Boundary element methods

Transcription

1 Dr. L. Banjai Institut für Mathematik Universität Zürich Contents Boundary element methods L. Banjai Herbstsemester 2007 Version: October 26, Introduction Integration by parts Single and double layer potentials Boundary integral formulation: Indirect approach Boundary integral formulation: Direct approach The boundary element method Elementary Functional Analysis Banach and Hilbert Spaces Embeddings Hilbert Spaces Dual Spaces Dual Space of a Normed, Linear Space Dual Operator Adjoint Operator Compact Operators Fredholm-Riesz-Schauder Theory Bilinear and Sesquilinear Forms Existence Theorems

2 3 Geometric Foundations Function Spaces Smoothness of Domains Normal Vector Boundary Integrals Sobolev Spaces on Domains Second definition of Sobolev spaces Sobolev Spaces on Surfaces Definition of Sobolev Spaces on Some further properties of Sobolev spaces The trace theorem Normal Derivatives and Green Formulas Mapping properties of integral operators The volume (Newton) potential Single layer potential Double layer potential The ellipticity of the single layer potential (for the Laplace operator) Mapping properties for ranges of Sobolev spaces The numerical solution: The boundary element method Piecewise constant boundary elements Piecewise linear boundary elements Interior Laplace model problem Abstract Variational Problem Galerkin Approximation Compact Perturbations Helmholtz equation as an example of a compact perturbation Regularity of the Solutions of the Boundary Integral Equations Aubin-Nitsche Duality Technique Errors in Functionals of the Solution Consistent Perturbations. Strang s Lemma Direct method for interior problems Exterior scattering problems The non-invertibility of layer potentials for special k Combined layer potential formulation Representation formula

3 Literature: S. Sauter, C. Schwab: Randelementmethoden W. McLean: Strongly Elliptic Systems and Boundary Integral Equations G. Chen, J. Zhou: Boundary Element Methods O. Steinbach: Numerische Näherungsverfahren für elliptische Randwertprobleme 3

4 1 Introduction In this lecture course we discuss the application of boundary element methods to the numerical solution of elliptic partial differential equations (PDE). To simplify the presentation we restrict the discussion to two fundamental partial differential operators: the Laplace operator and the Helmholtz operator Lu = u(x) = d j=1 2 u (x) (1.1) x 2 j L k u = u(x) k 2 u(x), (1.2) where k C, Im k 0, is a constant called the wave number. A model problem is given by: Find u which satisfies u ku = f in (1.3) and some boundary condition, e.g., u = g, on =. (1.4) The domain is either a bounded domain + := R \. R d, d = 2, 3, or its open complement 1.1 Integration by parts For the rest of the introduction, we assume that is a bounded domain in R d. The boundary = is assumed to be smooth enough so that the integration by parts formula (in multidimensional space) holds u x j v dx = n j uv ds u v x j dx, for u, v C 1 (), (1.5) where n j is the jth component of the outer normal vector of. Easy, but very useful, consequences of the integration by parts formula are The Gauss divergence theorem: For F = (f 1, f 2,...,f d ) C 1 () d, let divf := d f j j=1 x j denote the divergence operator. Then divf dx = F, n ds, for F C 1 () d. (1.6) The first Green formula: From the identity u = div ( u) we obtain u ( u)v dx = u, v dx n v ds, for u C2 () and v C 1 (). (1.7) The second Green formula: Using the first Green formula twice we obtain u ( u)v dx + n v ds = u( v) dx + u v n ds, for u, v C2 (). (1.8) 4

5 1.2 Single and double layer potentials A crucial concept for the theory of boundary integral equations is the fundamental solution of a PDE. For the Laplace operator the fundamental solution is given by { 1 log x, for d = 2, 2π G(x) := 1 1, for d = 3, (1.9) 4π x and for the Helmholtz operator { 1 2π G k (x) := H 0(k x ), for d = 2, 1 e ik x, for d = 3, 4π x (1.10) where H 0 ( ) is the Hankel function of first kind and zero order (in Matlab the function is given by besselh). Note that in 3D, G 0 G. The fundamental solutions satisfy LG = 0 and L k G k = 0, in R d \ {0}. (1.11) More exactly LG = L k G k = δ, where δ is the Dirac delta distribution 1. We can now define the single layer potential operator (Su)(x) := G(x y)u(y) ds y, x R d \, (1.12) and the double layer potential operator G(x y) (Du)(x) := u(y) ds y n y x R d \, (1.13) and investigate their properties. Analogously we define the corresponding Helmholtz potentials S k and D k. To simplify the discussion for the rest of the section we assume d = 3. Theorem 1.1. Let f L (). Then (S k f) is continuous and uniformly bounded throughout R 3. Proof. Main idea of the proof: For a given y, write (S k f)(x) = G k (x ζ)f(ζ) ds ζ + \ ε G k (x ζ)f(ζ) dζ, ε x R 3 \, where ε := {z ; z y ε}. For x y the first integral is continuous and the second can be shown to exist and converge to zero for ε 0. To prove uniform boundedness, we just need to consider the decay at infinity. Note that the continuity of the single layer potential can be strengthened to Hölder continuity. In view of the above result we define the single layer potential, mapping to functions on the boundary, as (V k f)(x) := G k (x ζ)f(ζ) ds ζ, x. (1.14) 1 The Dirac delta distribution is a generalized function whose defining property is R d δ(x)f(x)dx = f(0) for any sufficiently smooth f. 5

6 Lemma 1.2. For the twice continuously differentiable surface there exists a constant L > 0 such that n y, x y L x y 2, and n y n x L x y, (1.15) for all x, y. For a smooth surface, we can generate locally parallel surfaces h, by the representation x = z + hn z, z, (1.16) where h denotes the distance from x to z. For sufficiently small h, (z, h) establishes a local coordinate system near for z and h sufficiently small. Theorem 1.3. Let f C 0 (). Then (D k f)(x) can be continuously extended from + to + and from to with limiting values G k (x ζ) lim y x (D k f)(y) = f(ζ) ds ζ ± 1 f(x), x. (1.17) y ± n ζ 2 Proof. We prove the result for the case k = 0. Extension to k 0 follows from the extra smoothness of Since G k (x ζ) n ζ G 0(x ζ) n ζ = n ζ, x ζ 4π x ζ 3 (eik x ζ ik x ζ e ik x ζ 1). (1.18) by Lemma 1.2 we have that G(x ζ) n ζ G(x ζ) n ζ = n ζ, x ζ 4π x ζ 3 (1.19) L, x, ζ, x ζ. (1.20) 4π x ζ Hence the integral in (1.17) exists as an improper integral. We first prove (1.17) for f 1. From the divergence theorem, we have { 0, y c, D(1)(y) = 1, y. (1.21) For x and r > 0, let H x,r := {ζ : ζ x = r}, x,r := {ζ : ζ x r}. (1.22) Using the divergence theorem, and the properties of the fundamental solution we have G(x ζ) G(x ζ) G(x ζ) ds ζ = ds ζ ds ζ n ζ \ x,r n ζ H x,r n ζ G(x ζ) G(x ζ) + ds ζ + ds ζ H x,r n ζ x,r n ζ G(x ζ) G(x ζ) = ds ζ +. (1.23) H x,r n ζ x,r n ζ 6

7 The contribution from the integral on r tends to zero as r 0. On H x,r we have therefore We get n ζ, x ζ = G(x ζ) ds ζ = lim n ζ r 0 1 ζ x, x ζ = x ζ = r, (1.24) ζ x G(x ζ) n ζ = 1 4πr 2 on H x,r. (1.25) G(x ζ) H x,r n ζ ds ζ = lim 1 ds r 0 4πr 2 ζ = 1 H x,r 2. (1.26) With this, we have proved (1.17) for the case f 1. Let us consider the general case f C 0 (). Let h 0 > 0 be sufficiently small so that for each y h0 \ there exists a unique representation We write then G(y ζ) f(ζ)ds ζ = f(x) n ζ = f(x) y = x + h(y)n x, h 0 < h(y) < h 0, x. (1.27) G(y ζ) G(y ζ) ds ζ + [f(ζ) f(x)]ds ζ n ζ n ζ G(y ζ) ds ζ + u(y). (1.28) n ζ The next step in the proof is to show that u(y) can be extended to the whole neighbourhood h0 such that lim u(y) = lim u(x + h(y)n x) = u(x), y h0. (1.29) y x h(y) 0 This is done using the mean value theorem and Lemma 1.2. Once this result is settled (1.17) follows, since G(y ζ) lim y x f(ζ) ds ζ = f(x) (± 12 ) y ± n + G(x ζ) ds ζ + u(x) ζ n ζ = ± 1 2 f(x) + G(x ζ) f(ζ)ds ζ. n ζ So now we complete the proof, by proving (1.29). Using Lemma 1.2, we have y ζ 2 = x y y x, x ζ + x ζ 2 = x y 2 + 2h(y) n x, x ζ + x ζ 2 x y 2 2Ch 0 x ζ 2 + x ζ 2 (1.30) 1 ( x y 2 + x ζ 2), if h 0 sufficiently small. (1.31) 2 Using (1.30), we have 1 n ζ, x ζ 4π y ζ 3 23/2 n ζ, x ζ 4π ( x y 2 + x ζ 2 ) 3/2 x ζ 2 C x ζ = C 3 x ζ, ζ h 0. 7

8 Also by (1.30) we get n ζ, y x y ζ 3 C y x ( x y 2 + x ζ 2 ) 3/2. (1.32) Using the last two inequalities we then obtain G(y ζ) n ζ = 1 ( ) ( ) nζ, x ζ n ζ, y x 1 C 4π y ζ 3 y ζ 3 x ζ + y x. ( x y 2 + x ζ 2 ) 3/2 Therefore by projecting S x,r := {ζ : ζ x < r} into the tangent plane at x, we have ( G(y ζ) 2π r S x,r n ζ ds 1 2π ) ρ ζ C ρdρdθ + y x 0 0 ρ 0 0 (ρ 2 + x y 2 ) 3/2dρdθ = C(r + 1). (1.33) From the mean value theorem and (1.30), we have G(y ζ) G(x ζ) dn ζ dn ζ = 1 4π ( 3) n ζ, x ζ ( x ζ) + n ζ x ζ 5 x ζ 3 y x, (1.34) for some x = αy + (1 α)x, α (0, 1). This for y such that 2 y x x ζ, x y = (x ζ) + α(y x) x ζ α y x 1 x ζ, 2 and similarly x ζ 3 x ζ. 2 Using the above in (1.34), we have Therefore \S x,r G(y ζ) dn ζ G(y ζ) dn ζ G(x ζ) dn ζ G(x ζ) dn ζ C y x x ζ 3. ds ζ C x y r 3. (1.35) Combining (1.32) and (1.35), we obtain u(y) u(x) = G(y ζ) G(x ζ) (f(ζ) f(x)) (f(ζ) f(x))ds ζ dn ζ dn ζ 2 G(y ζ) S x,r dn ζ ds ζ sup f(ζ) f(x) ζ S x,r +2 G(y ζ) G(x ζ) \ dn ζ dn ζ ds ζ sup f(x) x ( ) C 0 sup f(ζ) f(x) + x y. ζ x r r 3 Given ε > 0, we can choose r > 0 such that f(ζ) f(x) ε C 0, ζ, x, ζ x < r, 8

9 because f is uniformly continuous on. Take δ < εr 3 /C 0. Then for all x, y, y x < δ, we have u(x) u(y) ε and the proof is complete. For the integral in (1.17) we will use the following notation G k (x ζ) (K k f)(x) := f(ζ)ds ζ, x, (1.36) n ζ and we also call this operator the double layer potential. 1.3 Boundary integral formulation: Indirect approach Let us return to the model problem u = 0 in u = g on. (1.37) Let us attempt to represent the solution u as a single layer potential u(x) = (Sϕ)(x) = G(x y)ϕ(y) ds y, x. (1.38) From the properties of the fundamental solution we know that for any sufficiently smooth ϕ, u = 0 in. Therefore it only remains to find the unknown density ϕ such that the boundary condition is satisfied: Find ϕ such that (V ϕ)(x) = G(x y)ϕ(y) ds y = g(x), for all x. (1.39) In this way we have arrived at a boundary integral formulation of the PDE (1.37). Once ϕ is computed, we can use (1.38) to compute u in. Alternatively we could have represented the solution as a double layer potential. In view of Theorem 1.3, in this case we would obtain the following boundary integral formulation of (1.37): Find ψ such that 1 2 ψ(x) + G(x ζ) ψ(ζ)ds ζ = g(x), for all x. (1.40) n ζ 1.4 Boundary integral formulation: Direct approach Apply the second Green formula with u the solution of (1.37) and v(y) = G(x y). The we obtain the representation formula u(x) = G(x y) u y G(x y) (y) ds y u(y) ds y, x. (1.41) n y n y (Note that Green formula cannot be directly applied since v / C 2 (), more work needs to be done to justify the above equation). Using the jump properties of the double layer potential we obtain now the direct boundary integral formulation of the problem: Find ϕ such that g(x) = G(x y)ϕ(y) ds y g(x) 9 y G(x y) n y g(y) ds y, x. (1.42)

10 Which can be rewritten as: Find ϕ such that (V ϕ)(x) = 1 g(x) + (Kg)(x), for all x, (1.43) 2 which is known as Symm s integral equation. The unknown density in the direct approach is a physical quantity ϕ = u n y. This is often a useful property and an advantage over the indirect method. 1.5 The boundary element method In the FEM, the PDE is written in a variational form and then the solution is approximated by a Galerkin method. We will use the same procedure, but applied to the boundary integral formulation. This will constitute the boundary element methods (BEM). For example, we write the SLP formulation of the model problem in a variational form as follows: Find ϕ H 1 such that a(ϕ, φ) := (V ϕ, φ) L 2 () = (g, φ) L 2 (), for all φ H 2. (1.44) We will later see that a convenient choice for the spaces is H 1 = H 2 = H 1/2 (). We will also be concerned with the existence and uniqueness of the solution of (1.44). To obtain a numerical solution, we chose a finite dimensional (boundary element space) S H 1/2 () with basis (b j ) N j=1. As in the FEM we find an approximation to ϕ ϕ(x) N α j b j (x) (1.45) j=1 by solving the linear system N α j a(b j, b k ) = (g, b k ) L 2 (), k = 1, 2,..., N. (1.46) j=1 We will investigate the uniqueness of the solution of the discrete systems and the dependence of the error on the various parameters. 2 Elementary Functional Analysis In this chapter we recall a few fundamental results from the area of functional analysis that we will need at a later stage. 2.1 Banach and Hilbert Spaces A complete (i.e., Cauchy sequences converge in the space), normed, linear space is called a Banach space. For two normed spaces let L(X, Y ) denote the space of all bounded linear operators T : X Y. It constitutes a normed, linear space (L(X, Y ), Y X ), with T Y X = sup{ Tx Y / x X : 0 x X} <. 10

11 We call the set A X dense in X if we have for the closure A = X. More specifically, this means that for all x X there exists a sequence (x n ) n A with x n x. If (X, X ) is normed but not complete, then the Banach space ( X, X) is the completion of X if X is dense in X, X is complete and we have x X = x X for all x X. Proposition 2.1. Let X 0 be a dense subset of (X, X ). An operator T 0 L(X 0, Y ) with T 0 Y X0 = sup{ T 0 x Y / x X : 0 x X 0 } < has a unique extension T L(X, Y ) that satisfies the following conditions: 1. For all x X 0 we have Tx = T 0 x. 2. For all sequences (x n ) n X 0 with x n x X we have Tx = lim n T 0 x n. 3. T Y X = T 0 Y X Embeddings Let X, Y be Banach spaces with X Y. The injection (or embedding) I : X Y is defined by Ix = x for all x X and is clearly linear. If I is bounded: x X : x Y C x X, (2.1) we have I L (X, Y ). If X is also dense in Y, we call X densely and continuously embedded in Y Hilbert Spaces Let X be a vector space. A mapping (, ) : X X K is called an inner product on X if (x, x) > 0 x X\ {0}, (λx + y, z) = λ(x, z) + (y, z) λ K, x, y, z X, (x, y) = (y, x) x, y X. (2.2a) (2.2b) (2.2c) A Banach space (X, X ) is called a Hilbert space if there exists an inner product on X, such that x X = (x, x) 1/2 for all x X. 2.2 Dual Spaces Dual Space of a Normed, Linear Space Let X be a normed, linear space over K {R, C}. The dual space X of X is the space of all bounded, linear mappings (functionals) X is a Banach space with norm For x (x) one also writes X = L(X, K). x X := x K X = sup { x (x) / x X : x X\ {0}}. (2.3) x, x X X = x, x X X = x (x), (2.4) where, X X,, X X are called the dual forms or duality pairings. 11

12 Lemma 2.2. Let X Y be continuously embedded. Then Y X is continuously embedded. Proof. For y Y, X Y gives us that y is defined on X. We therefore have Y X. Since I : X Y continuous, i.e. x Y C x X, we have that y Y = sup { y (x) / x Y } C 1 x Y \{0} sup { y (x) / x X } = C 1 y X x X\{0} and therefore that y X C y Y. This proves that the embedding Y X is continuous. The bidual space X of X is defined as X = L(X, K). In general we have the strict inclusion X X. However, in many cases X is isomorphic to X, i.e., every x X can be identified with an x X. We write X = X. In this case we call X reflexive. In particular, all Hilbert spaces are reflexive Dual Operator One of the most general principles in functional analysis is the extension of continuous linear operators which are defined on some subspace of a Banach space to the whole Banach space. We will need here the version of the Hahn-Banach extension theorem in Banach spaces. Theorem 2.3. Let X be a Banach space, M a subspace of X and f 0 a continuous linear functional defined on M. Then there exists a continuous linear functional f defined on X such that i) f is an extension of f 0 and ii) f 0 C M = f C X. Corollary 2.4. Let X be a Banach space and x 0 X\ {0}. Then, there exists a continuous linear functional f 0 on X such that f 0 (x 0 ) = x 0 X and f 0 X = 1. Proof. Let M := span {x 0 } and define f 0 : M R by f 0 (αx 0 ) := α x 0 X α C. Then, f is linear on M and f 0 (αx 0 ) = α x 0 X = αx 0 X, i.e., f 0 C M = 1. Theorem 2.3 implies that there is a continuous linear functional f defined on X such that f (x) = f 0 (x) x M and f C X = f 0 C M = 1. Proposition 2.5. Let X, Y be Banach spaces and let T L(X, Y ). For y Y, x (x) = x, x X X := Tx, y Y Y for all x X (2.5) defines a unique x X. The mapping y x is linear and defines the dual operator T : Y X as given by T y := x. Furthermore, we have T L(Y, X ) and T X Y = T Y X. (2.6) 12

13 Proof. The relation given in (2.5) can be written as y (Tx) = x (x) or x = y T. It follows from y L(Y, K), T L(X, Y ) that x L(X, K). From the defining relation Tx, y Y Y = x, T y X X we obtain T T y y, x X = sup X X Tx, y = sup Y Y x X\{0} x X x X\{0} x X y Tx Y sup Y = y x X\{0} x Y T Y X. X Hence, T X Y T Y X. The reverse inequality is proved next. Corollary 2.4 implies that for any x 0 X there exists a functional f 0 Y such that f 0 Y = 1 and f 0 (Tx 0 ) = Tx 0, f 0 Y Y = Tx 0 Y. Thus, f 0 := T f 0 X satisfies and so x 0, f 0 X X = Tx 0 Y Tx 0 Y = x 0, T f 0 X X T X Y x 0 X f 0 Y = T X Y x 0 X. We conclude that T Y X T X Y holds and (2.6) follows. Corollary 2.6. For two operators S L(X, Y ), T L(Y, Z) we have i) (TS) = S T, ii) S is surjective = S L(Y, X ) is injective Adjoint Operator Let X be a Hilbert space over K {R, C}. For all y X, f y ( ) := (, y) X : X K is continuous and linear: We have f y ( ) X and f y X = y X. The converse is a result of Riesz theorem. Theorem 2.7 (Riesz Representation Theorem). Let X be a Hilbert space. For all f X there exists a unique y f X such that f X = y f X and f(x) = (x, y f ) X x X. Definition 2.8. Let X, Y be Hilbert spaces and T L(X, Y ). The adjoint operator of T is given by its defining property Definition 2.9. T Y X = T X Y and (Tx, y) Y = (x, T y) X x X, y Y. (2.7) a. T L(X) is self adjoint if T = T. b. T L(X) is a projection if T 2 = T. 13

14 c. T L(X) is an orthogonal projection if T 2 = T and (im P) = ker P. Proposition Let X 0 X be a closed subspace of the Hilbert space X. For x X there exists a unique x 0 (x) X 0 with x x 0 X = min{ x y X : y X 0 }. (2.8) The mapping x x 0 =: Px is an orthogonal projection. Proof. Existence and uniqueness: The decomposition x = x 0 + x, x 0 X 0, x X 0 is unique. We will show that y = x 0 minimizes the right-hand side in (2.8). If we take x (x 0 z) into consideration, we have for every z X 0 x z 2 X = x x 0 + x 0 z 2 X = x x 0 2 X 2 Re(x x 0, x 0 z) X + x 0 z 2 X = x x 0 2 X + x 0 z 2 X x x 0 2 X. (2.9) This means that x 0 minimizes as required. The inequality in (2.9) only becomes an equality for x 0 = z, which gives us the uniqueness. Projection property: For x X 0 the first part of the proof implies Px = x and therefore P 2 = P. Orthogonality: Let P L (X) be the adjoint operator of P. For x, y X with x 0 := Px and y 0 = Py we have P = P, since (x, P y) X = (Px, y) X = (x 0, y) X = (x 0, y 0 + y ) X = (x 0, y 0 ) X = (x 0 + x, y 0 ) X = (x, Py) X. The assertion follows from: For all y X 0 we have (x Px, y) X = (x Px, Py) X = (P x P Px, y) X = ( Px P 2 x, y ) X = (Px Px, y) X = Compact Operators Definition The subset U X of the Banach space X is called precompact if every sequence (x n ) n N U has a convergent subsequence (x ni ) i N. It is compact if, furthermore, x = lim i x ni U. Definition Let X, Y be Banach spaces. T L(X, Y ) is called compact if {Tx : x X, x X 1} is precompact in Y. We will often consider operators that are compositions of several other operators. Lemma Let X, Y, Z be Banach spaces, let T 1 L(X, Y ), T 2 L(Y, Z) and let at least one of the operators T i be compact. Then T = T 2 T 1 L(X, Z) is also compact. Lemma T L(X, Y ) compact = T L(Y, X ) compact. Definition Let Y be a Banach space and X Y a subspace that is continuously embedded. The embedding is compact if the injection I L(X, Y ) is compact. We denote this by X Y. Remark X Y if every sequence (x i ) i N X with x i X 1 has a subsequence that converges in Y. Remark For dim(x) < or dim(y ) <, T L(X, Y ) is compact. 14

15 2.4 Fredholm-Riesz-Schauder Theory Let X be a Banach space and let T L(X) be a compact operator. In the following theorem we will establish the connection between the spectrum and the eigenvalues of T. Theorem σ(t) := {λ C : (T λi) 1 / L(X)} (2.10) i) For all λ C\{0} we have one of the alternatives a) (T λi) 1 L(X) or b) λ is an eigenvalue of T. The following alternatives are equivalent: (a ) The equation Tx λx = y has a unique solution x X for all y X. (b ) There exists a finite-dimensional eigenspace E(λ, T) = kern(t λi) {0}, i.e. 0 x E(λ, T) : Tx = λx. ii) σ(t) consists of all eigenvalues of T and of λ = 0 if T 1 / L(X, X). There are at most countably many eigenvalues {λ j } and the only possible accumulation point is zero. iii) λ σ(t) λ σ(t ). (2.11) iv) We have v) For λ σ(t)\{0} the equation dim(e(λ, T)) = dim(e(λ, T )) <. (2.12) (T λi)x = y has at least one solution if and only if the compatibility condition is satisfied. y, x X X = 0 x E(λ, T ) (2.13) The following corollary is a result of Theorem 2.18 and will play a significant role in later applications. Corollary Let X be a Banach space and let T L(X) be a compact operator. Then we have the following equivalence: I + T is injective I + T is an isomorphism. Corollary Let X, Y be Banach spaces, T L(X, Y ) be a compact linear operator, and A 0 L(X, Y ) be a bounded linear operator with a bounded inverse A 1 0 L(Y, X). Then A 0 + T is injective A 0 + T is an isomorphism. Proof. Let A := A 0 + T and à := A 1 0 A = I + A 1 0 T. According to Lemma 2.13 A 1 0 T is a compact operator and hence we can apply Corollary 2.19 to the operator Ã. The result for A is then easily deduced. 15

16 2.5 Bilinear and Sesquilinear Forms Let H 1, H 2 be Hilbert spaces with norms H1, H2 over K. A mapping a(, ) : H 1 H 2 K is called a sesquilinear form if u 1, u 2 H 1, v 1, v 2 H 2, λ K : a(u 1 + λu 2, v 1 ) = a(u 1, v 1 ) + λa(u 2, v 1 ), a(u 1, v 1 + λv 2 ) = a(u 1, v 1 ) + λa(u 1, v 2 ). (2.14) A sesquilinear form a : H H C is called Hermitian if if a (u, v) = a (v, u) u, v H. (2.15) If K = R we speak of a bilinear form. The bilinear form a : H H R is called symmetric a (u, v) = a (v, u) u, v H. A sesquilinear form a (, ) : H 1 H 2 K is continuous (or bounded) if there exists a constant C < with a(u, v) C u H1 v H2, (2.16) for all u H 1, v H 2. The smallest C in (2.16) is the norm of a(, ) and we write a := sup sup u H 1 \{0} v H 2 \{0} We can identify sesquilinear forms with linear operators. Lemma Let H 1, H 2 be Hilbert spaces over K. a(u, v) u H1 v H2 (2.17) a) Every sesquilinear form a(, ): H 1 H 2 C can be identified with a A L(H 1, H 2 ) such that a(u, v) = Au, v H 2 H 2 u H 1, v H 2. (2.18) and A H 2 H 1 = a. (2.19) b) Let S 1, S 2 be dense in H 1, H 2 and let the sesquilinear form a(, ) be defined on S 1 S 2. We assume that (2.16) holds for all u 1 S 1, v 1 S 2. Then a(, ) can be uniquely and continuously extended on H 1 H 2 and (2.16) holds on H 1 H 2 with the same constant C = a. Proof. a) For u H 1, ϕ u (v) := a(u, v) defines a linear functional ϕ u ( ) H 2 with ϕ u H 2 C u H1. We set Au := ϕ u for all u H 1. We then have Au H 2 C u H1 and as a consequence A H 2 H 1 a. Conversely, let A L(H 1, H 2). Then a(u, v) := Au, v H 2 H 2 is a sesquilinear form with Au, v H 2 H 2 Au H 2 v H2 A H 2 H 1 u H1 v H2. b) According to Proposition 2.1, for the above argument we only need to consider the definition of A on dense subspaces S 1 H 1, S 2 H 2 to extend the form a(u, v) to H 1 H 2. Au, v H 2 H 2 then denotes the extension. The operator A from Lemma 2.21 is called the associated operator of a (, ). 16

17 2.6 Existence Theorems Differential and integral equations can often be formulated as variational problems. In this section we will define abstract variational problems and prove the existence and uniqueness of solutions under suitable conditions. For this let H 1, H 2 be Hilbert spaces, a(, ): H 1 H 2 K a continuous sesquilinear form and l : H 2 K a continuous, linear functional. We consider the abstract problem: Find u H 1 with a(u, v) = l(v) v H 2. (2.20) The form a(, ) satisfies the inf-sup condition if inf sup u H 1 \{0} v H 2 \{0} a (u, v) u H1 v H2 γ > 0, (2.21a) v H 2 \ {0} : sup a (u, v) > 0. (2.21b) u H 1 \{0} If the sesquilinear form is Hermitian, i.e., a(u, v) = a(v, u), then (2.21a) implies (2.21b). Theorem The following statements are equivalent: (a) For every l (H 2 ) the abstract problem (2.20) has a unique solution u H 1 and we have u H1 1 γ l H. (2.22) 2 (b) The sesquilinear form a (, ) satisfies the inf-sup condition (2.21). Remark i) Let A L(H 1, H 2 ) be the operator that is associated with the form a(, ) (see Lemma 2.21). Let (2.21) hold. Then A 1 L(H 2, H 1 ) exists and A 1 H1 H 2 γ 1. (2.23) ii) Conversely, if A 1 L(H 2, H 1 ) exists and (2.23) holds, then we have (2.21). The sesquilinear form a(, ) : H H K is called H-elliptic if there exists a constant γ > 0 such that u H : Re (a(u, u)) γ u 2 H. (2.24) Lemma 2.24 (Lax-Milgram). Let H be a Hilbert space. Let the sesquilinear form a : H H C be H-elliptic. Then (2.21) holds and the variational problem (2.20), with H 1 = H 2 = H, has a unique solution u H for all l H with u H 1 γ l H. (2.25) 17

18 3 Geometric Foundations 3.1 Function Spaces Boundary integral equations are formulated on the surfaces of domains in R d. In order to define the relevant function spaces on the boundaries one has to characterize the smoothness of the boundaries. To do this we first need to introduce Hölder continuous parametrizations. Let k N 0 and R d be a domain. The space of all k times continuously differentiable functions on is denoted by C k ( ) := {f : C : f is k times continuously differentiable and α f can be extended as a continuous function on for all 0 α k}. Here α N d 0 is a multi-index and we use the following conventions: For µ Nd 0 µ! := d µ i!, µ 1 := µ := i=1 v =(v i ) d i=1 Cd : v µ := d i=1 v µ i d i=1 µ i, µ := max 1 i d µ i, i, µ f (x) := µ xf (x) := µ f (x) In the vector space C k ( ) we can define the following norms ϕ C0 () := sup { ϕ (x) }, x ϕ Ck () := max 0 α k A function ϕ C 0 ( ) is Hölder continuous of order λ (0, 1] in, if ϕ C 0,λ () ϕ (x) ϕ (y) := sup <. x,y x y λ µ 1 x 1 µ 2 x 2... µ d { } α ϕ C0 (). x d. we set (3.1) The set of all Hölder continuous functions is given by C 0,λ ( ). The space C k,λ ( ) contains all functions on with α ϕ C 0,λ ( ) for all α k. On C k,λ ( ) a norm is given by ϕ C k,λ () := ϕ C k () + max α =k α ϕ C 0,λ (). Remark 3.1. For all k N 0 and 0 < λ 1, C k,λ ( ) is a Banach space. The space of all infinitely differentiable functions is given by C ( ) := k N 0 C k ( ). All the functions we have considered so far are scalar, i.e., they map points from a domain to C. The definitions can, however, be generalized for vector-valued functions Φ = (Φ i ) d i=1 : 1 2 on domains 1, 2 R d. We set C k,λ ( 1, 2 ) := { Φ : i d : Φ i C k,λ ( 1 )}. (3.2) If the condition Φ i C k,λ ( 1 ) in (3.2) is replaced by Φi C k ( 1 ) we obtain the space C k ( 1, 2 ). For 1 = 2 we use the notation C k,λ ( 1 ) := C k,λ ( 1, 2 ) and similarly C k ( 1 ) := C k ( 1, 2 ). 18

19 Definition 3.2. Let 1, 2 R d be two domains and k N 0 { }. A mapping Φ : 1 2 is a C k -diffeomorphism if it satisfies the conditions (a)-(c). (a) Φ C k ( 1, 2 ). (b) The inverse mapping Φ 1 : 2 1 exists and satisfies Φ 1 C k ( 2, 1 ). (c) There exists a constant 0 < c < such that the Jacobian DΦ = the inequality ( Φ i x j )1 i,j d satisfies x 1 : 0 < c det (DΦ (x)) 1/c. (3.3) Remark 3.3 follows from the inverse mapping theorem. Remark 3.3. If R d is bounded (a) and (b) imply (c). If k 1 and Φ is surjective (a) and (c) imply (b). Definition 3.4. A function Φ : 1 2 is bi-lipschitz continuous if in Definition 3.2 we have C 0,1 ( i, j ) instead of C k ( i, j ) and instead of (3.3). Φ (x) Φ (y) 0 < c sup 1/c (3.4) x,y 1 x y x y The space of all Lebesgue measurable functions that are bounded almost everywhere on is denoted by L (). 3.2 Smoothness of Domains In order to describe the smoothness of domains one uses local as well as global criteria. Lipschitz domains represent a reasonably general class of domains, for the boundaries of which integral equations can be defined. Lipschitz domains are given by the existence of an atlas which consists of bi-lipschitz continuous charts. In general, we assume that R d is a domain with compact boundary =. For r > 0, we set B r := {ζ R d : ζ < r}, B + r := {ξ B r : ξ n > 0}, B r := {ξ B r : ξ n < 0}, B 0 r := {ξ B r : ξ n = 0}. (3.5) Definition 3.5. A domain R d is a Lipschitz domain ( C 0,1 ) if there exists a finite cover U of open subsets in R d such that the associated bijective mappings { χ U : B 1 U } U U have the following properties 1. χ U C 0,1 ( B 1, U ), χ 1 U C0,1 ( U, B 1 ), 2. χ U (B 0 1 ) = U, 3. χ U ( B + 1 ) = U, 19

20 4. χ U ( B 1 ) = U R d \. Let k N { }. A domain is a C k -domain if Property 1 can be substituted by χ U C k ( B 1, U ), χ 1 U Ck ( U, B 1 ). Remark 3.6. Properties 2-4 in Definition 3.5 express the fact that is locally situated on one side of the boundary. In order to describe the local smoothness of the surface we use surface meshes. For this let q N, Ŝ q := {ξ R q : 0 < ξ 1 < ξ 2 <...ξ q 1 < ξ q < 1} be the unit simplex and Q q := (0, 1) q the unit cube. In the following these domains will be called reference elements and will be abbreviated by τ q. If there is no confusion with regard to the dimension q we will simply write τ. Definition 3.7. Let R d (for d = 2, 3) be a bounded domain with boundary. 1. A subset τ is called a boundary element or panel of smoothness k N 0 { } in short C k -element if there exists a C k -diffeomorphism χ τ : τ τ. 2. A set G is called a paneling (of smoothness k N 0 ) if (a) all τ G panels are of smoothness k, (b) the elements of G are open and disjoint, (c) we have = τ G τ. Definition 3.8. A bounded domain R d, d = 2, 3, is piecewise smooth with the index k N { }, in short C k pw, if 1. there exists a paneling G of smoothness k, 2. is a Lipschitz domain, where the mapping χ U from Definition 3.5 can be chosen in such a way that we have χ U τ = χ τ. Similarly, the boundary = of a bounded C k pw -domain Rd, d = 2, 3 is also called piecewise smooth with the index k N { } and is denoted by C k pw. The definition of C k pw-domains that we have presented here has been chosen in such a way that we will not need to introduce a new notation for the discretization. A paneling allows us to define piecewise smooth functions on surfaces. Definition 3.9. Let k N 0 { } and Cpw k. A function f : C is called k times piecewise differentiable if there exists a paneling G of smoothness k with f χ τ C k (ˆτ ) τ G. The set of all k times piecewise differentiable mappings on is denoted by C k pw (). 20

21 3.3 Normal Vector Let R d with d = 2, 3 be a bounded domain of the type Cpw 1 and let G be the paneling from Definition 3.7 of smoothness k 1. The sphere in R d is denoted by S d 1. For x τ G we define a normal vector n (x) S d 1 by (χ τ (ξ)) d = 2 ñ (x) := ( χ τ (ξ)/ ξ 1 ) ( χ τ (ξ)/ ξ 2 ) d = 3 n (x) := ñ (x)/ ñ(x). ( ) with ξ = χ 1 τ (x) and v v2 = v 1 (3.6) In general, we assume that the orientation of the charts χ τ is chosen in such a way that the normal vector points towards the unbounded space outside of. Remark For domains of type Cpw 1 the set { x : x / } τ τ G has zero surface measure. Therefore, (3.6) defines an outer normal field on almost everywhere. 2. For domains of type C 1 there exists a normal vector for all x. 3.4 Boundary Integrals Let τ be a C 1 -panel with the parametrization χ τ : ˆτ τ and let f : τ C be a measurable function. Then the surface integral of f over τ can be written as f (x)ds x = ˆf (ˆx) g (ˆx)dˆx with ˆf := f χτ. (3.7) τ ˆτ Here g signifies the Gram determinant, which is defined as follows. The Jacobian of the parametrization χ τ is denoted by J τ := Dχ τ = ( χ i ˆx j ) 1 i d. The Gram matrix is given by 1 j d 1 G (ˆx) := J τ (ˆx)J τ (ˆx) R (d 1) (d 1). The surface element g (ˆx) in (3.7) is the square root of the determinant of the Gram matrix g (ˆx) := det G (ˆx). More generally, for piecewise smooth boundaries Cpw 1 and measurable functions f : C we have ˆf τ (ˆx) g τ (ˆx)dˆx f (x) ds x := τ G where ˆf τ := f χ τ and g τ is the surface element of the parametrization χ τ. For a measurable subset γ of a surface we denote the surface measure by γ := γ 1ds x. For measurable subsets ω R d we use the same notation and set ω := ω 1dx. ˆτ 21

22 4 Sobolev Spaces on Domains Results concerning existence and uniqueness can be formulated for elliptic boundary value problems with the help of Sobolev spaces on domains. We will briefly review some of the properties of the function space L 2 (), after which we will introduce Sobolev spaces. We consider an open subset R d. L 2 () denotes all Lebesgue measurable functions f : C that satisfy f 2 dx <. We do not distinguish between two functions u, v if these differ on a set of measure zero. Theorem 4.1. L 2 () is a Hilbert space with inner product (u, v) 0, := (u, v) L 2 () := u (x) v (x)dx and norm u 0, := u L 2 () := (u, u)1/2 0,. If there is no cause for confusion we will simply write (u, v) 0 and u 0 instead of (u, v) 0, and u 0,. It is not possible to define classical derivatives (e.g., pointwise as the limit of difference quotients) for functions from L 2 (). In order to define a generalized derivative we use the fact that every function from L 2 () can be approximated by smooth functions. For a continuous function u C 0 (), supp (u) := {x : u (x) 0} (4.1) denotes the support of the function u. The space of all infinitely differentiable functions on is denoted by C () and we set C 0 () := {u C () : supp (u) }. The space of all functions from C () with compact support is defined as C comp () := C 0 ( R d ) := { u : u C 0 ( R d )}. (4.2) Lemma 4.2. The spaces C () L 2 () and C 0 () are dense in L2 (). Definition 4.3. A function u L 2 () has a weak derivative g := α u L 2 () if the property (v, g) 0, = ( 1) α ( α v, u) 0,, v C 0 () is satisfied. Definition 4.4. Let R d be a bounded domain. For l = 0, 1, 2,... the Sobolev space H l () is given by H l () := { ϕ L 2 () : α ϕ L 2 () for all α l }. (4.3) On the space H l () we define the inner product (ϕ, ψ) l := ( α ϕ, α ψ) 0 = α l α l α ϕ α ψdx (4.4) and the norm ϕ l := (ϕ, ϕ) 1/2 l. (4.5) 22

23 The space H l () is sometimes denoted by W l,2 (). If in (4.4) we only sum over those multi-indices with α = l we can define a seminorm on H l () by ϕ 2 l := α ϕ 2 dx. (4.6) α =l Sobolev spaces can also be defined for non-integer exponents. For l R, l denotes the greatest integer for which l l. For a non-integer l 0, i.e., l = l + λ with λ (0, 1), we define (ϕ, ψ) l := ( α ϕ, α ψ) 0 (4.7) α l + α l ( α ϕ (x) α ϕ (y)) ( α ψ (x) α ψ (y)) dxdy x y d+2λ and ϕ l := (ϕ, ϕ) 1/2 l. (4.8) For a non-integer l the Sobolev space H l () is defined as the closure of with regard to the norm l from (4.8). {u C () : u l < } (4.9) Definition 4.5. H l 0 () is the closure of the space C 0 () with regard to the l norm. Proposition 4.6. Let R d be open and l 0. Then the space H l () C () is dense in H l (). For the dual spaces of H l () and H l 0() we use the following notation: H l () := (H l ()), H l () := (H l 0()). (4.10) 4.1 Second definition of Sobolev spaces For R d let D() = C0 () denote the space of test functions. Functions in the dual space D () we call distributions. For any u L 1 loc () the expression ψ u (ϕ) = u(x)ϕ(x) dx, for ϕ D() (4.11) defines a distribution ψ u D(). A distribution induced by (4.11) from an L 1 loc function, is said to be a regular distribution. In this way we can identify L 1 loc () with the subspace of regular distributions and for the sake of brevity, we often call functions in L 1 loc () distributions. This is a proper subspace since not all distributions are regular. For example, assuming 0, the Dirac delta distribution, defined by δ(ϕ) := ϕ(0), for ϕ D(), is a distribution but not a regular distribution. In analogy to the weak derivative, we can also define a distributional derivative. 23

24 Definition 4.7. Distributional derivative of ψ D () is defined by Exercise 4.8. Let = ( 1, 1) and ( α ψ)(ϕ) = ( 1) α ψ( α ϕ) for ϕ D(). v(x) = sign(x) = { 1, if x 0 1, if x > 0. Show that the distributional derivative of v (i.e., of the distribution ψ v induced by it) is given by x v = 2δ D (). Definition 4.9. The Schwarz space of rapidly decreasing, C functions is defined by S(R d ) = {φ C (R d ) : sup x R d x α β φ(x) <, for all multi-indices α and β}. The dual space S () is called the space of tempered distributions. Note that for ϕ(x) := e x 2, ϕ S(R d ) and ϕ / D(R d ). For functions ϕ S(R d ) we can define the Fourier transform ˆϕ(ξ) := (Fϕ)(ξ) = e 2πi x,ξ ϕ(x) dx S(R d ), for ξ R d. (4.12) R d The mapping F : S(R d ) S(R d ) is invertible with the inverse Fourier transform defined by (F 1 ˆϕ)(x) = e 2πi x,ξ ˆϕ(ξ) dξ S(R d ), for x R d. (4.13) R d From the definition of the Fourier transform it follows easily that for ϕ S(R d ) and α (Fϕ)(ξ) = ( 2πi) α F(x α ϕ)(ξ) (4.14) F( α ϕ)(ξ) = (2πiξ) α (Fϕ)(ξ). (4.15) For tempered distributions ψ S (R d ) we define the Fourier transform by ˆψ(ϕ) = (Fφ)(ϕ) := ψ(fϕ), for ϕ S(). (4.16) With this definition F : S (R d ) S (R d ) is invertible with inverse given analogously by (F 1 ψ)(ϕ) := ψ(f 1 ϕ), for ϕ S(). (4.17) For s R and u S(R d ) we define the Bessel potential J s : S(R d ) S(R d ) by J s u(x) := (1 + ξ 2 ) s/2 û(ξ)e 2πi x,ξ dξ, x R d. (4.18) R d In this way, F{J s u(x)} = (1 + ξ 2 ) s/2 û(ξ), (4.19) 24

25 so under Fourier transformation the action of J s is to multiply û(ξ) by a function that is O( ξ s ) for large ξ. In view of (4.15), we can therefore think of J s as a kind of differential operator of order s. Notice that for all s, t R, J s+t = J s J t, (J s ) 1 = J s, J 0 = identity operator. The action of the Bessel potential on temperate distributions ψ S (R d ) is defined by (J s ψ)(ϕ) := ψ(j s ϕ), for all ϕ S(R d ). (4.20) Definition The Hilbert space H s (R d ) is the space of all tempered distributions v S (R d ) with J s v L 2 (R d ) and the inner product The induced norm is hence given by (u, v) H s (R d ) := (J s u, J s v) L 2 (R d ). (4.21) u 2 H s (R d ) := J s u 2 L 2 (R d ) = R d (1 + ξ 2 ) s û(ξ) 2 dξ. (4.22) The following theorem tells us that we have in fact just given another definition of Sobolev spaces H s (R d ). Proof can be found in the book of McLean. Theorem For all s R with equivalent norms. H s (R d ) = H s (R d ), We now give also an alternative definition of Sobolev spaces on domains. Definition For a domain R d the Hilbert space H s () is defined by H s () := {u D () : u = U for some U H s (R d )}. The norm is given by u H s () = min U H s U =u,u H s (R d (R ). d ) Theorem For Lipschitz domains R d it holds H s () = H s () for all s R. In view of these result we will stick to using the notation H s (R d ) and H s () and use whichever definition is more convenient. 5 Sobolev Spaces on Surfaces In order to define boundary integral equations one needs Sobolev spaces on boundaries := of domains. These are defined with the help of Sobolev spaces on Euclidean (parameter-) domains by means of lifting. 25

26 5.1 Definition of Sobolev Spaces on Let R d be a bounded Lipschitz domain. We can define coordinates on the boundary := with the help of the partition U of with properties as in Definition 3.5. We introduce the following restrictions χ U,0 : B1 0 U 0 := U, χ U,0 := χ U B 0. 1 Now we can define a coordinate system a = ( ) U 0, χ 1 U,0 as well as a subordinate partition U U of unity {β U : R} U U by 1 = ( ) β U on, supp (β U ) U 0, β U χ U,0 C 0,1 0 B1 0. U U Functions ϕ : C can be localized with the help of this partition. The function ϕ U = ϕβ U : C satisfies supp (ϕ U ) U 0. If is a C k -domain with k 1 we can carry out the localization in an analogous way, in which case the functions χ U,0 are C k -diffeomorphisms. The smoothness of a function on the surface is characterized by the smoothness of the reverse transformed, localized function. The reverse transform on parameter domains is given by ϕ U := ϕ U χ U,0 : B 0 1 C, U U Therefore, it is obvious that the maximum smoothness of the domain constitutes an upper bound for the order of differentiability of the Sobolev spaces on. More specifically, for C 0,1 or C k -domains, only Sobolev spaces H l () with a maximum order of differentiability l, invariant under the choice of the coordinate system, that satisfy l 1 for Lipschitz domains, l k for C k -domains (5.1) can be defined. We use the previously introduced notation for the following definition. Definition 5.1. Let R d be a bounded C 0,1 or C k -domain with k 1. We assume that (5.1) is satisfied for s R 0. The space H s () contains all functions ϕ : C that satisfy ϕ U H0 s (B1) 0 for all U U. The Sobolev-norm is defined by { } 1/2 ϕ H s () := ϕ U 2 H s (B1 0). U U Formally, the Sobolev space H l () depends on the chosen coordinates. Should it be necessary, we will write H l a () instead of H l (). It can, however, be shown that H l () is defined invariantly on under the condition that there is a suitable relation between the order of differentiation l and the smoothness of the boundary. Proposition 5.2. Let be a bounded Lipschitz domain or a C k -domain with k 1. We assume that the index of differentiation l satisfies (5.1). Let a 1, a 2 be two coordinate systems on. Then the spaces H l a 1 () and H l a 2 () are equivalent: They correspond with regard to their sets and the norms are equivalent. 26

27 In the same way as in (4.7), we can also define a norm on H s () for ϕ H s () by ϕ 2 s, := ϕ α 2 L 2 () + ϕ α (x) ϕ α (y) 2 ds x y d 1+2λ x ds y, (5.2) α s where the functions ϕ α : C are given by α s ϕ α (x) := U U α ξ ( ϕ U ) (ξ) with x = χ U,0 (ξ) (5.3) and ξ α denotes the differentiation with regard to the variable ξ. If the order s is an integer the integral term in (5.2) has to be omitted. Then, for s N 0 we have, ϕ 2 s, = ϕ α 2 L 2 (). α s This definition of the norm is equivalent to the definition given in Definition 5.1, i.e., there exists c > 0 such that 0 < c ϕ s, ϕ H s () 1 c ϕ s,. We have introduced Sobolev spaces with non-negative differentiation indices for domains and their boundaries. The dual spaces of these Sobolev spaces contain all the continuous linear functionals defined thereon. Let X either be a domain or a surface. Then the following notation is used for the dual space H s (X) := (H s 0 (X)), s 0. (5.4) Note that in the case of closed surfaces (X = ) the boundary of X is the empty set and, therefore, H s 0 (X) = Hs (X). 5.2 Some further properties of Sobolev spaces Theorem 5.3. Let 0 s < t < and let be a bounded Lipschitz domain. Then (i) The embedding I : H t () H s () is compact. (ii) If s, t 1 or if is a C k domain and s, t k, then the embedding I : H t () H s () is compact. We end this section with some remarks on dual spaces. Since L 2 () is a Hilbert space we can identify it with its dual using Riesz s representation theorem and the meaning of the L 2 -inner product can then be extended to the dual pairing: (f, g) L 2 () = f, g L 2 () (L 2 ()). Let f H s (R d ) for s 0. Since (f, g) L 2 (R ) = d ˆf ĝ = (1 + ξ R R 2 ) s/2 ˆf(1 + ξ 2 ) s/2 ĝ f H s (R ) g d H s (R ), d d d we have that each f H s (R d ) by means of the L 2 -inner product defines an element of (H s (R d )). In fact H s (R d ) = (H s (R d )) and the L 2 -inner product can be extended to the dual pairing (, ) L 2 (R d ) =, H s (R d ) H s (R d ). In a similar way the L 2 -inner product on a closed boundary can be extended to a dual pairing between H s () and H s (). 27

28 5.3 The trace theorem To study boundary value problems we have to make sense of the restriction u as an element of a Sobolev space on when u belongs to a Sobolev space on. The main idea is contained in the following lemma. Lemma 5.4. Define the trace operator γ 0 : D(R d ) D(R d 1 ) by γ 0 u(x ) = u(x, 0), for x R d 1. If s > 1/2, then γ 0 has a unique extension to a bounded linear operator γ 0 : H s (R d ) H s 1/2 (R d 1 ). Proof. We use the following notation: x = (x 1, x 2,..., x d ), ξ = (ξ 1, ξ 2,...,ξ d ) R d and x = (x 1, x 2,...,x d 1 ), ξ = (ξ 1, ξ 2,...,ξ d 1 ) R d 1. For u D(R d ) the Fourier inversion formula gives ( + ) (γ 0 u)(x ) = u(x, 0) = û(ξ)e 2πi x,ξ dξ = û(ξ, ξ d )dξ d e 2πi x,ξ dξ, R d and so γ 0 u(x ) = + û(ξ, ξ d )dξ d = + R d 1 Applying the Cauchy-Schwarz inequality, we obtain the bound γ 0 u(x ) 2 M s (ξ ) (1 + ξ 2 ) s/2 (1 + ξ 2 ) s/2 û(ξ, ξ d )dξ d. where using the substitution ξ d = (1 + ξ 2 ) 1/2 t, M s (ξ ) = (1 + ξ 2 ) s û(ξ) 2 dξ d, dξ d (1 + ξ 2 + ξ d 2 ) = 1 s (1 + ξ 2 ) s 1/2 dt (1 + t 2 ) s. The integral with respect to t converges because s > 1/2, so if we write M s = M s (0) then Integrating over ξ R d 1 gives (1 + ξ 2 ) s 1/2 γ 0 u(x ) 2 M s (1 + ξ 2 ) s û(ξ) 2 dξ d. γ 0 u 2 H s 1/2 (R d 1 ) M s u 2 H s (R d ), and since D(R d ) is dense in H s (R d ), we obtain a unique continuous extension for γ 0. Theorem 5.5. Define the trace operator γ 0 by γ 0 u = u, for u C 0 (). If is a C k 1,1 domain, and if 1/2 < s k, then γ 0 has a unique extension to a bounded linear operator γ 0 : H s () H s 1/2 (). 28

29 Proof. The result is a consequence of Lemma 5.4 and a procedure called local flattening of the boundary. As in Definition 3.5, let U = {U} be a covering of the domain by open sets and {χ U : B 1 U} U U the associated diffeomorphisms. Define a corresponding partition of unity 1 = θ U on, θ U C0 (Rd ). (5.5) U U Let v H s () and v U = θ U v. Then v U χ U H s 0 (B 1) and can hence be extended by zero to the whole space v U H s (R d ). We can now apply the previous lemma to the functions v U and combine the result with the definition of the norm on the boundary and (5.5). Remark 5.6. An important consequence of the definition of the trace operator is that only the values of u in the vicinity of the boundary influence the values of its trace γ 0 u. Namely if v C 0 (Rd ) and v 1 in a neighbourhood of, then γ 0 u = γ 0 (vu). Furthermore we can see that γ 0 can also be defined as a continuous map from H s loc () Hs 1/2 (). There are a number of applications in which the inverse of this question plays a significant role. Can functions from H s 1/2 (), that are given on surfaces, be extended to H s ()? Theorem 5.7. Let be a C k 1,1 domain with compact boundary. Then for 1/2 < s k there exists a linear, continuous extension operator Z : H s 1/2 () H s () with (γ 0 Z )(ϕ) = ϕ on H s 1/2 (). An important corollary of Theorem 5.5 and Theorem 5.7 is that an equivalent definition of Sobolev spaces on the boundary can be given by the use of the trace operator. In particular, the space H 1/2 () is just the space of all traces of functions in H 1 (). Corollary 5.8. Let R d be a C k 1,1 domain with compact boundary and 1/2 < s k. Then, H s 1/2 () = {γ 0 û : û H s ()}, (5.6) and defines an equivalent norm on H s 1/2 (). u := inf{ û H s () : û H s () with γ 0 û = y} (5.7) 5.4 Normal Derivatives and Green Formulas Theorem 5.9 (Rademacher). Let be a bounded Lipschitz domain with boundary. Then there exists an outer normal vector almost everywhere on which satisfies n (L ()) d. Notation Let be a bounded Lipschitz domain with boundary and + := R d \. We assume that every one of these domains is connected and, furthermore, that the orientation of the normal field n : S d 1 is chosen in the direction of +. In the following denotes one of the domains, +, and the algebraic sign function σ is given by { 1 for = σ :=, 1 for = +. Therefore, σ n is the outer normal relative to. 29

30 An immediate consequence of the trace theorem is that the integration by parts formula (1.5) also holds for Sobolev functions u, v H 1 ( ). Corollary For all u, v H 1 ( ), there holds u v u dx + v dx = x j x j γ 0 uγ 0 vn j ds. (5.8) Proof. The above formula holds for u, v, C 1 ( ). All three terms define continuous bilinear forms on H 1 ( ) H 1 (). Therefore (5.8) follows for arbitrary u, v H 1 ( ) from the density of C 1 ( ) in H 1 ( ). We state now also Gauss s divergence theorem and the Green formulae in desired general form. Theorem 5.12 (Gauss s Theorem). Let {, + }. For all F H ( 1, R d) we have (divf) dx = σ n,f ds x. A direct result of Gauss theorem is the first of Green formulas. Theorem We have for all u H 2 () and v H 1 () Green first formula div (gradu)v dx = gradu, gradv dx σ n, grad u v ds x. (5.9) For v H 2 () one obtains Green second formula ( div (gradu)v dx u div (grad v)dx = σ n, gradu vds x Definition For u H 2 (), the conormal derivative is defined as n, gradv uds x ). (5.10) γ 1 u := n, γ 0 gradu. (5.11) Remark The boundary differential operator γ 1 is a continuous mapping from H 2 () to H 1/2 (). We can further enlarge the domain of definition of the γ 1 operator. 30