i=1 α i. Given an m-times continuously

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1 1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable function u C m (Ω), we denote by D α u(x), α m, the partial derivatives of u in x = (x 1,, x d ) T according to D α u(x) = α u(x) x α xα d d (1.1.1) Definition: Assume that F : Ω R N R, N N, is a nonlinear mapping and u C m (Ω). Then (1.1.2) F (x, (D α u(x)) α m ) = 0, x Ω is called a scalar nonlinear partial differential equation of m-th order in u. A nonlinear partial differential equation (1.1.2) is said to be quasilinear, if it is linear with respect to the partial derivatives D α u, α = m of highest order, i.e., if there exist functions a α : Ω R N 1 R and F 1 : Ω R N 2 R, N i N, 1 i 2, such that the function F in (1.1.2) admits the representation (1.1.3) F (x, (D α u(x)) α m ) = α =m a α (x, (D β u(x)) β m 1 )D α u(x) + F 1 (x, (D β u(x)) β m 1 ). A quasilinear partial differential equation (1.1.3) is called semilinear, if the coefficient functions a α, α = m, only depend on the independent variables x i, 1 i d, but do not depend neither on the function u nor on its partial derivatives. Finally, we call (1.1.2) a linear partial differential equation, if the function F is linear in u as well as in its partial derivatives D α u, α m.. In particular, given functions a α : R d R, α m, and f : R d R, a linear partial differential equation of m-th order has the representation (1.1.4) Lu := L(, D)u = f in R d, where L(, D) denotes the linear partial differential operator of m-th order given by.

2 2 1 Fundamentals (1.1.5) L(x, D)u := The differential operator α m (1.1.6) L H (, D)u := a α (x) D α, x R d. α =m is said to be the principal part of L(, D). a α ( ) D α The classification of a linear partial differential equation of m-th order is done by means of its symbol. (1.1.7) Definition: The symbol of the differential operator (1.1.2) is given by (1.1.8) L(x, iξ) := α m a α (x)(iξ) α, ξ α := where i stands for the imaginary unit. We refer to (1.1.9) L H (x, iξ) := a α (x)(iξ) α as the principal part of the symbol. α =m d ν=1 ξ αν ν, In the following, we restrict ourselves to m = 2. Given continuous functions a ij : R d R, 1 i, j d, and b i : R d R as well as c : R d R, a scalar, linear partial differential equation of second order can be written in the form d d (1.1.10) Lu = a ij u xi x j + b i u xi + c u = f, i,j=1 where u xi := u/ x i and u xi x j := 2 u/ x i x j. The principal partl H (x, iξ) of the associated symbol L(x, iξ) is given by i=1 (1.1.11) L H (x, iξ) = d a ij (x) ξ i ξ j. i,j=1 We introduce A(x) R d d as the matrix (1.1.12) A(x) := ( a ij (x)) d i,j=1. Then, in view of L H (x, iξ) = ξ T A(x)ξ the principal part L H (x, iξ) is a quadratic form in ξ. The classification of linear partial differential equations of second order is done according to the properties of the principal part of its symbol as a quadratic form in ξ.

3 1.1 Classification and characteristics 3 (1.1.13) Definition: A linear partial differential equation of second order (1.1.4) is said to be elliptic in x R d, if all eigenvalues of the matrix A(x) in (1.1.12) have the same sign. It is called parabolic in x R d, if A(x) is singular. It is called hyperbolic in x R d, if all eigenvalues of A(x) except one have the same sign and one eigenvalue has the opposite sign. The eigenvalues are counted according to their multipicities. The linear partial differential equation (1.1.4) is said to be elliptic (parabolic, hyperbolic) in a domain Ω R d, if its is elliptic (parabolic, hyperbolic) for all x Ω. Using a principal axis transformation, the principal part of a linear partial differential equation of second order (1.1.4) of the same type can be transformed to a canonical form. The canonical form of a linear second order elliptic differential equation is Poisson s equation (1.1.14) u = f, where denotes the Laplace operator (1.1.15) := d 2 x 2 i=1 i. In case of a vanishing right-hand side, i.e., f 0, we refer to (1.1.14) as Laplace s equation. The symbol of is given by d i=1 ξ2 i. Hence, the matrix A = A(x) has the eigenvalue +1 of multiplicity d. The canonical form of a linear second order parabolic differential equation is the heat equation (1.1.16) u x1 d u xi x i = f. i=2 The symbol is given by iξ 1 + d i=2 ξ2 i. In this case, the matrix A = A(x) has the simple eigenvalue 0 and the eigenvalue +1 of multiplicity d 1. The name heat equation is motivated by the fact that it describes the temporal and spatial temperature distribution of a heat conducting material. The canonical form of a linear second order hyperbolic differential equation is the wave equation (1.1.17) u x1x 1 d u xix i = f. i=2 The associated symbol is given by ξ1 2 + d i=2 ξ2 i. The matrix A = A(x) has the simple eigenvalue 1 and the eigenvalue +1 of multiplicity d 1. The name wave equation is due to the fact that it describes the temporal and spatial distribution of acoustic waves.

4 4 1 Fundamentals In general, partial differential equations may change their type with their domain of definition. A classical example is Tricomi s equation (1.1.18) x 2 u x1x 1 + u x2x 2 = 0. Its symbol is given by x 2 ξ 2 1 ξ 2 2. Hence, Tricomi s equation is elliptic for x 2 > 0, parabolic for x 2 = 0 and hyperbolic for x 2 < 0. The classification (1.1.7) of linear second order partial differential equations goes along with the notion of characteristics. (1.1.19) Definition: Assume that the function φ is continuously differentiable in R d. Then, the d 1-dimensional manifold given by C := {x R d φ(x) = 0} is called characteristic in x R d, if for the matrix A(x) given by (1.1.12) there holds (1.1.20) φ(x ) = 0 und ( φ(x )) T A(x ) φ(x ) = 0. The manifold C is said to be a characteristics, or a characteristic surface, if it is characteristic in each of its points. An immediate consequence of the classification (1.1.7) and the definition (1.1.19) is that elliptic partial differential equations do not admit real characteristics. For d = 2, the characteristics of the heat equation (1.1.16) are given by the straight lines x 2 = const. parallel to the x 1 -axis, whereas the characteristics of the wave equation (1.1.17) are given by the straight lines x 2 ± x 1 = const.. The generalization of (1.1.1) to systems of partial differential equations can be easily done. (1.1.21) Definition: Let Ω R d be a domain, let F : Ω R N 1+ +N k R k, k N, N i N, 1 i k, be a nonlinear mapping and u i C m (Ω), 1 i k. Then (1.1.22) F i (x, (D α u 1 (x)) α m,, (D α u k (x)) α m ) = 0, 1 i k is called a system of nonlinear partial differential equations of order m. The adoption of the notion system of quasilinear (semilinear, linear) partial differential equations of order m is obvious. In particular, a system of linear partial differential equations is of the form (1.1.23) k L ij (x, D)u j = f i, 1 i k, j=1 where f i : Ω R, 1 i k, are given functions and L ij (, D), 1 i, j k, are linear partial differential operators according to (1.1.5). We set (1.1.24) L(, D) := (L ij (, D)) k i,j=1.

5 1.1 Classification and characteristics 5 We note that a scalar partial differential equation of order m (1.1.2) can be formally written as a system of d + 1 partial differential equations of first order in the variables u 1 := u, u i+1 := u xi, 1 i d. As an example, we consider Poisson s equation (1.1.14) in case d = 2. We obtain the first order system (1.1.25) (u 1 ) x1 u 2 = 0, (u 1 ) x2 u 3 = 0, (u 2 ) x1 + (u 3 ) x2 = f. However, the example (1.1.25) shows that the definition of the principal part L H (, D) of a system of partial differential equations can not be directly adopted from the scalar case. Indeed, if we define L H (, D) as that part of L(, D) which only contains first order terms, for (1.1.25) we would obtain det(l H (iξ)) = 0 in contradiction to L H (iξ) = (xi ξ 2 2) for Poisson s equation. With regard to a consistent definition of the principal part of systems of partial differential equations and the associated classification as well as the definition of the characteristics we refer to Renardy, Rogers (1993). A classical existence and uniqueness result is the Theorem of Cauchy- Kovalevskaya, which deals with the local existence and uniqueness of a realanalytical solution of the so-called Cauchy problem. We consider the following system of quasilinear partial differential equations of first order (1.1.26) (u i ) xd = k j=1 α 1 a (ij) α (x, u)d α u j, 1 i k, where x := (x 1,, x d 1 ) T and u := (u 1,, u k ) T. (1.1.27) Definition: A function g : R d R is called real-analytical in x R d, if there exists a neighborhood U(x ) such that for all x U(x ) the function g can be expanded into a a Taylor series g(x) = c α (x x ) α, α N d 0 It is called real-analytical in R d, if it is real-analytical in any point x R d. If the coefficient functions a (ij) α, α 1, 1 i, j k, in (1.1.26) are real-analytical in the origin p 0 of R d+k 1, a vector-valued function u = (u 1,, u k ) T with u i : R d R, 1 i k, is called a real-analytical solution of (1.1.26) in p 0, if the functions u i, 1 i k, are real-analytical in p 0 and (1.1.26) holds true in p 0. In case of real-analytical data a (ij) α in R d, a vector-valued function u is called a real-analytical solution of (1.1.26) in R d, if the functions u i, 1 i k, are real-analytical in R d and (1.1.26) is satisfied in any point x R d. The Cauchy problem associated with the system (1.1.26) amounts to the computation of a locally unique solution when initial conditions are prescribed on a (d 1)-dimensional hyperplane

6 6 1 Fundamentals (1.1.28) H := {x R d x d = 0}, which is not a characteristics of (1.1.26). (1.1.29) Definition: Let g i : H R, 1 i k. The problem to compute a vector-valued function u = (u 1,, u k ) T such that (1.1.26) is satisfied and (1.1.30) u i (x) = g i (x), x H, 1 i k, holds true, is called a Cauchy problem for the system of quasilinear partial differential equations of first order (1.1.26). The following Theorem of Cauchy-Kovalevskaya provides sufficient conditions for the local existence and uniqueness of a real-analytical solution of the Cauchy problem. (1.1.31) Theorem: Assume that a (ij) α, α 1, 1 i, j k, in (1.1.26) are real-analytical functions in the origin of R d+k 1, and that g i, 1 i k, in (1.1.28) are real-analytical-functions in the origin of the hyperplane H given by (1.1.28). Then, the Cauchy problem (1.1.26),(1.1.30) has a locally unique solution which is a real-analytical function in the origin of R d. Proof. The proof uses the technique of real-analytical majorants, i.e., real-analytical functions with larger coefficients in its representation as a Taylor series than a given real-analytical function. For details we refer to Renardy, Rogers (1993). 1.2 Sobolev spaces In this section, we give a short overview on Sobolev spaces and their associated dual spaces and trace spaces. For the proofs of the subsequent results we refer to the literature (cf., e.g., Adams (1975)). Let Ω be a bounded domain in the Euclidean space R d. We denote by Ω the closure and by Γ = Ω := Ω \ Ω the boundary of Ω. Moreover, we refer to Ω e := R d \ Ω as the associated exterior domain. We define C m (Ω), m N 0, as the linear space of continuous functions in Ω with continuous partial derivatives D α u, α m, up to order m. C m (Ω) is a Banach space with respect to the norm u C m (Ω) := max sup D α u(x). 0 α m Further, we refer to C m,α (Ω), m N 0, 0 < α < 1, as the linear space of all functions in C m (Ω), whose m-th partial derivatives are Hölder continuous, i.e., for all β N d 0 with β = m there exist constants Γ β > 0, such that for all x, y Ω x Ω

7 D β u(x) D β u(y) Γ β x y α. C m,α (Ω) is a Banach space with respect to the norm u C m,α (Ω) := u C m (Ω) + max sup β =m x,y Ω 1.2 Sobolev spaces 7 D β u(x) D β u(y) x y α. We denote by C m 0 (Ω) and C m,α 0 (Ω) the subspaces of C m,α (Ω) and C m,α (Ω) of all functions with compact support in Ω). Finally, C (Ω) stands for the space of all arbitrarily often continuously differentiable functions and C 0 (Ω) for the subspace of all functions in C (Ω) with compact support in Ω. In the sequel, we mostly restrict ourselves to Lipschitz domains, which are defined as follows: (1.2.1) Definition: A bounded domain Ω R d with boundary Γ is said to be a Lipschitz domain, if there exist constants α > 0, β > 0, and a finite number of local coordinate systems (x r 1, x r 2,..., x r d ), 1 r R, as well as locally Lipschitz continuous mappings such that Γ = a r : {ˆx r = (x r 2,..., x r d) R d 1 x r i α, 2 i d} R, R r=1 {(x r 1, ˆx r ) x r 1 = a r (ˆx r ), ˆx r < α}, {(x r 1, ˆx r ) a r (ˆx r ) < x r 1 < a r (ˆx r ) + β, ˆx r < α} Ω, 1 r R, {(x r 1, ˆx r ) a r (ˆx r ) β < x r 1 < a r (ˆx r ), ˆx r < α} Ω e, 1 r R. The geometrical interpretation of the conditions in Definition is that both Ω and the exterior domain Ω e are located on exactly one side of the boundary Γ. In case of smoother mappings in Definition 1.2.1, the domain is called a C m domain or a C m,α domain: (1.2.2) Definition: A Lipschitz domain Ω R d is called a C m domain (C m,α domain), if the functions a r, 1 r R, in Definition (1.2.1) are C m functions (C m,α functions). We denote by L p (Ω), p [1, ), the linear space of functions whose p- th power is Lebesgue integrable in Ω and by L (Ω) the linear space of essentially bounded functions. L p (Ω), p [1, ], is a Banach space with respect to the norm ( 1/p v p,ω := v(x) dx) p bzw. Ω v,ω := ess sup x Ω v(x).

8 8 1 Fundamentals For p = 2, L 2 (Ω) is a Hilbert space with respect to the inner product (v, w) 0,Ω := vw dx. Sobolev spaces are based on the concept of generalized (distributional) derivatives: (1.2.3) Definition: Let u L 1 (Ω) and α N d 0. The function u is said to be differentiable in the weak (distributional) sense, if there exists a function v L 1 (Ω) such that ud α ϕ dx = ( 1) α vϕ dx, ϕ C0 (Ω). Ω Ω In this case, we set D α wu := v and call D α wu the weak (distributional) derivative of u. The notion weak derivative suggests that it represents a generalization of the classical derivative and that there exist functions which are weakly differentiable but not differentiable in the classical sense. Example. Let d = 1 and Ω := ( 1, +1) and consider the function u(x) := x, x Ω,. In the origin, it is not differentiable in the classical sense. However, it has a weak derivative Dwu 1 which is given by { Dwu 1 1, x < 0 = +1, x > 0. For ϕ C 0 (Ω) partial integration yields +1 1 = u(x)d 1 ϕ(x) dx = Ω +1 u(x)d 1 ϕ(x) dx + u(x)d 1 ϕ(x) dx = +1 Dwu(x)ϕ(x) 1 dx + (uϕ) 0 1 Dwu(x)ϕ(x) 1 dx + (uϕ) 1 0 = +1 = Dwu(x)ϕ(x) 1 dx [u(0)]ϕ(0), 1 where [u(0)] := u(0+) u(0 ) denotes the jump of u in x = 0. Due to the continuity of u, it follows that [u(0)] = 0 and we conclude. 0 0

9 1.2 Sobolev spaces 9 (1.2.4) Definition: Let m N 0 and p [1, ]. The linear space W m,p (Ω) given by (1.2.5) W m,p (Ω) := { u L p (Ω) D α wu L p (Ω), α m } is called a Sobolev space. It is a Banach space with respect to the norm (1.2.6) ( 1/p v m,p,ω := Dwv p,ω) α p, p [1, ), α m v m,,ω := max α m Dα wv,ω. For p = 2, W m,2 (Ω) is a Hilbert space with respect to the inner product (1.2.7) (u, v) m,2,ω := DwuD α wv α dx. α m Ω The associated norm will be denoted by m,2,ω. For simplicity, we will use the notations (, ) m,ω and m,ω, if it is evident from the context that we mean the W m,2 (Ω) inner product and the W m,2 (Ω) norm, respectively. Example. There exist functions in W 1,p (Ω), 1 < p < 2, which do not live in W 1,2 (Ω). Consider Ω := {x R 2 x x 2 2 < 1}, u(x) := ln(r), r := (x x 2 2) 1/2. There holds u W 1,p (Ω) if and only if 1 < p < 2. The proof is left as an exercise. An obvious question is whether C m (Ω) is dense in W m,p (Ω). We consider the linear space C m, (Ω) := { u C m (Ω) u m,p,ω < }, which is a normed linear space with respect to m,p,ω, but not complete, i.e., it is a pre-hilbert space. Its completion with respect to the m,p,ω norm is denoted by H m,p (Ω): (1.2.8) H m,p (Ω) := C m, (Ω) m,p,ω. For p =, we have H m, (Ω) = C m (Ω), and hence, W m, (Ω) H m, (Ω). On the other hand, a well-known result by Meyers and Serrin, states that the spaces W m,p (Ω) and H m,p (Ω) coincide for p [1, ). (1.2.9) Theorem: Let m N 0 and p [1, ). Then, for any domain Ω R d (1.2.10) W m,p (Ω) = H m,p (Ω).

10 10 1 Fundamentals Remark.We point out that in general C m (Ω) is not dense in W m,p (Ω). However, if Ω has the so-called segment property (cf., e.g., Adams (1975)), then C m (Ω) is dense in W m,p (Ω). In particular, Lipschitz domains are domains with the segment property. Theorem says that the space C m (Ω) W m,p (Ω) is dense in W m,p (Ω) for p [1, ). An obvious question is whether functions in W m,p (Ω), m 1, belong to the Banach space L q (Ω) or are even continuous functions. We will see that the latter holds true for sufficiently large m. However, before we give an example which shows that in general we can not expect such a result. Example. For d 2 and Ω := {x R d x < 1 2 } consider the function u(x) := ln( ln( x )). The function u has square integrable first weak derivatives D α u(x) = x α x 2 ln( x ), α = 1, since u admits a square integrable majorant according to D α u(x) d ρ( x ) := 1 x d ln( x ) d, α = 1. On the other hand, obviously u is not essentially bounded. The following Sobolev embedding theorems hold true: (1.2.11) Theorem: Let Ω R d be a Lipschitz domain, m N 0 and p [1, ]. Then, the following mappings represent continuous embeddings: W m,p (Ω) L p (Ω), 1 p = 1 p m d, if m < d p, W m,p (Ω) L q (Ω), q [1, ), falls m = d p, W m,p (Ω) C 0,m d p (Ω), if d p < m < d p + 1, W m,p (Ω) C 0,α (Ω), 0 < α < 1, if m = d p + 1, W m,p (Ω) C 0,1 (Ω), if m > d p + 1. In the subsequent chapters we will occasionally use the compactness of embeddings of Sobolev spaces. The corresponding results are known as Rellich embedding theorems.

11 1.2 Sobolev spaces 11 (1.2.13) Theorem: Let Ω R d be a Lipschitz domain, m N 0 and p [1, ]. Then, the following mappings represent compact embeddings: W m,p (Ω) L q (Ω), 1 q p, 1 p = 1 p m d, if m < d p, W m,p (Ω) L q (Ω), q [1, ), if m = d p, W m,p (Ω) C 0 (Ω), if m > d p. We now consider restrictions and extensions of functions in Sobolev spaces: If u W m,p (R d ), p [1, ],, then for any domain Ω R d the restriction Ru of u to Ω as given by Ru(x) := u(x) f.a.a. x Ω belongs to W m,p (Ω), and R : W m,p (R d ) W m,p (Ω) is a bounded linear operator. On the other hand, in general it is not possible to extend a function u W m,p (Ω) continuously to a function in W m,p (R d ). Example. Let d = 2 and Ω := {x = (x 1, x 2 ) R 2 0 < x 1 < 1, x 2 < x r 1}, r > 1,. We consider the function u(x) := x ε p 1, 0 < ε < r. We note that Ω has a cusp in the origin and hence, does not represent a Lipschitz domain. For ε < r + 1 p we have u W 1,p (Ω), since D α u p dx = C ε,p α =1 Ω 1 0 x ε p+r 1 dx 1. On the other hand, u does not live in L (Ω). If we choose ε such that p > 2 is admissible, we see that a Lipschitz domain is necessary for the Sobolev embedding theorem to hold true. Since the Sobolev embedding theorem also holds true in R d, we deduce that u can not be extended to a function in W 1,2 (R 2 ). In case of a Lipschitz domain, the following Sobolev extension theorem holds true: (1.2.15) Theorem: Let Ω R d be a Lipschitz domain, m N 0, and p [1, ]. Then, there exists a bounded linear extension operator E : W m,p (Ω) W m,p (R d ), i.e., Eu = u for all u W m,p (Ω), and there exists a constant C 0 such that for all u W m,p (Ω) Eu m,p,ω C u m,p,r d.

12 12 1 Fundamentals We recall that the algebraic and topological dual space V of a Banach space V is the linear space of all bounded linear functionals on V. V is a Banch space with respect to the norm l V l(v) := sup. v 0 v V Therefore, we can define the dual spaces of the Sobolev spaces W m,p (Ω), p [1, ]. (1.2.16) Definition: Assume Ω R d, let m be a negative integer and let p [1, ]. Then, the Sobolev space W m,p (Ω) is defined as the dual space of W m,q (Ω), where q is conjugate to p, i.e., 1 q + 1 p = 1. For Ω = R d we define the Sobolev space H s (R d ) with broken index s R + by means of the Fourier transform ˆv of a function v C0 (R d ) ˆv(ξ) := ( 1 2π )d/2 exp( iξ x)v(x) dx. The associated Sobolev norm is given by (1.2.17) v s,r d := (1 + 2 ) s/2 ˆv( ) 0,R d. We define (1.2.18) H s (R d ) := C 0 (Rd ) s,r d. R d If Ω R d is a Lipschitz domain, the Sobolev space H s (Ω) is either implicitly defined according to (1.2.19) v s,ω := inf z s,r d, z=ev H s (R d ) where Ev is the extension of v to H s (R d ), or explicitly by means of (observe s = m + λ, m N 0, 0 λ < 1,) ( 1 (1.2.20) v s,ω := ( diam(ω) )2s v 2 0,Ω + v 2 s,ω where λ,ω stands for the semi-norm ) 1/2, v 2 s,ω := D α v 2 0,Ω + 1 α m α =m Ω Ω D α v(x) D α v(y) 2 x y d+2λ dx dy. For Σ Γ = Ω we define H s (Σ) as the linear space according to

13 (1.2.21) H s (Σ) := {v L 2 (Σ) v s,σ < }, 1.2 Sobolev spaces 13 equipped with the norm ( 1 v s,σ := ( diam(σ) )2s v 2 0,Σ + v 2 s,σ ) 1/2, where v 2 s,σ := D α v 2 0,Σ + 1 α m α =m Σ Σ D α v(x) D α v(y) 2 x y d 1+2λ dσ x dσ y. For Σ Γ and v H s (Σ), s < 1, we denote by ṽ the extension of v by zero, i.e., { v(x), x Σ ṽ(x) := 0, x Γ \ Σ. We define H s 00(Σ) by (1.2.22) H s 00(Σ) := {v L 2 (Σ) ṽ H s (Γ )}, equipped with the norm where and v H s 00 (Σ) := v 2 H00 s (Σ) := v 2 s,σ + v 2 s,σ := Σ Σ ( v 2 0,Σ + v H s 00 (Σ)) 1/2, Σ v 2 (x) dist(x, Σ) dσ v(x) v(y) 2 x y d 1+2s dσ x dσ y. For s < 0 we define H s (G), G {Σ, Ω} as the dual space of H s (G), equipped with the negative norm v s,g := < v, w > G sup, w H s (G),w 0 w s,g where <, > G refers to the dual product. For 1 < s < 0 we define H s (G), G {Σ, Ω}, Σ Γ, as the dual space of H00 s (G). For s = 1 we further define H 1 (G) as the dual space of H0 1 (G). For Σ = Γ and 1 s < 0 we define H s (Γ ) as the dual space of H s (Γ ). For details we refer to Grisvard (1985). For a bounded Lipschitz domain Ω R d with boundary Γ and a function u C(Ω) the restriction of u to the boundary Γ is well defined in the sense

14 1.2 Sobolev spaces 15 of pointwise restriction. However, in view of the Sobolev embedding theorem (1.2.11), for functions u W m,p (Ω) the pointwise restriction is well defined, if m > d/p. For a general m, the trace mapping u γ(u) = u γ, which is well defined for u C(Ω), can be extended to a continuous mapping W m,p (Ω) W m 1/p,p (Γ ). In case of a sufficiently smooth boundary Γ, in this way we can also define normal derivatives of Sobolev space functions. Finally, this approach can be generalized to Sobolev spaces with broken indices. (1.2.23) Theorem: Let Ω R d be a bounded Lipschitz domain with a C m,1 boundary Γ, m N 0,. For p [1, ] and l N 0 let s R such that s m, s 1/p = l + σ, σ (0, 1). Then, the mapping u {γ(u), γ( u ν ),, γ( l u ν l )} which is well defined for u C m,1 (Ω), can be uniquely extended to a continuous surjective mapping (1.2.24) W s,p (Ω) l W s i 1/p,p (Γ ) i=0 with a continuous right inverse that is independent of p. Proof. We refer to Theorem in Grisvard (1985). (1.2.25) Definition: Under the assumptions of Theorem (1.2.23) functions γ( k u), 0 k l, are called the traces of u W s,p (Ω) on Γ. The spaces ν k W s i 1/p,p (Γ ) are called trace spaces. Remark.Theorem (1.2.23) obviously does not hold true in the important special case of polyhedral domains, since the boundary of such domains does not represent a C m,1 boundary. On the other hand, the boundary Γ of bounded polyhedral domains consists of a finite number of polyhedral sub-boundaries which satisfy the smoothness assumption. With regard to the extension of Theorem (1.2.23) to bounded polyhedral domains Ω R 2 with polygonal boundaries (including curved polygonal boundaries) we refer to Chapter in Grisvard (1985).