Domain perturbation and invariant manifolds

We consider a reaction-diffusion equation defined on a sequence of bounded open sets ( n)n∈N, converging to in the sense of Mosco, and we prove stability of invariant manifolds of the flux with respect to domain perturbation.


Introduction
Consider the following reaction-diffusion problem P ω (u 0 ) : where ω is a bounded open set in R N without any smoothness conditions on the boundary, and f is a function of class C 1 (R), globally Lipschitz continuous. The problem P ω generates then a flux of solutions on R + . In this paper, we study the stability of this flux with respect to geometrical perturbations of the domain ω. More precisely, we consider a sequence of open sets ( n ) n∈N uniformly included in D, where D is a fixed ball. This sequence converges to an open set ⊂ D in the sense of Mosco, defined by the following two conditions: ∀u ∈ H 1 0 ( ), ∃u n ∈ H 1 0 ( n ) such that u n Let u n k ∈ H 1 0 ( n k ), u n k This convergence is exactly equivalent to the convergence of solutions of the elliptic problem [20]: This convergence is general enough, and in particular, it covers convergence used by , which was defined in a geometrical way. For instance, in dimension two, if the number of connected components of c n is uniformly bounded and if c n converge to c for the Hausdorff metric, then n converge to in the sense of Mosco [22]. In dimension N > 2, this result has been generalized by a condition of flat cone on the boundary of n [7,8]. It is well known that if ( n ) n∈N converges to in the sense of Mosco, the solutions u n of the system P n (u 0n ) converge for all T > 0 in L 2 ((0, T ), H 1 0 (D)) to the solution u of P (u 0 ) as soon as (u 0n ) n∈N weakly converges in L 2 (D) to u 0 [13,14,21]; here we agree to extend the function u n (respectively u) by zero outside n (resp. ) with the same notation. Of course, this convergence is not uniform with respect to time. We are now interested in studying the stability of the structure of the flux when t → ∞. It was noted by Dancer [13] that if the problem P (u 0 ) admits a hyperbolic pointū, then for n large enough, P n (u 0n ) admits a hyperbolic pointū n , and the sequence of these stationary points converges toū with respect to n. Moreover, the stable and unstable manifolds associated toū n converge for the Hausdorff metric to the manifolds-respectively stable and unstable-ofū. The works of Bates et al. [4][5][6] deal with existence and persistence of invariant manifolds for more general dynamical systems. Applied to our problem, this allows to prove persistence and stability of the local stable and unstable manifolds around one hyperbolic point [4,5], and also persistence and stability of the local central unstable manifolds if the flux on the manifold is attractive or repulsive. The work of Kostin [19] on the invariant manifold for discrete semigroup implies in our situation persistence of local central unstable manifolds if we suppose existence of a stationary point for each problem P n , such that the sequence of those points converges to the stationary point of P . Vol. 12 (2012) Domain perturbation and invariant manifolds 549 In this paper, we suppose that P (.) has a stationary pointū, which is not necessarily hyperbolic, and we do not make any assumption on the problems P n . We know that there exists M, a local central unstable manifold that containsū [18,Theorem 8.5.1], and we prove (Theorem 4.1) that for n large enough, there exists a local central unstable manifold M n for P n such that the sequence M n converges to M in the Hausdorff metric. Of course, there does not necessarily exist a sequence (ū n ) n∈N of stationary points for P n , converging toū. Using the argument of the proof of this result, we find again the result concerning the perturbed manifold for hyperbolic points, stated by Dancer.
The plan of this paper is as follows: in Sect. 2, we introduce the notation, in particular the extension operator and the degenerate semigroup. Section 3 is devoted to the application of the result of Chow and Lu [11] to the present situation. In Sect. 4, we present and prove the main result about stability of the manifold. Finally, Sect. 5 contains remarks and open problems.

Notations
Let D be a ball in R N , which will contain all open sets considered here. Let ω be an open set. Since we consider perturbation of domains, it is necessary to extend the functions of L 2 (ω) (respectively H 1 0 (ω)) to functions of L 2 (D) (resp. H 1 0 (D)) so that all the solutions belong to the same space. Let p be the canonic extension operator defined as It is well-known that for all domains ω, p is an isometric continuous linear operator from L 2 (ω) into L 2 (D) and from H 1 0 (ω) into H 1 0 (D). Thus, it allows us to identify L 2 (ω) (resp. H 1 0 (ω)) with a closed subspace of L 2 (D) (resp. H 1 0 (D)) endowed with the induced topology. The convergence in the sense of Mosco (1) is then well defined. Let B be a sectorial operator in L 2 (ω) with its domain included in H 1 0 (ω), see [18] for a definition. The operator −B is then a generator of a semigroup e −Bt . The resolvent operator of B at λ, denoted by R(λ, B), which is in L(L 2 (ω)), can be extended to an operator R p (λ, B) in L(L 2 (D)) in the following way: let f ∈ L 2 (D) where is a contour in ρ(−B) with argλ → ±θ as |λ| → ∞ for some θ ∈ ( π 2 , π). We call {T (t)} t≥0 a degenerate semigroup [1]. Moreover, for all u ∈ L 2 (D), we have In this paper, we will denote by T (t) the degenerate semigroup of B. In the rest of this section and in Sect. 4, we will always identify L 2 (ω) (resp. H 1 0 (ω)) with the closed subspace of L 2 (D) (resp. H 1 0 (D)), and in order to simplify the notation, we will denote by v the function p(v), omitting the symbol p. On the contrary, in the Sect. 3, we will make the difference between the two notations in order to describe how we extend the manifolds into the space H 1 0 (D) (see 3.3). We consider − ω the Dirichlet Laplacian in its domain We denote by R(λ, − ω ) the corresponding resolvent operator at λ. We will call a solution to P ω (u 0 ) a continuous function of [0, +∞) into L 2 (ω), which satisfies in (0, +∞) the following integral equation: It is known that that for every u 0 ∈ L 2 (ω), there exists a unique solution u(t, u 0 ) of P ω (u 0 ) [18]. The flux of solutions is the application S ω : . Following the definition of Henry [18, Definition 6.1.1], M ⊂ L 2 (ω) will be called a local invariant manifold of the problem P ω if for all x 0 ∈ M, there exists a solution u(.) of (4) on an open interval (t 1 , t 2 ) containing 0 such that u(0) = x 0 and u(t) ∈ M for t 1 < t < t 2 . Sometimes M will be called invariant manifold of the flux of Eq. (4), or just of the flux (4). We will say that u(t) is a solution of (4) on an interval J of R if for all t 0 , t ∈ J, t 0 ≤ t, y(t) = u(t + t 0 ) is solution of (4) with u 0 = u(t 0 ). The set of stationary solutions of the problem P ω , denoted by S P(ω), is the set of solutions in H 1 0 (ω) of the elliptic equation We will use in the sequel the notation If ( n ) n∈N is a sequence of open sets converging as n → ∞ in the sense of Mosco to an open set as we will consider in the sequel, then R(λ, − n ) converge to R(λ, − ) in the uniform operator topology of L(L 2 (D)) [10]. Also, for each h ∈ L 2 (D) we have: The compact injection of H 1 0 (ω) into H 1 0 (D) guarantees that the spectral set of (− , D ω (− )) is composed of eigenvalues, which can be arranged in a nondecreasing sequence  [9,10]. Let J ⊂ R be an interval. For any η ∈ R and any Banach space E, we denote by C η (J, E) the following Banach space The distance between a point and a set M of E is given by The ball in E of center x and radius ρ will be denoted by B E (x, ρ). Let ( ρ ), 0 < ρ < ρ 0 be the family of applications, which to a pair of sets M 1 , M 2 in E associates the real number Let (ρ i ) i∈N be a strictly increasing sequence that converges to ρ 0 , the application is the distance between closed subsets of B E (x, ρ) for the topology induced from E. The spaces L 2 (ω) and H 1 0 (ω) are endowed with their usual norms, and we denote by (., .) the scalar product in L 2 (D).

The Chow and Lu's theorem
Givenū a solution of the stationary problemū ∈ S P( ), we shall prove now, using the result of Chow and Lu [11,Theorem 4.4], that there exists a local invariant manifold containingū, and this manifold is the limit in the sense of Hausdorff of a sequence of local invariant manifolds for the dynamical system P n .
Let us recall the Chow and Lu result, restricted to our situation. Let X, Y be Banach spaces such that X is continuously embedded in Y and let S(t) be a strongly continuous semigroup of bounded operators on Y . Consider the following assumptions: 552 N. Varchon J. Evol. Equ.
Let F ∈ C 1 (X, Y ) and consider the following integral equation Then x(t, x 0 ) is the solution of (9), which is equal to x 0 at time t = 0. We have the following result.
THEOREM 3.1 (Chow and Lu, see [11], Theorem 4.4). Let η < 0. Assume that Then there exists a C 1 invariant manifold M for the flow defined by (9) and M satisfies: Under the additional condition that We apply now the result of the previous section to our flow. Letv be any point in Here L ω is the linearized operator at pointv defined by and e −L ω t is the semigroup generated by −L ω . The mapping g associates to [18] with compact resolvent R(λ, L ω ). The spectrum of L ω again consists of eigenvalues, which can be arranged into a nondecreasing sequence (λ k (L ω )) k∈n∈N that are greater than is the first eigenvalue of the Dirichlet Laplacian on D. The semigroup e −L ω t enjoys the following properties. PROPOSITION 3.4. e −L ω t is a strongly continuous semigroup of bounded operators which, for x ∈ L 2 (ω), t > 0, and δ > 0, satisfies the following estimates where Proof. Since L ω is a sectorial operator, the semigroup generated by −L ω is strongly continuous and bounded for all t ≥ 0 [18]. Let (e i ) i∈N be the sequence of eigenvectors of L ω associated to the eigenvalue λ i = λ i (L ω ). In this basis, we write (13) and We obtain the first inequality by writing that To show the second one, recall that and that Then, we obtain the second inequality by combining the previous two equations and using the fact that |e −λt − 1| ≤ e ct for t ≥ 0 and λ > −c.
Let l be an integer such that λ l−1 (L ω ) < λ l (L ω ). The spectrum of L ω can be written as a partition Let P ω 1 , P ω 2 be the spectral projections associated to this decomposition [18]. The operator P ω 1 is nothing else than the orthogonal projection from L 2 (D) into its proper subspace generated by the l − 1 first eigenvectors, and P ω be the trivial extensions of those projections to L 2 (D), defined by The following proposition states that the semigroup verifies the conditions required for the application of Chow and Lu's theorem. Proof. The first four assumptions come directly from the properties of the spectral projections [18]. We will show that the fifth one is also satisfied. In this proof, we omit the index ω to simplify the notations. We first note that for all t ≤ 0 We then obtain the first estimate of (H 5 ) by using the fact that e −L ω t P 1 y 2 . For the second one, we observe that for t > 0 (Le −Lt P 2 y, e −Lt P 2 y) ≤ be −2bt (L ω P 2 y, P 2 y), and (e −Lt P 2 y, e −Lt P 2 y) ≤ e −2bt y 2 Vol. 12 (2012) Domain perturbation and invariant manifolds 555 and the result follows. For proving the last one, we recall the for all λ > b With the second inequality of (16), we obtain The estimates are then satisfied with REMARK 3.6. • Once the constants a, b, c, η, α, β are chosen, the terms M, M * are independent of the choice of ω and ofv as soon as the following inequality is satisfied • If we substitute T ω (t), the degenerate semigroup of L ω , to e −L ω t , replace the spaces L 2 (ω), H 1 0 (ω) by L 2 (D), H 1 0 (D), respectively, and the operators P ω 1 , P ω 2 by Q ω 1 , Q ω 2 , respectively, then the conditions (H 2 ), (H 3 ), (H 5 ) are still satisfied, with the same constants. The condition (H 1 ) should then be written as: "T ω (0)L 2 (D) = Y 1 ⊕ Y 2 ," and the condition (H 5 ): "T ω (t) can be extended to a degenerate group" that is defined in the same way as the degenerate semigroup.
As it will be proved in Lemma 3.11, G ρ ∈ C 1 (H 1 0 (ω), L 2 (ω)) and Lip (G ρ ) → 0 as ρ → 0. For all ρ > 0, the integral equation  (0, ρ), then for all t > 0 such that where, as usually, w(t, w 0 ) is the solution of (18) equal to w 0 at t = 0. The next theorem states that the flux (18) has a l − 1-dimensional global invariant manifold M ρ for l, verifying the condition (C 1 ) below and that the flow of P ω has a l − 1-dimensional local invariant manifoldv + M loc ρ .
THEOREM 3.7. Assume that ω andv are such that then there exists M ρ , global C 1 invariant manifold for the flow (18), which satisfies: is a nonempty local C 1 invariant manifold for the flow defined by the Eq. (10), and thenv + M loc ρ is a local invariant manifold of P ω . REMARK 3.8. • We will show in Lemma 3.11 that the assumption (C 2 ) is realized for ρ small enough. • We note that in the previous theorem, the manifolds M ρ , M loc ρ andv + M loc ρ are (l − 1)-dimensional. REMARK 3.9. Existence of the manifold M ρ is given by Theorem 3.1 above. In Chow and Lu [11], the manifold is constructed in the following way: for every ζ ∈ P ω 1 L 2 (ω), we have h ω (ζ ) = P ω 2 (ϕ(ζ )(0)) where ϕ(ζ ) is the unique fixed point belonging to C η (R − , H 1 0 (ω)) of the mapping J ω defined by Vol. 12 (2012)

Domain perturbation and invariant manifolds 557
The mapping J ω is a uniform contraction in C η (R − , H 1 0 (ω)) with respect to the variable ζ with a contraction coefficient smaller than Lip (G ρ )K (α, β, M, M * ). Hence for every w 0 ∈ M ρ , there exists w(t, w 0 ) solution of (18) in R − such that w(t, w 0 ) ∈ C η (R − , H 1 0 (D)) and w(t, w 0 ) is a solution of the integral equation We now prove Theorem 3.7.
Proof. We fix α, β, η to be as in the hypotheses of Proposition 3.5, and for ρ small enough for (C 2 ) to be realized, we apply Theorem 3.1. Then we know that there exists a global C 1 invariant manifold M ρ for the flow (18), which satisfies (G 1 ) and (G 2 ). It is now sufficient to prove that M loc ρ is nonempty. With the Eq. (21), for So for every ζ ∈ P ω 1 L 2 (ω) such that we infer that w(0, w 0 ) H 1 0 (ω) < ρ, and then M loc ρ is not empty. REMARK 3.10. Let us make three remarks about this theorem.
1. In the proof, when we show that M loc ρ is nonempty, we prove actually that where θ is such that one has Lip (G ρ )K (α, β, M, M * ) < θ < 1. We have then the same result. In this theorem, we choose θ = 1 2 that is less general, but does not change the final result and simplifies the notation. 3. For every two-part spectral decomposition of L ω such that λ l (L ω ) > 0 (see (15)

Extension of the manifolds
The manifold M ρ is a subset of H 1 0 (ω). This space becomes a closed subspace of H 1 0 (D) by trivial extension by zero outside ω. We wish that solutions of (21) were fixed points of a contraction in C η (R − , H 1 0 (D)). In this aim, we introduce the mapping This mapping is Lipschitz continuous and verifies It follows that J p ω is a contraction in C η (R − , H 1 0 (D)) with a contraction coefficient smaller than Lip (G p ρ )K (α, β, M, M * ). We also have the following lemma. LEMMA 3.11. For all ρ > 0 and all n ∈ N, the mappings G ρ and G n ρ belong to C 1 (H 1 0 (D), L 2 (D)). Moreover, for all ε > 0, there exist ρ > 0 and N ρ , which depend only on ρ such that To prove the first part of the lemma, it is sufficient to show e. to 0 as t goes to 0 and are bounded by 2c|ϕ|. So, We prove by contradiction that f is a continuous mapping from By the compact embedding of H 1 0 (D) into L 2 (D), ( f (u n ) − f (u))h n converges, up to a subsequence, a.e. to zero and is uniformly bounded by a L 2 (D) function. (25) is then a contradiction with the dominated convergence theorem. Using the same argument, we show that f is uniformly continuous in the bounded subset of H 1 0 (D), and so, for all ε > 0 there exists ρ > 0 and N ρ , which depend only on ρ such that for all v ∈ B H 1 0 (D) (0, ρ) sup and ∀n > N ρ , sup Here, as in the rest of the proof, We infer that If we choose ρ such that g (v) L(H 1 0 (D),L 2 (D)) < ε(1 + 2 sup |ψ |), as g(0) = 0, we have sup B(0,2ρ) g(v) L 2 (D) ≤ ε(1 + 2 sup |ψ |). We thus obtain Let now N ρ be such that for all n > N ρ , g n (v) L(H 1 0 (D),L 2 (D)) < ε(1 + 2 sup |ψ |) then for n > N ρ we have also and the result follows.
We infer that for every ζ ∈ Q ω 1 L 2 (D), Thanks to (24), if J p ω is strictly contractive, then J ω is also strictly contractive. Hence, by Lemma 3.11 and since for every ζ ∈ Q ω 1 L 2 (D) we have J p ω (ζ, p(ϕ)) = J p ω ( p(ζ | ω ), p(ϕ)), we conclude that the unique fixed point of J p ω (ζ, .) is p(ϕ), where ϕ is the fixed point of J ω (ζ | ω , .). Consequently, M ρ considered as a subset of In the following section, we will omit the symbol p to simplify the notations, but we always work in the functional space defined on D. The mapping G ρ will be always replaced by the mapping G p ρ , it does not involve any problem because the condition (C 2 ) occurs for G p ρ as soon as it occurs for G ρ .

Stability of the manifolds
In what follows,ū ∈ S P( ). We denote by L the operator L = − − f (ū) with the domain D(L) = D(− ). The main result of this paper is the following theorem.  We will prove this result in two steps. In the first one, we show the existence of local manifolds of P and P n . In the second one, we prove the convergence result.
Step 1: existence. Let a, b be two nonnegative real numbers such that and let α, β, η be real numbers verifying the hypotheses of Proposition 3.5. Now M and M * are fixed. We apply Theorem 3.7 at ω = andv =ū. There exists ρ 0 such Vol. 12 (2012) Domain perturbation and invariant manifolds 561 that the condition (C 2 ) is satisfied (see Lemma 3.11). So there exists M ρ 0 (l − 1)dimensional global manifold of the flow (18), which satisfies (G 1 ) and (G 2 ). We will note in what follows H instead of H . Sinceū is a fixed point (K = 0), the condition (C 3 ) is obviously satisfied and 0 ∈ M ρ 0 . We then conclude that M, defined by   For all the sequence (v n ) n∈N , we have f (v n ) −−−→ f (ū), and by passing to the limit in the equation above we infer that (ū n ) n∈N converges in H 1 0 (D) toū. So, we also have f (ū n ) because v n = u n a.e. outside n . We infer the result. Before showing convergence of the manifolds, we introduce some new notations. We note L n instead of L n = − n − f (ū n ), T n (t) for the degenerate semigroup of L n , Q n i instead of Q n i (i = 1, 2), G n ρ for the nonlinearity in the Eq. (18) and K n instead of K n = − nū n − f (ū n ).
Step 2: convergence. Assume that the following lemma is proved: we postpone its proof to Sect. 4.3. LEMMA 4.4. Let (ζ n ) n∈N be sequence such that ζ n ∈ Q n 1 H 1 0 (D), converging in H 1 0 (D) to ζ . Then ζ ∈ Q 1 H 1 0 (D) and H n (ζ n ) In order to end the proof of Theorem 4.1, it is sufficient to show that for all ρ < ρ 0 , we have ρ (M n , M) → 0. We begin with showing that the first term in the maximization converges to zero. Let (x n ) n∈N be a maximizing sequence, there exists ζ n ∈ H 1 0 (D) such that ζ n H 1 0 (D) ≤ ρ and such that x n =ū n + ζ n + H n (ζ n ). Up to a subsequence, and thanks to the fact that Q n 1 converge uniformly to Q 1 (Corrolary 4.7), The end of this section is devoted to the proof of results that are used in the proof of Theorem 4.1.

Fundamental stability results
In this part, we will prove on one hand the theorem about the convergence of eigenvalues and eigenspaces that are used in the proof of the Theorem 4.1 and, on the other hand, we will prove various results about the convergence of the semigroup (T n (t)), which will be used in the proof of Lemma 4.4. We keep the notations and hypotheses of the previous part.
In the following Theorem 4.5, we state essential facts about spectral convergence on which we base our method. We refer, for example, to [15] for the proof (see Corollaries 4.3 and 4.7). Note also that we can find a proof of a similar theorem in [3,9]. Analogous results with different hypotheses on convergence are proven in [1], and in [2] for Neumann boundary conditions. In what follows, we will note  Proof. The first statement comes from the second one by the triangle inequality. Let us thus prove the second statement. Let B be a ball in the complex plane containing uniquely the eigenvalues λ i (L) for i from 1 to l − 1. Then By Theorem 4.5, this last equality is also true with Q n 1 instead of Q 1 and L n instead of L for n large enough. The uniform convergence of resolvent operators implies that Proof. First of all, in Lemma 4.4, the fact that ζ ∈ Q 1 H 1 0 (D) is an immediate consequence of Corollary 4.7. Indeed, Q n 1 converge uniformly to Q 1 so Q n 1 ζ n L 2 (D) −−−→ Q 1 ζ and ζ = Q 1 ζ . We show now a key result of the proof of (29). Recall that with the notation of Chow and Lu, the operator F defined by is a bounded linear operator from C η (R − , L 2 (D)) into C η (R − , H 1 0 (D)) such that F ≤ K (α, β, M, M * ).
We define F n in the same way for the semigroup T n (t). Then we have the following result.