arXiv:1609.00040v1 [math.FA] 31 Aug 2016 .Isiu ¨rAayi .Dprmn fMathematics of Department 2. f¨ur Analysis Institut 1. institutions: Home conv resolvent weak convergence, group resolvent Strong 65N30. Keywords: 47A05, 47D03, 35B40, Classification: Subject AMS 2016 2, September emn ukad1142 Auckland 92019 bag Auckland Private of University Germany Dresden 01062 Dresden TU ekadsrn prxmto fsemigroups of approximation strong and Weak r qiaetudrcranntrlasmtoswihaefrequ oper are no applications. which weak integral in assumptions certain met the natural in in certain semi- or under generators, topology equivalent the are operator the of strong of convergence the limit resolvents in that a the topology, show to of we convergences a or of others, groups, on types Among semigroups various degenerate compare semigroup. bounded, we uniformly space, of Hilbert sequence a For .Chill R. nHletspaces Hilbert on e Zealand New 1 n ...trElst ter A.F.M. and Abstract 2 rec,dgnrt semi- degenerate ergence, ently ator rms 1 Introduction

The subject of approximation of one-parameter semigroups of operators in various is a fundamental topic in semigroup theory. The Trotter–Kato theorem for the approximation of a C0-semigroup in the strong operator topology is a classical result which can be found in many textbooks. More recently, the question of approximation in the has been studied, too; see, for example, Kr´ol [Kr´o09], Eisner & Sereny [ES10] and Furuya [Fur10]. The purpose of this article is to study the relation between convergence of a sequence of semigroups in the weak operator topology and the convergence in the strong operator topology. We concentrate on semigroups in Hilbert spaces whose generators are associated with m-sectorial forms. We show that convergence in the weak operator topology and in the strong operator topology are equivalent in the case where all involved semigroups are selfadjoint, while they are not equivalent in the general case, even when all semigroups are analytic and contractive on the same sector. In the case where all involved semigroups are analytic and contractive on the same sector we give additional conditions under which equivalence of convergence in the weak operator topology and the strong operator topology does hold. In fact, equivalence between the two types of convergences holds if in addition the semigroups generated by the real parts of the associated forms converge in the weak operator topology, or if a monotonicity condition holds which is for example satisfied in the context of the Galerkin approximation. Motivated by applications to numerical analysis (the Galerkin approximation) or the stability of parabolic partial differential equations with respect to the underlying (un- bounded) domain, we consider not only C0-semigroups but general degenerate semigroups, that is, semigroups which are merely defined and strongly continuous on the open interval (0, ), and bounded on the open interval (0, 1). In this more general context, it is for ∞ example possible to study the approximation of a (C0-) semigroup on an infinite dimen- sional by degenerate semigroups acting on finite dimensional subspaces. Due to the variational character of the applications which we describe in Sections 4, 5 and 6, convergence in the weak operator topology is often easy to establish while convergence in the strong operator topology is comparatively more involved, especially when compactness arguments (obtained by compact embeddings of domains of generators) are not at hand. In principle, the additional arguments which allow one to pass from convergence in the weak operator topology to convergence in the strong operator topology exist in a scattered way in the literature. In the case of the Galerkin method, these additional arguments are sometimes given, but sometimes the reader is left with a statement which does not give the full (strong) convergence properties, especially if one is only interested in abstract existence results for solutions of parabolic partial differential equations. The purpose of this article is to gather the arguments in an abstract context and to show that the equivalence between convergence in the weak and strong operator topology is a general principle independent from the concrete application in numerical analysis, the study of parabolic equations on varying domains or in homogenization.

1 2 Preliminaries on degenerate semigroups

Let X be a . We call a function S : (0, ) (X), t S , a degenerate ∞ → L 7→ t semigroup if

(i) S is strongly continuous on (0, ), ∞ (ii) S = S S for every t, s (0, ), and t+s t s ∈ ∞ (iii) sup S < . t∈(0,1) k tk ∞ It is an exercise to show, using properties (ii) and (iii) above, that every degenerate semi- group is exponentially bounded, that is, there exist constants M 0 and ω R such ≥ ∈ that S M eωt k tk ≤ for all t (0, ). ∈ ∞ Let A X X be a graph in X. Then A is called the generator of a degenerate ⊆ × − semigroup S if there exists a ω R such that λI + A is boundedly invertible for every ∈ λ (ω, ) and if ∈ ∞ ∞ −1 −λt (λI + A) x = e Stx dt Z0 for all x X. In particular, ω is chosen large enough so that the Laplace integral on the ∈ right-hand side converges. A crude estimate of the Laplace integral then yields that the pseudoresolvent λ (λI + A)−1 satisfies the Hille–Yosida condition 7→ (λ ω)k (λI + A)−k M k − k ≤ uniformly for all k N and λ (ω, ). There seems to be no characterisation of degen- ∈ ∈ ∞ erate semigroups on Banach spaces known in the literature, for example solely in terms of the Hille–Yosida condition, the problem being that pseudoresolvents need not have dense range. However, there are two important situations in which one has a positive result. The first is the situation of degenerate semigroups on reflexive spaces. If A is a graph on a reflexive Banach space X, if there exists an ω R such that λI + A is boundedly ∈ invertible for every λ (ω, ) and if the pseudoresolvent λ (λI + A)−1 satisfies the ∈ ∞ 7→ Hille–Yosida condition above, then A is the generator of a degenerate semigroup S for − which, in addition, the limit

P x := lim Stx t↓0 exists for every x X and defines a bounded projection P . Moreover, range P = ∈ range(λI + A)−1 and ker P = ker (λI + A)−1. In particular, the range and the kernel of (λI + A)−1 do not depend on λ, and P is a projection onto the closure of the domain of A. The second situation where a characterisation of the generator is available is the sit- uation of analytic degenerate semigroups, that is, of degenerate semigroups which extend analytically to a sector of the form

Σ := z C 0 : arg z < θ , θ { ∈ \{ } | | }

2 for some θ (0, π ]. If A is a graph on a Banach space X, if there exist ω R and θ (0, π ] ∈ 2 ∈ ∈ 2 such that λI + A is boundedly invertible for every λ ω + Σ π , and if ∈ 2 +θ sup (λ ω)(λI + A)−1 < , λ∈ω+Σ π θ k − k ∞ 2 + then A generates a degenerate semigroup S which extends analytically to a semigroup − on the sector Σθ. For the results stated above, see Arendt [Are01] or Baskakov [Bas04]. Analytic degenerate semigroups on Hilbert spaces are, for example, generated by graphs associated with closed, (quasi-) sectorial forms, and our main results concern indeed solely this particular situation, with the exception of Lemma 3.2, where we consider general analytic semigroups on Banach spaces. By a form on a Hilbert space H we mean here a sesquilinear mapping a: V V C, where the form domain V is a linear subspace of × → H. We point out that the form domain V need not be dense in H. The real part a of ℜ a form a is defined by ( a)(u, v) := 1 (a(u, v)+ a(v,u)), and similarly one may define the ℜ 2 imaginary part which is, however, not used in this article. A form a is called sectorial if there are θ (0, π ) and γ R such that ∈ 2 ∈ a(u) γ u 2 Σ − k kH ∈ θ for all u V , where a(u) = a(u,u). We call γ a vertex of a. Finally, a sectorial form a ∈ on H is called closed if there exists an ω R such that (u, v) ( a)(u, v)+ ω(u, v) is ∈ 7→ ℜ H a complete inner product on V . For a closed, sectorial form we define the associated graph

A := (u, f) H H : u V and a(u, v)=(f, v) for all v V . { ∈ × ∈ H ∈ } Then this graph is m-sectorial in the sense that there is an ω > 0 such that λI + A is invertible and λ(λI +A)−1 1 for every λ ω+Σ π . If a is symmetric in the sense that k k ≤ ∈ θ+ 2 a = a, then the associated graph is self-adjoint. By applying [Kat80, Theorem VI.1.27] ℜ to the part of an m-sectorial graph in the closure of its domain one can see that every m-sectorial graph is associated to a closed, sectorial form.

3 Semigroup convergence

The first main result of this note is the following theorem for self-adjoint graphs and semigroups. It asserts that pointwise convergence of the resolvents in the weak operator topology and in the strong operator topology are equivalent, and that the same is true for pointwise convergence of semigroups in the weak operator topology and pointwise con- vergence of semigroups in the strong operator topology, uniformly for times in compact subsets of (0, ). ∞ Theorem 3.1. Let H be a Hilbert space. For all n N let A and A be positive self- ∈ n adjoint graphs on H. Let S(n) and S be the degenerate semigroups generated by A and − n A. Then the following are equivalent. − (n) (i) lim →∞ S = S in ( (H), WOT) for all t (0, ). n t t L ∈ ∞ 3 T (n) T (ii) lim →∞ (S f,g) dt = (S f,g) dt for all T > 0 and f, g H. n 0 t H 0 t H ∈ R T (n) R T (iii) lim →∞ (S f,g(t)) dt = (S f,g(t)) dt for all T > 0, f H and g n 0 t H 0 t H ∈ ∈ L1([0, T ]R,H). R T (n) (iv) lim →∞ (S S )f dt =0 for all T > 0 and f H. n 0 k t − t kH ∈ R −1 −1 (v) lim →∞(λI + A ) =(λI + A) in ( (H), SOT) for all λ C with Re λ> 0. n n L ∈ −1 −1 (vi) There exists a λ C with Re λ> 0 such that lim →∞(λI + A ) =(λI + A) in ∈ n n ( (H), SOT). L −1 −1 (vii) lim →∞(λI + A ) =(λI + A) in ( (H), WOT) for all λ C with Re λ> 0. n n L ∈ (viii) There exists a set D C with accumulation point in the open right half-plane µ ⊆ −1 −1 { ∈ C : Re µ> 0 such that lim →∞(λI + A ) =(λI + A) in ( (H), WOT) for all } n n L λ D. ∈ −1 −1 (ix) There exists a λ C R with Re λ> 0 such that lim →∞(λI + A ) =(λI + A) ∈ \ n n in ( (H), WOT). L (x) For all f H and δ, T > 0 with δ < T it follows that ∈ (n) lim sup (St St)f H =0. n→∞ t∈[δ,T ] k − k

Theorem 3.1 is for self-adjoint graphs on Hilbert spaces. Statements (iv), (v), (vi) and (x) are, however, also equivalent for general analytic degenerate semigroups on Banach spaces. This is the contents of the next lemma.

Lemma 3.2. Let X be a Banach space. For all n N let A and A be graphs on X, such ∈ n that A and A generate analytic degenerate semigroups S(n) and S, respectively. As- − n − sume there exists a θ (0, π ] such that the degenerate semigroups S(n) and S are uniformly ∈ 2 bounded on the same sector Σ with a bound independent of n N. Then the following θ ∈ assertions are equivalent.

T (n) (i) lim →∞ (S S )f dt =0 for all T > 0 and f X. n 0 k t − t kX ∈ R −1 −1 (ii) lim →∞(λI + A ) =(λI + A) in ( (X), SOT) for all λ C with Re λ> 0. n n L ∈ −1 −1 (iii) There exists a λ C with Re λ> 0 such that lim →∞(λI + A ) =(λI + A) in ∈ n n ( (X), SOT). L (iv) For all f X and δ, T > 0 with δ < T it follows that ∈ (n) lim sup (St St)f X =0. n→∞ t∈[δ,T ] k − k

Proof. ‘(i) (ii)’. This follows by taking Laplace transforms of the semigroups S(n) and S. ⇒ ‘(ii) (iii)’. Trivial. ⇒ −k −k ‘(iii) (ii)’. First Condition (iii) implies that lim →∞(λI + A ) = (λI + A) in ⇒ n n ( (X), SOT) for every k N. Since the powers of (λI + A )−1 coincide, up to a scalar L ∈ n factor, with the derivatives of the holomorphic function µ (µI + A )−1 at the point λ, 7→ n Statement (ii) follows by an application of Vitali’s theorem [AN00, Theorem 2.1]. (Compare also with [Are01, Remark 3.8] for a slightly different argument).

4 ‘(ii) (iv)’. This follows from Arendt [Are01, Theorem 5.2]. A different proof is as ⇒ −1 −1 follows. First it follows again from Vitali’s theorem that limn→∞(λI +An) =(λI +A) in ( (X), SOT) for all λ Σ π . Secondly, for every n N, f X and t > 0, one has L ∈ 2 +θ ∈ ∈ the integral representation

(n) 1 tλ −1 St f = e (λI + An) f dλ, 2πi ZΓ

′ π e±iθ θ′ π , π θ where Γ is an appropriately chosen curve in Σ 2 +θ connecting for some ( 2 2 + ). (n) ∞ ∈ Of course, this integral representation also holds when S and An are replaced by S and (n) A, respectively. The strong convergence of (St ), uniformly for t in intervals of the form −1 −1 [δ, T ] now follows from the locally uniform convergence of (λI + An) f to (λI + A) f and a rough estimate of the resolvents for large λ. ‘(iv) (i)’. This follows from Lebesgue’s dominated convergence theorem. ⇒ Now we turn to the proof of the main theorem of this section.

Proof of Theorem 3.1. As mentioned above, the equivalence of the statements (iv), (v), (vi) and (x) follows from Lemma 3.2. ‘(i) (ii)’. This follows from the Lebesgue dominated convergence theorem. ⇒ ‘(ii) (iii)’. Let T > 0. It follows from Statement (ii) that ⇒ T (n) 1 lim ((St St)f, [a,b]g)H dt =0 →∞ n Z0 − for all f,g H and a, b R with 0 a < b T . Since the step functions are dense in ∈ ∈ ≤ ≤ L1([0, T ],H) and the Sn are contractive, Statement (iii) follows by a 3ε-argument. ‘(iii) (iv)’. Let T > 0 and f H. Then symmetry and the semigroup property give ⇒ ∈ T (n) 2 (St St)f H dt Z0 k − k T T (n) (n) = ((S2t S2t)f, f)H dt 2 Re ((St St)f,Stf)H dt Z0 − − Z0 − 2T T 1 (n) (n) = ((St St)f, f)H dt 2 Re ((St St)f,Stf)H dt (1) 2 Z0 − − Z0 − for all n N. Applying the hypothesis with g(t) = f to the first term of (1) and with ∈ g(t)= Stf to the second term, one deduces that both terms on the right-hand side of (1) T (n) 2 tend to 0 as n . Hence lim →∞ (S S )f dt = 0. A simple application of → ∞ n 0 k t − t kH the Cauchy–Schwarz inequality, usingR the contractivity of all involved semigroups, yields also convergence in the L1-sense. ‘(iv) (v) (vi) (x)’. This is a special case of Lemma 3.2. ⇔ ⇔ ⇔ ‘(x) (i)’, ‘(v) (vii) (viii)’ and ‘(vii) (ix)’. Trivial. ⇒ ⇒ ⇒ ⇒ ‘(viii) (vii)’. This follows from Vitali’s theorem. ⇒ −1 −1 ‘(ix) (vi)’. Let λ C R with Re λ> 0 be such that lim →∞(λI +A ) =(λI +A) ⇒ ∈ \ n n

5 in ( (H), WOT). Let f H. Then L ∈ −1 2 −1 −1 lim (λI + An) f H = lim ((λI + An) (λI + An) f, f)H n→∞ k k n→∞

−1 −1 −1 = lim (λ λ) ((λI + An) f, f)H ((λI + An) f, f)H n→∞ −  −  −1 −1 −1 =(λ λ) ((λI + A) f, f)H ((λI + A) f, f)H −  −  = (λI + A)−1f 2 . k kH Then Statement (vi) is valid.

One might hope that the convergence in Statement (x) of Theorem 3.1 is valid with [δ, T ] replaced by (0, T ]. However, a counterexample has been provided by Daners [Dan05, Example 6.7]. There is the following characterisation of uniform convergence on (0, T ] in the strong operator topology.

Lemma 3.3. Assume the assumptions and notation as in Lemma 3.2, and assume in addition that the Banach space X is reflexive. Suppose the four equivalent statements in Lemma 3.2 are valid. For all n N let P and P be the projections given by ∈ n (n) Pnf := lim St f and P f := lim Stf (f X). t↓0 t↓0 ∈ Then the following are equivalent.

(n) (i) lim →∞ sup (S S )f =0 for all T > 0 and f X. n t∈(0,T ] k t − t kX ∈ (n) (ii) There exists a T > 0 such that lim →∞ sup (S S )f =0 for all f X. n t∈(0,T ] k t − t kX ∈ (iii) lim →∞ P = P in ( (X), SOT). n n L Proof. ‘(i) (ii)’. Trivial. ⇒ (n) ‘(ii) (iii)’. Since lim ↓ S f = P f for all f X and n N, with a similar identity ⇒ t 0 t n ∈ ∈ for S and P , the implication (ii) (iii) follows by a 3ε-argument. ⇒ ‘(iii) (i)’. It follows from the strong resolvent convergence and [Are01, Theorem 4.2 ⇒ (n) (b)] that lim →∞ sup (S S )f = 0 for all T > 0 and f dom A. Recall that n t∈(0,T ] k t − t kX ∈ P is a projection onto dom A. Now let T > 0 and f X. Then ∈ S(n)f S f = S(n) P f S P f k t − t kX k t n − t kX S(n) (P f P f) + (S(n) S )P f ≤k t n − kX k t − t kX M P f P f + (S(n) S )P f ≤ k n − kX k t − t kX (m) for all t (0, T ] and n N, where M = sup N Ss < , from which State- ∈ ∈ m∈ ,s∈(0,t] k k ∞ ment (i) follows.

In Theorem 3.1(ix) it is essential that λ R. In the next example there is convergence 6∈ in ( (H), WOT) for λ = 1, but clearly not in ( (H), SOT). The example is part of [ES10, L L Example 2.3].

6 Example 3.4. Let H = ℓ . For all n N define U (H) by 2 ∈ n ∈L

Un(x1, x2,...)=(xn+1,...,x2n, x1,...,xn, x2n+1,...).

Then Un is self-adjoint and limn→∞ Un = 0 in ( (H), WOT), but (Un) does not converge L 1 N to 0 in ( (H), SOT) since the Un are also unitary. Let Vn = (1 n )Un for all n . Then L −1 − ∈ Vn < 1 and the Cayley transform An =(I+Vn)(I Vn) (H) is a positive self-adjoint k k −1 1 1 − ∈L1 −1 operator. Moreover, (I + An) = 2 (I Vn)= 2 (I (1 n )Un). So limn→∞(I + An) = 1 −1 − − − 2 I =(I +A) in ( (H), WOT), where A = I is a positive self-adjoint operator. However, −1 L lim →∞(I + A ) does not exist in ( (H), SOT), that is Statement (vi) in Theorem 3.1 n n L is not valid.

We next present an example that symmetry of the generators in Theorem 3.1 cannot be replaced by m-sectoriality, even not with a uniform sector.

Example 3.5. Let H be an infinite dimensional separable Hilbert space and let S be −t the contraction semigroup defined by St = e I for all t > 0. By [Kr´o09, Theorem 2.1] (n) there exists a sequence (S )n∈N of unitary C0-groups on H such that for all T > 0 one (n) N has limn→∞ St = St in ( (H), WOT) uniformly for all t (0, T ]. For all n let L(n) ∈ ∈ Bn be the generator of S and set B = I. Then, by taking Laplace transforms of the − −1 −1 respective semigroups, limn→∞(λI + Bn) = (λI + B) in ( (H), WOT) for all λ C L (n) ∈ with Re λ> 0. Clearly, for each t> 0 one does not have lim →∞ S = S in ( (H), SOT), n t t L since the S(n) are isometric while S is not. Hence for all λ C with Re λ> 0 one does not −1 −1 ∈ have lim →∞(λI + B ) =(λI + B) in ( (H), SOT), see Lemma 3.2. n n L The operator Bn is not invertible in general, but Cn = I + Bn is m-accretive and −1 −1 invertible for all n N. Set C = I + B = 2I. Then lim →∞(λI + C ) = (λI + C) ∈ n n in ( (H), WOT) for all λ C with Re λ > 0. Moreover, for all λ C with Re λ > 0 one L ∈ −1 −1 ∈ 1/2 does not have lim →∞(λI + C ) =(λI + C) in ( (H), SOT). Finally, let A = Cn n n L n for all n N. Then A is m-sectorial with vertex zero and semiangle π by [Kat80, ∈ n 4 Theorem V.3.35] and [ABHN01, Theorem 3.8.3]. Since

1 ∞ √µ λI A −1 µI C −1 dµ ( + n) = 2 ( + n) π Z0 λ + µ

−1 −1 for all λ (0, ) by [Kat61, (A1)], it follows that limn→∞(λI + An) = (λI + √2I) ∈ ∞ 2 −1 in ( (H), WOT) for all λ (0, ). Note that Cn = An and therefore (I + Cn) = L −1 −1 ∈ ∞ −1 (iI +An) ( iI +An) for all n N. So if limn→∞(iI +An) converges in ( (H), SOT), − −1 ∈ L then also lim →∞( iI+A ) converges in ( (H), SOT) (compare with the argument in the n − n L proof of Lemma 3.2 using Vitali’s theorem and the fact that i lie in the same component −±1 of analyticity of the resolvent) and hence lim →∞(I + C ) converges in ( (H), SOT), n n L which is a contradiction. Thus Theorem 3.1 cannot be extended to m-sectorial operators.

1 2 Instead of considering the square roots An = Cn one could also consider the fractional powers A = Cα for arbitrary α (0, 1). The argument which allows one to pass from n n ∈ the convergence of the resolvents of Cn in the weak operator topology to the convergence of the resolvents of An in the weak operator topology (and back) then simply relies on a representation of the resolvent of An in terms of a contour integral over

7 the resolvent of Cn and vice versa [Haa06]. This means that the angle of sectoriality in the above counterexample can be chosen arbitrarily small. A variant of Theorem 3.1 is true if, in addition, one also requires weak resolvent conver- gence for the real parts of the generators, or, more precisely, for the operators associated with the real parts of the involved forms. Under this additional assumption one again has that weak resolvent convergence implies strong resolvent convergence.

Theorem 3.6. Let H be a Hilbert space. For all n N let a , a be closed sectorial ∈ n sesquilinear forms in H with vertex zero. Let An, A, Rn and R be the m-sectorial graphs associated with a , a, a and a, respectively. Suppose there exist λ,λ′ C with Re λ> 0, n ℜ n ℜ ∈ Re λ′ > 0 and λ′ R such that 6∈ −1 −1 lim (λI + An) =(λI + A) in ( (H), WOT) n→∞ L and ′ −1 ′ −1 lim (λ I + Rn) =(λ I + R) in ( (H), WOT). n→∞ L −1 −1 Then lim →∞(I + A ) = (I + A) in ( (H), SOT) and, for all f X and δ, T > 0 n n L ∈ with δ < T , (n) lim sup (St St)f X =0. n→∞ t∈[δ,T ] k − k

−1 −1 Proof. It suffices to show that limn→∞(λI + An) = (λI + A) in ( (H), SOT). Let −1 −1 L f H. Write u := (λI+A ) f and u := (λI+A) f for all n N. Then lim →∞ u = u ∈ n n ∈ n n weakly in H by the assumed convergence of resolvents in the weak operator topology.

Therefore we have to prove that limn→∞ un = u (strongly) in H. Let n N. Then a (u , v)+ λ (u , v) =(f, v) for all v V . Choosing v = u and ∈ n n n H H ∈ n n taking the real part gives

Re a (u )+(Re λ) u 2 = Re(f,u ). (2) n n k nkH n Similarly Re a(u)+(Re λ) u 2 = Re(f,u). (3) k kH Since limn→∞ un = u weakly in H, it follows from (2) that

2 a a (Re λ) lim sup un H = lim sup Re(f,un) Re n(un) = Re(f,u) lim inf Re n(un). n→∞ k k n→∞  −  − n→∞

Now an and a are symmetric closed sesquilinear forms. Moreover, by assumption and by ℜ ℜ ′ −1 ′ −1 Theorem 3.1(ix) (v) one has lim →∞(λ I + R ) =(λ I + R) in ( (H), SOT). Using ⇒ n n L again that limn→∞ un = u weakly in H, one deduces from Attouch [Att84, Theorem 3.26] (or for a shorter proof for forms, see Mosco [Mos94, Theorem 2.4.1]) the bound ( a)(u) ℜ ≤ lim inf →∞( a )(u ). Therefore n ℜ n n 2 a a 2 (Re λ) lim sup un H = Re(f,u) lim inf Re n(un) Re(f,u) Re (u) = (Re λ) u H, n→∞ k k − n→∞ ≤ − k k where we used (3) in the last step. So limn→∞ un = u in H and the convergence of resolvents in the strong operator topology follows. The remaining assertion on the convergence of semigroups follows from Lemma 3.2.

8 We finish this section by presenting a theorem in which we deal with a single form a which does not have to be symmetric and where the approximation is connected to a space approximation of the form domain. It is not a corollary to the main theorem (Theorem 3.1) nor is it an immediate consequence of Theorem 3.6. We rather give a variant of the proof of the latter which does not use the Mosco convergence hidden in the references to Attouch [Att84] or Mosco [Mos94].

Theorem 3.7. Let V and H be Hilbert spaces such that V is continuously embedded in H. Let a: V V C be a closed, sectorial, sesquilinear form. Let A be the m-sectorial × → graph in H associated with a. Let (Vn) be an increasing sequence of closed subspaces of V N a a such that n∈N Vn is dense in V . For all n let n = Vn×Vn . Further let An be the ∈ (n) | m-sectorialS graph in H associated with an. Let S and S be the semigroups generated by A and A, respectively. Then − n − (n) lim sup (St St)f H =0 n→∞ t∈(0,T ] k − k for every T > 0 and every f H. ∈ Proof. Without loss of generality we may assume that the form a is coercive, that is, that there exists a µ> 0 such that µ u 2 Re a(u) for all u V . k kV ≤ ∈ Let f H. Let n N. Set u =(I + A )−1f. Then u V and ∈ ∈ n n n ∈ n

a(un, v)+(un, v)H =(f, v)H

a 2 for all v Vn. Choose v = un. Then Re (un)+ un H = Re(f,un)H f H un H . ∈ 2 2k k ≤ k k k k So u f and µ u Re a(u ) f . Therefore the sequence (u ) ∈N is k nkH ≤ k kH k nkV ≤ n ≤ k kH n n bounded in V . Passing to a subsequence if necessary, there exists a u V such that ∈ lim →∞ u = u weakly in V . Let m N and v V . Then n n ∈ ∈ m

a(un, v)+(un, v)H =(f, v)H for all n N with n m. Take the limit n . Then ∈ ≥ →∞

a(u, v)+(u, v)H =(f, v)H . (4)

Since V is dense in V one deduces that (4) is valid for all v V . So u D(A) and n∈N n ∈ ∈ u =(IS+ A)−1f. Since

2 a lim sup un H = lim sup Re(f,un)H Re (un) n→∞ k k n→∞  − 

= Re(f,u)H lim inf Re a(un) − n→∞ Re(f,u) Re a(u)= u 2 ≤ H − k kH −1 −1 it follows that lim →∞ u = u in H. So lim →∞(I + A ) = (I + A) in ( (H), SOT). n n n n L In particular, the four equivalent statements of Lemma 3.2 hold. For every n N the limits ∈ (n) Pnf := lim St f and P f := lim Stf t↓0 t↓0

9 exist and define projections onto the closures (in H) of the domains of the graphs An and A, respectively, and thus by [Kat80, Theorem VI.2.1 ii)] onto the closures of Vn and V in H. Since the graphs An and A are associated with forms, one can easily see from their definition that the projections Pn and P are orthogonal. Using again that (Vn) is increasing and that V is dense in V , we find that lim →∞ P = P in ( (H), SOT). The claim n∈N n n n L follows fromS Lemma 3.2 and Lemma 3.3.

The situation of Theorem 3.7 has a flavour of the situation of a monotonically decreas- ing sequence of forms, if one can speak of monotonicity in the context of sectorial forms.

If the conclusion was strong resolvent convergence of the operators An, it may be seen as a generalisation of Simon [Sim78, Theorems 3.2 and 4.1]. By Lemma 3.2, the strong resolvent convergence is equivalent to the convergence of the semigroups in the strong operator topology, uniformly in times from compact subsets of (0, ). The uniform con- ∞ vergence up to t = 0 is an additional feature of Theorem 3.7. The situation of Theorem 3.7 is somewhat opposite to the situation of a monotonically increasing sequence of forms for which strong resolvent convergence of associated operators follows from Kato [Kat80, Theorem VIII.3.13a] and Simon [Sim78, Theorems 3.1 and 4.1] in the symmetric case and from Batty & ter Elst [BtE14] in a somewhat more general case of sectorial forms.

4 Galerkin approximation

One popular situation in which an analytic C0- or degenerate semigroup is approximated by degenerate semigroups arises in the numerical analysis of parabolic partial differential equations, namely in the Galerkin approximation, that is, the space discretization via finite element spaces. It is not necessary to state a separate corollary for this situation since Theorem 3.7 is precisely designed for it. Given two Hilbert spaces V and H such that V is continuously embedded in H, and given a closed, sectorial, sesquilinear form a: V V C, × → it suffices to chose an increasing sequence (Vn) of finite dimensional subspaces of V (finite element spaces) such that n∈N Vn is dense in V . Then Theorem 3.7 asserts that the semigroups generated by theS graphs A associated with the forms a := a n× n converge n n |V V in the strong operator topology to the semigroup generated by the graph A associated with a, uniformly for times in intervals of the form (0, T ]. In some textbooks on linear and nonlinear analysis, the Galerkin approximation is used as a method of proof of existence of solutions of abstract elliptic, parabolic or hyperbolic equations. The approximate solutions living in finite dimensional subspaces of the energy space usually fulfill some a priori estimates, that is, they live in bounded subsets of the form domain or in a function space with values in the form domain. Then an argument using weak compactness allows one to find limit points (in a ), and these limit points are shown to be solutions of the original equation. The question of convergence of the approximate solutions in a norm topology is, however, not systematically discussed. In Dr´abek–Milota [DM13, Proposition 7.2.41] (see also Evans [Eva90, Theorem 2 in Chap- ter 2], the strong convergence of approximate solutions of a (nonlinear) stationary problem is explicitly stated, while this is not done in the case of approximate solutions of a (non- linear) gradient system in reflexive spaces [DM13, Theorem 8.2.5]. A statement on strong

10 convergence is also missing in Evans [Eva98, Theorem 3 in 7.1.2] in the context of an ab- § stract linear parabolic problem, even under restrictive assumptions on the finite elements. Yet, the strong convergence of approximate solutions in the Galerkin approximation of abstract parabolic equations is known and Theorem 3.7 is not new in this context: see the monograph by Dautray–Lions [DL92, Remark 5 in Section XVIII.3, p. 520] with uniform convergence in time on bounded intervals of (0, ) or the survey by Fujita–Suzuki [FS91, ∞ Theorem 7.1] with uniform convergence in time on compact intervals of (0, ), in order to ∞ mention only two references.

Note, however, that the subspaces Vn in Theorem 3.7 do not have to be finite dimen- sional.

5 Elliptic and parabolic problems on varying domains

In this section we illustrate Theorems 3.6 and 3.7 by considering a sequence of diffusion equations on varying open sets Ωn which converge monotonically from below to an open set Ω. We provide new proofs for the next results, which have been studied also in Simon [Sim78, Example 1 and Theorem 4.1 in Section 4]. In the following, given an open set Rd 1 1 Rd Ω , we consider the H0 (Ω) as a subspace of H ( ) by identifying ⊆ 1 1 Rd functions in H0 (Ω) with functions in H ( ) which are equal to 0 almost everywhere on Rd Ω, and similarly we consider the space L (Ω) as a subspace of L (Rd). \ 2 2 d Theorem 5.1. Let (Ωn)n∈N be an increasing sequence of open subsets of R and let Ω := d×d Ω . Let a L∞(Ω; C ) be uniformly elliptic in the sense that there exists η > 0 n∈N n ∈ suchS that d a (x)ξ ξ η ξ 2 for every ξ Cd and x Ω. ij j i ≥ | | ∈ ∈ i,jX=1 For all n N let u L (Ω ) L (Ω) and u L (Ω). Further, for all n N let ∈ 0,n ∈ 2 n ⊆ 2 0 ∈ 2 ∈ u C([0, ); L (Ω )) and u C([0, ); L (Ω)) be the solutions of the diffusion equations n ∈ ∞ 2 n ∈ ∞ 2 ∂ u div (a(x) u )=0 in (0, ) Ω , t n − ∇ n ∞ × n u =0 on (0, ) ∂Ω , n ∞ × n u (0, )= u in Ω , n · 0,n n and ∂ u div (a(x) u)=0 in (0, ) Ω, t − ∇ ∞ × u =0 on (0, ) ∂Ω, ∞ × u(0, )= u in Ω, · 0 respectively. If lim →∞ u u Rd =0, then n k 0,n − 0kL2( )

d lim sup un(t) u(t) L2(R ) =0. n→∞ t∈[0,T ] k − k

Proof. For all n N we consider the sectorial sesquilinear form a : H1(Ω ) H1(Ω ) C ∈ n 0 n × 0 n → a u, v a x u v A L defined by n( )= Ωn ( ( ) ) and we denote by n the sectorial graph on 2(Ω) a ∇ · ∇ a 1 associated to n. Similarly,R we define the form on H0 (Ω) and the associated operator A,

11 by replacing Ωn by Ω in the above definition. Observe that an is the restriction of the form a 1 (n) (n) to the space H0 (Ωn). Observe in addition that un(t)= St u0,n for all t> 0, where S is the semigroup generated by A , and similarly u(t)= S u for all t> 0, where S is the − n t 0 semigroup generated by A. − ∞ Since the sequence (Ωn) is monotonically increasing to Ω, it is easy to see that Cc (Ω) 1 1 is a subspace of n H0 (Ωn) and thus the latter space is dense in H0 (Ω). The claim then follows from TheoremS 3.7.

As a consequence of Theorem 5.1 and Lemma 3.2, we obtain strong convergence of solutions of elliptic problems, that is, convergence of resolvents in the strong operator topology.

d Corollary 5.2. Let (Ωn)n∈N be an increasing sequence of open subsets of R and let Ω := Ω . Fix λ > 0 and f L (Ω). For all n N let u H1(Ω ) be the weak solution n∈N n ∈ 2 ∈ n ∈ 0 n ofS the Dirichlet problem λu ∆u = f in Ω , n − n n un =0 on ∂Ωn. Further, let u H1(Ω) be the weak solution of ∈ 0 λu ∆u = f in Ω, − u =0 on ∂Ω.

Then lim →∞ u u Rd =0. n k n − kL2( ) There is a characterisation for convergence of resolvents of the Dirichlet Laplacian or resolvents of more general operators as in Theorem 5.1 in the strong operator topology. 1 One defines that a sequence (H0 (Ωn))n∈N of Sobolev spaces converges in the sense of Mosco [Mos69] if the following two conditions are valid:

(i) if u u weakly in H1(Rd) and u H1(Ω ) for all n N, then u H1(Ω), and n → n ∈ 0 n ∈ ∈ 0 (ii) for every u H1(Ω) there exists a sequence (u ) with u H1(Ω ) for all n N ∈ 0 n n ∈ 0 n ∈ and u u in H1(Rd); n → see, for example, [Dan05, Assumption 6.2]. Then convergence of resolvents in the strong 1 1 operator topology is valid if and only if the sequence (H0 (Ωn)) converges to H0 (Ω) in the sense of Mosco, see Daners [Dan03, Theorem 5.3] for the implication ‘ ’ and for the ⇐ implication ‘ ’ see Attouch [Att84, Theorem 3.26] or Mosco [Mos94, Theorem 2.4.1]. ⇒ Let us sketch a proof of the implication that Mosco convergence of the Sobolev spaces implies strong resolvent convergence. In fact, it is not difficult to prove the convergence (λI + A )−1 (λI + A)−1 in the weak operator topology whenever λ > 0. Similarly, n → (λI + R )−1 (λI + R)−1 in the weak operator topology, where R and R are the n → n operators associated with the real parts a and a, respectively, simply because the ℜ n ℜ latter sesquilinear forms have a similar structure as the forms an and a and the same arguments apply. Once, the two convergences in the weak operator topology are shown, the convergence in the strong operator topology follows from Theorem 3.6.

12 Of course, by Lemma 3.2 again, the convergence of resolvents in the strong operator topology implies convergence of the semigroups in the strong operator topology, uniformly in time intervals of the form [δ, T ] with δ,T > 0. This is weaker than the convergence of the semigroups in the strong operator topology, uniformly in time intervals of the form (0, T ], as stated in Theorem 5.1. In the situation of merely Mosco convergence of the 1 1 spaces H0 (Ωn) to H0 (Ω), one cannot expect uniform convergence on (0, T ] in general, as the example from [Dan05, Example 6.7] shows. The problem of stability of solutions of elliptic equations with respect to the domain has been studied extensively in the works by Bucur [Buc99], Bucur & Butazzo [BB02], Bucur & Varchon [BV00], Daners [Dan03], Daners, Hauer & Dancer [DHD15], Arrieta & Barbatis [AB14], Arendt & Daners [AD07], [AD08], Biegert & Daners [BD06], Dal Maso & Toader [DMT96], Sa Ngiamsunthorn [SN12a], [SN12b] and Mugnolo, Nittka & Post [MNP13].

6 Homogenization on (unbounded) open sets

We next illustrate Theorem 3.1 and consider the classical problem of homogenization of second-order elliptic operators with periodic coefficients. For all k, l 1,...,d let c : Rd R be measurable and bounded. Suppose that ∈ { } kl → these coefficients are

(i) symmetric, that is, c = c for all k, l 1,...,d , kl lk ∈{ } (ii) periodic, that is, c (x + γ)= c (x) for all k, l 1,...,d , x Rd and γ Zd, and kl kl ∈{ } ∈ ∈ (iii) uniformly elliptic, that is, there exists a µ > 0 such that d c (x) ξ ξ µ ξ 2 k,l=1 kl k l ≥ | | for all x Rd and ξ Cd. P ∈ ∈ For all ε > 0 and k, l 1,...,d define c(ε) : Rd R by c(ε)(x) = c ( 1 x). Let Ω Rd ∈ { } kl → kl kl ε ⊆ be open. We emphasise that we do not assume that Ω is bounded. Let V be a closed 1 ∞ subspace of H (Ω) which contains Cc (Ω). For all ε> 0 let Aε be the self-adjoint operator in L (Ω) associated to the form a : V V C defined by 2 ε × → d a (u, v)= c(ε) (∂ u) ∂ v. ε Z kl k l Ω k,lX=1

We shall prove that there exists a positive self-adjoint operator A in L2(Ω) such that

−1 −1 b lim(λI + Aε) =(λI + A) ε↓0 b in the strong operator topology for all λ C with Re λ > 0. In fact, we determine the ∈ operator A and we only need to prove convergence in the weak operator topology. 1 Rd An explicitb description of A is as follows. Consider the space Hper( ) of all functions u H1 (Rd) which satisfy u(x + γ) = u(x) for a.e. x Rd and all γ Zd. For all ∈ loc b ∈ ∈ j 1,...,d there exists a χ H1 (Rd) such that ∈{ } j ∈ per d d

ckl (∂kχj) ∂lv = cjl ∂lv (5) Z d − Z d [0,1] k,lX=1 [0,1] k,lX=1

13 1 d d for all v H (R ). Then χ L∞(R ) by Stampacchia [Sta60, Teorema 4.1] since ∈ per j ∈ c L with p>d. For all k, l 1,...,d define jl ∈ p ∈{ } d

cˆkl = ckl ckj ∂jχl. Z d − Z d [0,1] Xj=1 [0,1]

It follows from Bensoussan, Lions & Papanicolau [BLP78, Remark 1.2.6] that there exists a µ′ > 0 such that d cˆ ξ ξ µ′ ξ 2 for all ξ Cd. Let A be the operator in L (Ω) k,l=1 kl k l ≥ | | ∈ 2 associated to the formP a: V V C defined by × → b d a(u, v)= cˆ (∂ u) ∂ v. Z kl k l Ω k,lX=1

The alluded theorem is the following.

Theorem 6.1. Let λ C with Re λ> 0. Then ∈ −1 −1 lim(λI + Aε) =(λI + A) in ( (H), SOT). ε↓0 L b Proof. Let f L2(Ω) and let λ C with Re λ > 0. Let (εn) be a sequence of positive ∈ ∈ −1 real numbers such that lim →∞ ε = 0. Let n N. Set u =(λI + A n ) f. Then n n ∈ n ε d c(εn) (∂ u ) ∂ v + λ u v = f v (6) Z kl k n l Z n Z Ω k,lX=1 Ω Ω for all v V . Choosing v = u gives ∈ n

2 2 µ un + (Re λ) un Re f un f L2(Ω) un L2(Ω). ZΩ |∇ | ZΩ | | ≤ ZΩ ≤k k k k So u (Re λ)−1 f and µ u 2 (Re λ)−1 f 2 . Hence the sequence k nkL2(Ω) ≤ k kL2(Ω) Ω |∇ n| ≤ k kL2(Ω) (u ) ∈N is bounded in V . For all k 1R,...,d and n N define n n ∈{ } ∈ d (εn) wkn = ckl ∂lun. Xl=1

Then the sequence (w ) ∈N is bounded in L (Ω) for all k 1,...,d . Passing to a kn n 2 ∈ { } subsequence, if necessary, there exist u V and w ,...,w L (Ω) such that lim →∞ u = ∈ 1 d ∈ 2 n n u weakly in V and lim →∞ w = w weakly in L (Ω) for all k 1,...,d . n kn k 2 ∈{ } By (6) one has d w ∂ v + λ u v = f v Z kn k Z n Z Xk=1 Ω Ω Ω for all v V and n N. Take the limit n . Then ∈ ∈ →∞ d w ∂ v + λ u v = f v (7) Z k k Z Z Xk=1 Ω Ω Ω for all v V . We next determine the w , which requires some work. ∈ i 14 Let i 1,...,d . Let ϕ C∞(Ω). For all n N define χ(n) H1 (Rd) by χ(n)(x)= ∈{ } ∈ c ∈ i ∈ loc i χ ( 1 x). We denote by π : Rd R the k-th coordinate function for all k 1,...,d . i εn k → ∈ { } Let n N. Then ϕ(π ε χ(n)) H1(Ω) V . Moreover, (6) gives ∈ i − n i ∈ 0 ⊆ d f ϕ(π ε χ(n))= w ∂ (ϕ(π ε χ(n))) + λ u ϕ (π ε χ(n)) Z i − n i Z ln l i − n i Z n i − n i Ω Ω Xl=1 Ω

d d = w ∂ ϕ (π ε χ(n))+ w ϕ ∂ (π ε χ(n)) Z ln l i − n i Z ln l i − n i Ω Xl=1 Ω Xl=1

(n) + λ un ϕ (πi εnχi ). (8) ZΩ − The second term on the right hand side of (8) can be rewritten as

d w ϕ ∂ (π ε χ(n)) Z ln l i − n i Ω Xl=1

d = c(εn) (∂ u ) ϕ ∂ (π ε χ(n)) Z kl k n l i − n i k,lX=1 Ω

d d = c(εn) (∂ (ϕu )) ∂ (π ε χ(n)) c(εn) u ∂ ϕ ∂ (π ε χ(n)). Z kl k n l i − n i − Z kl n k l i − n i k,lX=1 Ω k,lX=1 Ω 1 1 Rd Note that ϕun H0 (Ω) H ( ) by extending the function with zero. Define vn 1 d ∈ ⊆ ∈ H (R ) by vn(x)=(ϕun)(εn x). Then vn has compact support. The first term can be simplified since

d d (εn) (n) 1 ckl (∂k(ϕun)) ∂l(πi εnχi )= ckl (∂kvn) ∂l(πi χi)=0 Z − εn ZRd − k,lX=1 Ω k,lX=1 by (5). So (8) gives

d d f ϕ(π ε χ(n))= w ∂ ϕ (π ε χ(n)) c(εn) u ∂ ϕ ∂ (π ε χ(n)) Z i − n i Z ln l i − n i − Z kl n k l i − n i Ω Ω Xl=1 k,lX=1 Ω

(n) + λ un ϕ (πi εnχi ). (9) ZΩ − d Now take the limit n . Since χ L∞(R ) it follows that →∞ i ∈ (n) lim f ϕ(πi εnχ )= f ϕ πi. →∞ i n ZΩ − ZΩ Also (n) wln(∂lϕ) χ χi ∞ wln L2(Ω) ∂lϕ L2(Ω) Z i ≤k k k k k k Ω for all n N and the sequence (w ) ∈ N is bounded in L (Ω). So ∈ ln n 2 d d (n) lim wln ∂lϕ (πi εnχi )= wl ∂lϕ πi. n→∞ Z − Z Ω Xl=1 Ω Xl=1

15 In order to evaluate the limit of the second term on the right hand side of (9), we need a lemma. Lemma 6.2. Let τ : Rd R be measurable and periodic, i.e. τ(x + γ) = τ(x) for all d d → d d x R and γ Z . Suppose that τ d L ([0, 1] ). Let Ω R be open. Let ∈ ∈ |[0,1] ∈ 2 ⊆ v, v , v ,... L (Ω) and K Ω compact. Suppose that lim →∞ v = v in L (Ω) and 1 2 ∈ 2 ⊆ n n 2 supp v K for all n N. Let (ε ) be a sequence of positive real numbers such that n ⊆ ∈ n limn→∞ εn =0. Then 1 lim τ( n x) vn = τ v. →∞ ε d n ZΩ  Z[0,1]  ZΩ Proof. The proof is similar to the proof in the one dimensional case in [Bra02, Exam- ple 2.4].

We continue with the proof of Theorem 6.1. Let k 1,...,d . Then lim un ∂kϕ = 1 d ∈ { } u ∂kϕ weakly in V , hence weakly in H (R ). Since supp ∂kϕ is a compact subset of Ω, it follows that lim u ∂ ϕ = u ∂ ϕ in L (Ω). Apply Lemma 6.2 with τ = c ∂ (π χ ). Then n k k 2 kl l i − i d d (εn) (n) lim ckl un ∂kϕ ∂l(πi εnχi )= ckl ∂l(πi χi) u ∂kϕ n→∞ Z − Z d − Z k,lX=1 Ω k,lX=1  [0,1]  Ω

d = cˆ (∂ u) ϕ, − ki Z k Xk=1 Ω where we used the definition of the homogenized coefficients and integrated by parts. The last term in (9) is easy and

(n) lim λ un ϕ (πi εnχ )= λ u ϕ πi. →∞ i n ZΩ − ZΩ Combining the limits, it follows from (9) that

d d f ϕ π = w ∂ ϕ π + cˆ (∂ u) ϕ + λ u ϕ π . Z i Z l l i ki Z k Z i Ω Ω Xl=1 Xk=1 Ω Ω

Next, choosing v = ϕ πi in (7) gives

d d f ϕ π = w ∂ (ϕ π )+ λ u ϕ π = w ∂ ϕ π + w ϕ + λ u ϕ π . Z i Z k k i Z i Z k k i Z i Z i Ω Xk=1 Ω Ω Xk=1 Ω Ω Ω Hence d w ϕ = cˆ (∂ u) ϕ. Z i ki Z k Ω Xk=1 Ω This equality is valid for all ϕ C∞(Ω). So w = d cˆ ∂ u. Then (7) gives ∈ c i k=1 ki k P d cˆ (∂ u) ∂ v + λ u v = f v Z kl k l Z Z k,lX=1 Ω Ω Ω for all v V . Therefore u dom A and (λI + A)u = f. ∈ ∈ b b 16 −1 −1 We showed that limn→∞(λI + Aεn ) =(λI + A) in the weak operator topology for all λ C with Re λ> 0. Then Theorem 3.1 gives that the limit is also valid in the strong ∈ b operator topology.

We emphasise once again that we do not assume that Ω is bounded. Strong resol- vent convergence with bounded Ω has been obtained in Bensoussan, Lions & Papanicolau [BLP78, Theorem 1.5.1] using much more work involving additional correctors. Strong resolvent convergence on Rd has been proved in [ZP05, Theorem 1.7] and for bounded Ω with Dirichlet or Neumann boundary conditions in [ZP05, Theorems 2.3 and 2.8]. A strong convergence of a slightly different nature can be found in Allaire [All92]. 1 In the case V = H0 (Ω) one has the following consequence of Theorems 6.1 and 3.1.

Corollary 6.3. For all ε (0, 1] let u L (Ω) and let u L (Ω). Further, for all ∈ 0,ε ∈ 2 0 ∈ 2 ε (0, 1] let u C(([0, ); L (Ω)) and u C(([0, ); L (Ω)) be the solutions of the ∈ ε ∈ ∞ 2 ∈ ∞ 2 diffusion equations ∂ u + A u =0 in (0, ) Ω, t n ε ε ∞ × u =0 on (0, ) ∂Ω, ε ∞ × u (0, )= u in Ω, n · 0,ε and ∂ u + Au =0 in (0, ) Ω, t ∞ × bu =0 on (0, ) ∂Ω, ∞ × u(0, )= u in Ω, · 0 respectively. Let δ, T > 0 with δ T . If lim ↓ u u =0, then ≤ ε 0 k 0,ε − 0kL2(Ω)

lim sup un(t) u(t) L2(Ω) =0. ε↓0 t∈[δ,T ] k − k

A similar corollary is valid for Neumann boundary conditions.

Acknowledgements The second-named author is most grateful for the hospitality extended to him during a fruitful stay at the TU Dresden. He wishes to thank the Institut f¨ur Analysis for financial support. Part of this work is supported by an NZ-EU IRSES counterpart fund and the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. Part of this work is supported by the EU Marie Curie IRSES program, project ‘AOS’, No. 318910.

References

[All92] G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23(6):1482–1518, 1992.

[Are01] W. Arendt. Approximation of degenerate semigroups. Taiwanese J. Math., 5:279–295, 2001.

17 [ABHN01] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander. Vector-valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Math- ematics. Birkh¨auser, Basel, 2001.

[AD07] W. Arendt and D. Daners. Uniform convergence for elliptic problems on varying domains. Math. Nachr., 280(1-2):28–49, 2007.

[AD08] W. Arendt and D. Daners. Varying domains: stability of the Dirichlet and the Poisson problem. Discrete Contin. Dyn. Syst., 21(1):21–39, 2008.

[AN00] W. Arendt and N. Nikolski. Vector-valued holomorphic functions revisited. Math. Z., 234(4):777–805, 2000.

1 [AB14] J. M. Arrieta and G. Barbatis. Stability estimates in H0 for solutions of elliptic equations in varying domains. Math. Methods Appl. Sci., 37(2):180–186, 2014.

[Att84] H. Attouch. Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.

[Bas04] A. G. Baskakov. Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators. Sovrem. Mat. Fundam. Napravl., 9:3–151 (electronic), 2004.

[BtE14] C. J. K. Batty and A. F. M. ter Elst. On series of sectorial forms. J. Evol. Equ., 14(1):29–47, 2014.

[BLP78] A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York, 1978.

[BD06] M. Biegert and D. Daners. Local and global uniform convergence for elliptic problems on varying domains. J. Differential Equations, 223(1):1–32, 2006.

[Bra02] A. Braides. Γ-convergence for beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002.

[Buc99] D. Bucur. Characterization for the Kuratowski limits of a sequence of Sobolev spaces. J. Differential Equations, 151(1):1–19, 1999.

[BB02] D. Bucur and G. Buttazzo. Variational methods in some shape optimization problems. Appunti dei Corsi Tenuti da Docenti della Scuola. [Notes of Courses Given by Teachers at the School]. Scuola Normale Superiore, Pisa, 2002.

[BV00] D. Bucur and N. Varchon. Boundary variation for a Neumann problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29(4):807–821, 2000.

[DMT96] G. Dal Maso and R. Toader. A capacity method for the study of Dirichlet prob- lems for elliptic systems in varying domains. Rend. Sem. Mat. Univ. Padova, 96:257–277, 1996.

18 [Dan03] D. Daners. Dirichlet problems on varying domains. J. Differential Equations, 188(2):591–624, 2003.

[Dan05] D. Daners. Perturbation of semi-linear evolution equations under weak assump- tions at initial time. J. Differential Equations, 210(2):352–382, 2005.

[DHD15] D. Daners, D. Hauer, and E. N. Dancer. Uniform convergence of solutions to elliptic equations on domains with shrinking holes. Adv. Differential Equations, 20(5-6):463–494, 2015.

[DL92] R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5. Springer-Verlag, Berlin, 1992. Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and H´el`ene Lanchon, Translated from the French by Alan Craig.

[DM13] P. Dr´abek and J. Milota. Methods of nonlinear analysis. Birkh¨auser Advanced Texts: Basler Lehrb¨ucher. [Birkh¨auser Advanced Texts: Basel Textbooks]. Birkh¨auser/Springer Basel AG, Basel, second edition, 2013. Applications to differential equations.

[ES10] T. Eisner and A. Ser´eny. On the weak analogue of the Trotter-Kato theorem. Taiwanese J. Math., 14(4):1411–1416, 2010.

[Eva90] L. C. Evans. Weak convergence methods for nonlinear partial differential equa- tions, volume 74 of CBMS Regional Conference Series in Mathematics. Pub- lished for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990.

[Eva98] L. C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.

[FS91] H. Fujita and T. Suzuki. Evolution problems. In Handbook of numerical anal- ysis, Vol. II, Handb. Numer. Anal., II, pages 789–928. North-Holland, Amster- dam, 1991.

[Fur10] K. Furuya. Trotter-Kato theorem for weak convergence on Hilbert space cases. Adv. Math. Sci. Appl., 20(1):143–152, 2010.

[Haa06] M. Haase. The functional calculus for sectorial operators, volume 169 of Oper- ator Theory: Advances and Applications. Birkh¨auser Verlag, Basel, 2006.

[Kat61] T. Kato. Fractional powers of dissipative operators. J. Math. Soc. Japan, 13:246–274, 1961.

[Kat80] T. Kato. Perturbation theory for linear operators, volume 132 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1980.

[Kr´o09] S. Kr´ol. A note on approximation of semigroups of contractions on Hilbert spaces. Semigroup Forum, 79(2):369–376, 2009.

19 [Mos69] U. Mosco. Convergence of convex sets and of solutions of variational inequali- ties. Advances in Math., 3:510–585, 1969.

[Mos94] U. Mosco. Composite media and asymptotic Dirichlet forms. J. Funct. Anal., 123(2):368–421, 1994.

[MNP13] D. Mugnolo, R. Nittka, and O. Post. Norm convergence of sectorial operators on varying Hilbert spaces. Oper. Matrices, 7(4):955–995, 2013.

[SN12a] P. Sa Ngiamsunthorn. Domain perturbation for parabolic equations. Bull. Aust. Math. Soc., 85(1):174–176, 2012.

[SN12b] P. Sa Ngiamsunthorn. Persistence of bounded solutions of parabolic equations under domain perturbation. J. Evol. Equ., 12(1):1–26, 2012.

[Sim78] B. Simon. A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal., 28(3):377–385, 1978.

[Sta60] G. Stampacchia. Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie h¨olderiane. Ann. Mat. Pura Appl. (4), 51:1–37, 1960.

[ZP05] V. Zhikov and S. Pastukhova. On operator estimates for some problems in homogenization theory. Russ. J. Math. Phys., 12(4):515–524, 2005.

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