m;p(­) ! C0;®(­) ; 0 < ® < 1 ; if m = + 1 ; p d (2.11) W m;p(­) ! C0;1(­) ; if m > + 1 : p Proof. For a proof, we refer to [1]. ¤ In the following chapters, we will frequently take advantage of com- pact imbeddings of Sobolev spaces. Finite Element Methods; Chapter 2 17 Theorem 2.3 Kondrasov imbedding theorems d Let ­ ½ lR be a Lipschitz domain, m 2 lN0 and p 2 [1; 1]. Then, the following mappings are compact imbeddings 1 1 m d (2.12) W m;p(­) ! Lq(­) ; 1 · q · p¤ ; = ¡ ; if m < ; p¤ p d p d (2.13) W m;p(­) ! Lq(­) ; q 2 [1; 1) ; if m = ; p d (2.14) W m;p(­) ! C0(­) ; if m > : p Proof. The interested reader is referred to [1]. ¤ We ¯nally deal with restrictions and extensions of Sobolev func- tions: If u 2 W m;p(lRd); p 2 [1; 1];, then for any domain ­ ½ lRd the restric- tion Ru of u to ­ Ru(x) := u(x) f.a.a.