Stone s theorem and the Laplacian

Transcription

1 LECTRE 5 Stone s theorem and the Laplacian In the previous lecture we have shown the Lumer-Phillips Theorem 4.12, which says among other things that if A is closed, dissipative, D(A) = X and R(λ 0 I A) = X for some λ 0 > 0, then A generates a contraction semigroup. In this lecture we first establish several variants of this result and deduce a characterization of generators of isometric groups. This characterization then implies Stone s famous theorem stating that precisely the skewadjoint operators generate unitary C 0 groups on Hilbert spaces. In the remainder of the lecture we then discuss the Laplacian in two simple settings: in L 2 (R d ) and in L 2 () with Dirichlet boundary conditions, where R d is open and bounded. We show that these realizations of the Laplacian are dissipative and selfadjoint (and thus generate C 0 semigroups). These facts play a crucial role in our main applications to wave and Schrödinger equations. Here the Hilbert space setting arises naturally in view of the physical background. It also allows to establish the desired properties of the Laplacian by elegant functional analytic tools. This approach relies on Sobolev spaces and weak derivatives. Actually, we do not need many deep results about these spaces. The relevant information is recalled below in an intermezzo. We also provide a rather long appendix containing the proofs and an introduction to this large subject. We further use basic facts about the Fourier transform and selfadjoint operators which are collected and proved in two more appendices. Again we list the relevant results in the intermezzo. We first come back to the Lumer-Phillips Theorem 4.12 and discuss the range condition via duality theory. To this aim, for a linear operator A in X with dense domain, we define its adjoint A by A x := y for all x D(A ), where y is taken from D(A ) := { x X y X x D(A) : Ax, x = x, y } (5.1). This means that Ax, x = x, A x for all x D(A) and all x D(A ), and D(A ) consist of all x X for which this operation works. It turns out that this is the correct definition to develop the theory. To get some experience with adjoints, we discuss several basic properties of adjoints in the next remark. Remark 5.1. Let A be a densely defined linear operator in X. (a) Since D(A) is dense, there is at most one vector y = A x as in (5.1), so that A : D(A ) X is a map. It is clear that A is linear. If A B(X), then D(A ) = X and (5.1) coincides with the definition of A usually given in courses on functional analysis. 41

2 (b) The operator A is closed in X (thought A does not need to be closed). In fact, let x n D(A ), x X, and y X such that x n x and A x n y in X as n. For every x D(A) we then compute x, y = lim x, n A x n = lim Ax, n x n = Ax, x. As a result, x D(A ) and A x = y. (c) If T B(X), then the sum A + T with D(A + T ) = D(A) has the adjoint (A+T ) = A +T with D((A+T ) ) = D(A ). To verify this fact, let x D(A) and x X. We obtain (A + T )x, x = Ax, x + x, T x. Hence, x D((A + T ) ) if and only if x D(A ), and then (A + T ) x = A x + T x. (d) If T B(X), then the product T A with domain D(T A) = D(A) has the adjoint A T with domain D(A T ) = { x X T x D(A ) }. This fact can be shown analogously as part (c). From (c) and (d) it follows that (λi A) = λi A for λ C. The next result allows to replace the range condition in the Lumer-Phillips theorem by the injectivity of λi A for some λ > 0. If one knows the adjoint of A, the injectivity should be much easier to check than the range condition. However, in applications it is often very hard to compute A (namely D(A )) directly. Nevertheless, Corollary 5.2 is used in the proof of Stone s theorem. Corollary 5.2. Let A be dissipative and densely defined. If λi A is injective for some λ > 0, then A generates a contraction semigroup. (By Proposition 4.11, λi A is injective for all λ > 0 if A is dissipative.) Proof. Due to Theorem 4.12, it suffices to show that λi A has dense range. Let x X annihilate R(λI A); i.e., (λi A)x, x = 0 = x, 0 for all x D(A). This fact leads to x D(A ) and λx A x = 0. The injectivity assumption now yields x = 0. A corollary to the Hahn-Banach theorem then implies the density of R(λI A). sually the density of the domain of an operator is relatively easy to check. Still it is a nice fact that one gets it for free in the context of the Hille-Yosida theorem if X is reflexive. Proposition 5.3. Let X be a reflexive Banach space and A be a closed operator in X such that (ω, ) ρ(a) and λr(λ, A) M for all λ > ω and some constants M, ω 0. Then D(A) is dense in X. In particular, if A is a dissipative operator on a reflexive Banach space such that λ 0 I A is surjective for some λ 0 > 0, then A generates a contraction semigroup. Proof. Let x X and n N with n > ω. Then the vectors x n = nr(n, A)x belong to D(A) and are uniformly bounded in X. Since X is reflexive, the Banach-Alaoglu theorem gives a subsequence (x nj ) j and a vector z X such that x nj converges weakly to z as j. Since D(A) is closed and convex, we obtain z D(A), see e.g. Theorem 3.7 in [Bre11]. We now show x = z to deduce D(A) = X. We first note that the vectors y n = R(1, A)x n = 42

3 nr(n, A)R(1, A)x also belong to D(A) and converge to R(1, A)x =: y in X as n due to Lemma 3.7 (because of y D(A)). We thus obtain the weak limits y nj y and Ay nj = y nj x nj y z as j. As above, we infer that (y, y z) is contained in gr(a) = gr(a) so that Ay = y z. Consequently, x = y Ay = z belongs to D(A). If A is dissipative and R(λ 0 I A) = X for some λ 0 > 0, then A is closed, (0, ) ρ(a) and λr(λ, A) 1 for all λ > 0 due to Proposition The first part thus yields that D(A) = X and the second assertion now follows from the Hille-Yosida Theorem 3.8. We next use the Lumer-Phillips theorem to characterize the generators of isometric C 0 groups. Corollary 5.4. For a linear operator A, the following assertions are equivalent. (a) The operator A generates an isometric C 0 group T ( ), i.e., T (t)x = x for all x X and t R. (b) The operator A is closed and densely defined, A and A are dissipative and λi A and λi + A are surjective for some λ > 0. (c) The operator A is closed and densely defined, R \ {0} ρ(a) and R(λ, A) 1 λ for all λ R \ {0}. Proof. The Lumer-Phillips Theorem 4.12 says that (b) holds if and only if A and A generate contraction semigroups. Theorem 4.2 then implies the equivalence of assertion (b) and (c), and it shows that (b) holds if and only if A generates a C 0 group of contractions T (t), t R. It remains to prove that a contractive C 0 group T ( ) is already isometric. Indeed, we have T (t)x x = T ( t)t (t)x T ( t) T (t)x T (t)x for all x X and all t R, so that T (t) is isometric. In the Hilbert space setting, the above result leads to Stone s theorem on unitary groups. To this aim, we have to recall a few more concepts. Definition 5.5. Let X be a Hilbert space with scalar product ( ). A linear operator A on X is called symmetric if x, y D(A) : (Ax y) = (x Ay). Let A be a densely defined linear operator on X. Then its Hilbert space adjoint A is given by A y := z for all y D(A), where z is taken from D(A ) := { y X z X x D(A) : (Ax y) = (x z) }, cf. (5.1). A densely defined linear operator A is called selfadjoint if A = A and skewadjoint if A = A. Finally, an operator T B(X) is called unitary if it is invertible with T 1 = T B(X). 43

4 We remark that besides A there also exists the adjoint A in X. These two operators are related via Riesz representation theorem, cf. (C.1) in Appendix C. Observe that a densely defined linear operator A is symmetric if and only if A A. Moreover, A is skewadjoint if and only if ia is selfadjoint. We point out that selfadjointness and skewadjointness means in particular that D(A) = D(A ). We further remark that symmetry is relatively easy to check in many examples, whereas selfadjointness or skewadjointness (mainly the equality D(A) = D(A )) often is very hard to verify. On the other hand, these properties have far reaching consequences as in Stone s theorem below. We collect several properties of the Hilbert space adjoints and of selfadjoint operators in the next remark, see Appendix C for more details. Property (f) is used below to prove the selfadjointness of the Dirichlet Laplacian. Remark 5.6. Let A be a densely defined operator in a Hilbert space X. Then the following assertions hold. (a) As in Remark 5.1 one sees that A is closed in X. In particular, a symmetric, densely defined, linear operator is closable with A A (since A A ). (b) Parts (c) and (d) of Remark 5.1 hold for A in a similar way. But recall that (ai) = ai for a C. (c) Let A be closed. Then σ(a ) = { λ λ σ(a) } by (C.2) in Appendix C. (d) If A is symmetric, then also A is symmetric. In fact, for x, y D(A) there are x n, y n D(A) with x n x, y n y, Ax n Ax and Ay n Ay in X as n. It follows (Ax y) = lim n (Ax n y n ) = lim n (x n Ay n ) = (x Ay). (e) There are symmetric closed operators which are not selfadjoint, see Example C.10 in Appendix C. (f) Let A be closed and symmetric. Then A is selfadjoint if and only if σ(a) R. Moreover, if ρ(a) R, then σ(a) R. See Theorem C.9. We can now derive Stone s theorem from the Lumer-Phillips theorem. Alternatively Stone s theorem also follows from the spectral theorem, see Section VII.4 in [RS72]. We will use Stone s theorem to solve the linear Schrödinger equation in a later lecture. In fact, this theorem is a cornerstone of the mathematical foundations of quantum mechanics. Theorem 5.7 (Stone, 1930). Let X be a Hilbert space and A be a linear operator on X with dense domain. Then A generates a C 0 group of unitary operators if and only if A is skewadjoint. Proof. 1) Let A = A. For x D(A) = D(A ), we have J(x) = {ϕ x } where ϕ x := ( x) (see Example 4.6). We compute Ax, ϕ x = (Ax x) = (x Ax) = (Ax x) = Ax, ϕ x, so that Re Ax, ϕ x = 0. Hence, A and A = A are dissipative as well as A = ( A) = A. Corollary 5.2 thus shows that A and A generate contraction semigroups. Therefore, A generates a C 0 group (T (t)) t R of invertible isometric 44

5 operators due to Corollary 5.4. A result from functional analysis then implies that each T (t) is unitary (see Proposition C.7 in Appendix C). 2) Let A generate a unitary C 0 group (T (t)) t R. Since T (t) = T (t) 1 = T ( t) for t 0, the family (T (t) ) t 0 is a contraction semigroup with the generator A by Theorem 4.2. For x, y D(A) we thus obtain (Ax y) = lim( 1 t 0 t (T (t)x x) y) = lim (x 1 t 0 t (T (t) y y)) = (x Ay), i.e., A A. We further know from Theorem 4.2 that σ(a) ir. Since σ(a ) = { λ λ σ(a) } ir by Remark 5.6 (c), Lemma 3.6 yields A = A. In our main applications the Laplacian in an L 2 setting plays a crucial role. To investigate this operator, we need weak derivatives, Sobolev spaces and the Fourier transform. These topics are discussed in the next intermezzo. The proofs and much more background material can be found in the corresponding appendices D and E. Intermezzo 3: Weak derivatives, Sobolev spaces and the Fourier transform The classical derivative does not fit well to L p spaces since it is based on a pointwise limit. Instead, one uses on L p spaces the so called weak derivative. In its definition one requires that one can integrate by parts against functions ϕ Cc, which is well adapted to integrable functions. Let R d be open, k N and p [1, ]. A function u L p () has a weak derivative v in L p () with respect to the jth coordinate for some j {1,..., d} if there is a function v L p () such that u j ϕ dx = vϕ dx (5.2) holds for all ϕ Cc (). The function v is uniquely determined (up to a null function) due to Lemma D.5 in Appendix D, and we set j u := v. The set Wp 1 () of all functions u in L p () possessing weak derivatives in L p () with respect to all coordinates is called a Sobolev space. The linear space Wp 1 () becomes a Banach space when endowed with the norm ( u u 1,p := p p + d j=1 j u p p)1/p, if p <, max j=1,...,d { u, j u }, if p =, see Proposition D.3. Here, as usual in the L p context, we identify functions which are equal almost everywhere. For each p [1, ], this norm is equivalent to the norm given by d u p + j u p. j=1 Weak derivatives of higher order and the Sobolev spaces Wp k () are defined analogously. We set Wp 0 () = L p () and H k () = W2 k (). Observe that (5.2) leads to u, j ϕ L p = u j ϕ dx = ( j u)ϕ dx = j u, ϕ L p 45

6 for all u Wp 1 () and ϕ Cc (). This definition via duality already allows to deduce various properties of weak derivatives (e.g. their linearity). Other properties follow by approximation. (The corresponding density results are proved via cut-off functions and mollifiers, see Appendix D.) In the next remark we summarize the properties of weak derivatives and Sobolev spaces needed later on. Roughly speaking, many of the simple facts about derivatives can be extended to weak ones if properly translated into the L p setting. But, of course, there are several new phenomena also discussed below. Remark 5.8. Let R d be open, k N and p [1, ]. Then the following assertions hold. (a) If u C k () and u and all its derivatives up to order k belong to L p (), then u Wp k () and the classical and the weak derivatives coincide (see Remark D.2). (b) Let p <. A function u L p () belongs to Wp k () if and only if there are u n C k () Wp k () such that u n u in L p () and all derivatives of u n up to order k converge in L p (). We then have j u = lim n j u n in L p () and analogously for higher derivatives up to order k (see Lemma D.6 and Theorem D.13). (c) The space Cc (R d ) is dense in Wp k (R d ) if p < (see Theorem D.13). (d) A function u L p (a, b) (where a < b ) belongs to Wp 1 (a, b) := Wp 1 ((a, b)) if and only if u is continuous and there is a function v L p (a, b) such that t u(t) = u(s) + v(τ) dτ for all t, s (a, b). (5.3) s It then holds u := 1 u = v and u has a continuous extension to a (or b) if a > (or b < ). See Theorem D.10. For instance, let u(t) = t and v(t) = 1 for t > 0 and v(t) = 1 for t < 0. Then (5.3) holds, and thus u Wp 1 ( 1, 1) with u = v. See also Example D.4 for further explicit examples. (e) (Product rule) If u Wp 1 () and v Wp 1 () with 1 p + 1 p = 1, then uv W1 1() and j(uv) = u j v + v j u for j = 1,..., d (see Proposition D.7). Let p [1, ) and k N. In view of Remark 5.8 (c), we define W k p () = closure of C c () in W k p (), where we set H k () = W 2 k(). Remark 5.8 (c) shows that W p k (R d ) = Wp k (R d ). We say that functions u W p 1 () have the trace 0 on. This definition is justified by the following result: If is sufficiently smooth (e.g. is compact and of class C 1 ), then the restriction map u u from Wp 1 () C() to L p (, σ) can continuously be extended to the trace operator tr : Wp 1 () L p (, σ), where σ is the surface measure on. Moreover, W 1 p () is the kernel of tr. (See Theorem D.27 in Appendix D.) We continue with an upgraded version of (5.2) which will allow us to prove the symmetry of the Laplacian on L 2 (R d ). Let p (1, ). For F W p 1 () d 46

7 and ϕ W p 1 () we have Gauß formula ϕ div F dx = F ϕ dx. (5.4) In fact, this identity holds for F C c () d and ϕ C c () and thus (5.4) follows by approximation. (See Theorem D.28 for a much more precise result.) One of the most important features of Sobolev spaces is that their elements enjoy better regularity properties than their definition directly implies. This behavior is encoded in the embedding theorems of Sobolev and Morrey (see Theorem D.15 and Corollary D.21). Here we only state certain basic versions: Wp k (R d ) L q (R d ) if k d p d q, q [p, ), (5.5) Wp k (R d ) C 0 (R d ) if k d p > 0. (5.6) For open sets R d it further holds W p k () L q () if k d p d q, q [p, ), (5.7) W p k () C 0 () if k d p > 0. (5.8) As a byproduct, one further obtains Poincaré s inequality u p dx δ u p p (5.9) for each bounded open subset R d, p [1, ), some δ > 0 and all u W 1 p (). (See Theorem D.15 and Corollary D.19 in Appendix D.) The Sobolev space W2 k(rd ) = H k (R d ) can be treated by means of the Fourier transform in a very efficient way. To that purpose, we recall that for a function f L 1 (R d ) L 2 (R d ) the Fourier transform is given by Ff(ξ) = f(ξ) := (2π) d 2 e iξ x f(x) dx, ξ R d, (5.10) R d where ξ x = ξ 1 x ξ d x d. Clearly, f (2π) d 2 f 1. In fact, it holds f C 0 (R d ) (see Corollary E.8 in Appendix E). Plancherel s theorem says that one can extend F to a unitary operator on L 2 (R d ) which is again denoted by F. Note that formula (5.10) does not hold with a Lebesgue integral if f is not integrable on R d. Theorem 5.9. The Fourier transform extends to a unitary operator F : L 2 (R d ) L 2 (R d ) satisfying (F 1 g)(y) = (Fg)( y) for g L 2 (R d ) and y R d. Let f, g L 2 (R d ), h L 1 (R d ), k N and j {1,..., d}. We then obtain the following assertions. (a) (Plancherel) (Ff Fg) L 2 = (f g) L 2, R d fĝ dx = fg R d dx. (b) F(h f) = (2π) d 2 ĥ f, F 1 (ĥ f) = (2π) d 2 { h f. (c) H k (R d ) = u L 2 (R d ) } ξ k 2û L2 (R d ). (d) j u = if 1 (ξ j û) for u H 1 (R d ). 47

8 This result is proved in Theorems E.11 and E.14 in Appendix E. Here the symbols ξ k 2 and ξ j denote the functions ξ ξ k 2 and ξ ξ j, respectively. The convolution h f L 2 (R d ) is given by (h f)(x) = h(x y)f(y) dy, R d x R d, see (E.3) in Appendix E. Based on the Fourier transform we now treat the Laplacian on L 2 (R d ) and show in particular its selfadjointness. Therefore, i is skewadjoint which is crucial for the investigation of Schrödinger equations. Example We set X = L 2 (R d ), m(ξ) = ξ 2 2 for ξ Rd, D(A) := { u X mû X } = H 2 (R d ), and Au := F 1 mû = u. The latter identities follow from Theorem 5.9. Observe that u is a sum of weak derivatives of second order and that u = div( u). We want to show that A is dissipative, selfadjoint and σ(a) = R. Recall that H 2 (R d ) = H 2 (R d ) by Remark 5.8 (c), so that (5.4) implies that u v dx = u v dx = u v dx, R d R d R d u u dx = u 2 dx 0 R d R d for all u, v D(A). This means that A is symmetric and dissipative. Let λ C \ R. To show the surjectivity of λi A, we set r λ (ξ) = 1 for λ+ ξ 2 ξ R d. Clearly, r λ and mr λ are bounded. We define u = F 1 r f λ for f X. Theorem 5.9 yields u X, mû = mr f λ X and λu u = F 1 (λ + m)û = F 1 (λ + m)r λ f = f. Hence, u H 2 (R d ) and λi A is surjective. Proposition 5.3 now shows that A generates a contraction semigroup, and thus A is closed, densely defined and satisfies (0, ) ρ(a). Since A is symmetric, Remark 5.6 (f) then implies that A is selfadjoint and that σ(a) R. So we arrive at σ(a) R. The equality σ(a) = R is shown in Exercise 5.1. We state two important consequences of the above result. First, the graph norm of the operator A of Example 5.10 is complete on H 2 (R d ) since A is closed on D(A) = H 2 (R d ). The open mapping theorem then implies that A is equivalent to the H 2 norm. There thus exists a constant c > 0 with u d k u k=1 d k,l=1 kl u 2 2 c( u u 2 2) (5.11) holds for all u H 2 (R d ). In other words, the graph norm of the Laplacian dominates in the L 2 sense all derivatives of second and first order. This is a truly astonishing result in view of the possible cancellations in the sum u. Thanks to the Lumer-Phillips theorem, the operator A generates a C 0 semigroup T ( ). Hence, the function u given by u(t) = T (t)u 0 for t 0 and 48

9 u 0 H 2 (R d ) belongs to C 1 (R +, L 2 (R d )) C(R +, H 2 (R d )) and it is the unique solution of the diffusion equation u (t) = u(t), t 0, u(0) = u 0, in this function class. We will not study this or similar parabolic equations in detail. In fact, their solutions have much better regularity properties than provided by the theory presented here. For the relevant theory on analytic semigroups we refer in particular to the monograph [Lun95]. To handle the Dirichlet Laplacian on a domain, we need another famous result (due to Lax and Milgram) which turns Riesz representation theorem for Hilbert space duals into one of the most useful tools for applied analysis. A map T on X is called antilinear if T (αx+y) = αt x+y for α C and x, y X. Theorem 5.11 (Lax-Milgram lemma). Let Y be a Hilbert space and a : Y Y C be a sesquilinear form (i.e., x a(x, y) is linear and y a(x, y) is antilinear for x, y Y ) such that a(x, y) c x y Re a(x, x) δ x 2 (boundedness), (strict accretivity) hold for all x, y Y and some constants c, δ > 0. Then for each ψ Y there is a unique z Y such that a(y, z) = ψ(y) for all y Y. The map ψ z is antilinear and bounded. Proof. We first establish a connection between the form a and the scalar product on Y. The map ϕ y := a(, y) belongs to Y with ϕ y c y for each y Y since a is bounded. Riesz representation theorem gives a unique Sy Y such that (x Sy) = ϕ y (x) = a(x, y) for all x Y and Sy = ϕ y c y. As a result, S B(Y ). Moreover, the strict accretivity yields δ y 2 Re a(y, y) = Re(y Sy) (y Sy) c y Sy for every y Y which implies δ c y Sy. It is then easy to see that S is injective and has a closed range R(S). If x Y is orthogonal to R(S), we conclude 0 = (x Sx) = Re(x Sx) = Re a(x, x) δ x 2, so that x = 0. Thus, R(S) = R(S) = Y and S is invertible with S 1 c δ. Let ψ Y. There is a unique v Y such that ψ = ( v) thanks to Riesz theorem. Hence, a(y, S 1 v) = (y SS 1 v) = (y v) = ψ(y) for all y Y and the vector z := S 1 v = S 1 T ψ Y is the desired solution, where T : Y Y denotes the antilinear isomorphism from Riesz theorem. If also z Y satisfies a(y, z) = ψ(y) for all y Y, then 0 = a(z z, z z) δ z z 2 as above, and thus z = z. We now use the Lax-Milgram lemma to define the Dirichlet Laplacian on a domain by means of a corresponding form. 49

10 Example 5.12 (Dirichlet Laplacian). Let R d be open and bounded and X = L 2 (). sesquilinear form a(u, v) = u v dx We define the for u, v H 1 () =: Y. We construct a selfadjoint, dissipative and invertible operator A corresponding to a. Due to Hölder s inequality and Poincaré s estimate (5.9), the form a satisfies the conditions of the Lax-Milgram lemma. We next introduce D(A) := { u Y f X v Y : a(u, v) = (f v) L 2}, Au := f, where f is given by D(A). Observe that here the function f is unique since Y is dense in X. Clearly, A is linear. The map ϕ f : v (v f) L 2 belongs to Y if f L 2 () and ϕ f Y = sup (v f) L 2 sup v 2 f 2 f 2. v 1,2 1 v 1,2 1 Theorem 5.11 now gives a unique u Y such that u 2 u 1,2 c ϕ f Y c f 2 and a(u, v) = a(v, u) = ϕ f (v) = (f v) L 2, i.e., a(u, v) = (ϕ v) L 2 for all v Y. Hence, u D(A) and Au = f so that A is bijective with a bounded inverse. It follows that A is closed and that there is a point λ 0 > 0 in ρ(a) since ρ(a) is open. For u, v D(A) we further compute (Au v) L 2 = a(u, v) = a(v, u) = (Av u) L 2 = (u Av) L 2, (Au u) L 2 = a(u, u) δ u 2 1,2 0, using (5.9). Consequently A is symmetric, dissipative and λ 0 I A is surjective. Proposition 5.3 and Remark 5.6 now imply that A generates a contraction semigroup and that it is selfadjoint. Below, we write D instead of A. In the above example we have constructed a selfadjoint operator that we call Dirichlet Laplacian. Let us justify this name. First, we recall that a function u C() satisfies the homogeneous Dirichlet boundary condition if it vanishes on. As explained after Remark 5.8 this condition is replaced by u H 1 () in the L 2 setting. To explain the operator A itself, we assume that is of class C 1. We set A 0 u = u for u D(A 0 ) = H 2 () H 1 (). Gauß formula then yields ( A 0 u v) = ( u)v dx = u v dx = a(u, v) for all u D(A 0 ) and v H 1 () = Y, see Theorem D.28 in Appendix D. So, if C 1, the operator A extends A 0, and one may consider A 0 as the natural Dirichlet Laplacian in L 2 (). If we assume more, namely C 2, known results about elliptic partial differential equations imply that I A 0 is surjective in L 2 (), see Theorems 8.3 and 8.12 in [GT01]. Therefore, A = A 0 if C 2. In this case the graph norm again controls the 2 norms of all derivatives of first and second order, cf. (5.11). 50

11 The above generation results can be extended to more general elliptic differential operators with suitable boundary, acting in L p spaces or in spaces of continuous functions. Among the vast literature we refer to [Lun95] and [Tan97], and to [Ouh05] for form methods. Example 5.12 allows to solve the heat equation on the domain with Dirichlet boundary conditions. As in Example 5.10, we omit this result. We rather consider an operator matrix involving the Dirichlet Laplacian which will later be used to investigate the wave equation (cf. Lecture 1). Example Let R d be open and bounded. We use the Dirichlet Laplacian D in L 2 () introduced in the previous example. We recall that D( D ) consists of those u H 1 () such that there is a function f L 2 () with u H 1 () : u v dx = fv dx, and then D u = f. To treat the wave equation (1.3) (at first with b = 0), we introduce the Hilbert space X = H 1 () L 2 () endowed with the scalar product (( ) ( )) u1 v1 = ( u 1 v 1 + u 2 v 2 ) dx. u 2 v 2 By Poincaré s estimate (5.9) the corresponding norm is equivalent to the usual norm on X given by ( u 1 2 1,2 + u ) 1 2. On X we define the operator ( ) 0 I A = with D(A) = D( D 0 D ) H 1 (). As mentioned in the introduction, the Cauchy problem for A in X should correspond to the wave equation (1.3) with b = 0. We explain this in detail in the next lecture. In this example we show that A is skewadjoint. For (u 1, u 2 ), (v 1, v 2 ) D(A) we first compute ( ( ) ( )) (( ) ( )) u1 v1 u2 v1 A = = ( u u 2 v 2 D u 1 v 2 v 1 + ( D u 1 )v 2 ) dx 2 = ( u 2 v 1 u 1 v 2 ) dx (( ) ( )) (( ) ( )) u1 v2 u1 v1 = = A. D v 1 u 2 v 2 u 2 Hence, A is skewsymmetric (i.e., ia is symmetric). Moreover, Re(Aw w) = 0 for all w D(A), so that A is dissipative. We define the bounded operator ( ) 0 R = 1 D I 0 on X, where the inverse 1 D exists by Example It is easy to see that RX D(A) and AR = I, as well as RAw = w for all w D(A). As a result, A is invertible, and thus λi A is surjective for sufficiently small λ > 0. Proposition 5.3 now implies that A is densely defined. Since also ia is invertible, Remark 5.6 (f) yields the selfadjointness of ia and so A is skewadjoint. 51

12 Exercises Exercise 5.1. On X = L 2 (R d ) let Au = u with D(A) = H 2 (R d ). Show that R σ(a) (and hence σ(a) = R by Example 5.10). { Exercise 5.2. Let X be a Hilbert space with an orthonormal basis un n N }. Let a n C be given. Define the operator A on X by Ax = n=1 a n (x u n )u n for x D(A) = { x X (a n (x u n )) n l 2}. Compute the spectrum and the resolvent of A. Show that (i) A generates a C 0 semigroup T ( ) if and only if sup n N Re a n <, (ii) A generates a unitary group T ( ) if and only if a n ir for all n N. How does T ( ) look like? Exercise 5.3. Let A generate a C 0 semigroup T ( ) on a reflexive Banach space X. Show that A generates a C 0 semigroup S( ) on X and that S(t) = T (t) for all t 0. What can be said if X is not reflexive? Exercise 5.4. Let X = L 2 (0, ) and Au = u for u D(A) = H 1 (0, ). Show that A is dissipative, closed and σ(a) = { λ C Re λ 0 }. Compute A. (We note that A generates a contraction semigroup T ( ) by the Lumer-Phillips theorem. As in Exercise 3.3 one can show that T (t)f(s) = f(s t) if s t > 0 and T (t)f(s) = 0 if s t 0, where f X, t 0 and s > 0.) 52

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