BOUNDARY FEEDBACK AS A SINGULAR LIMIT OF DAMPED HYPERBOLIC PROBLEMS WITH TERMS CONCENTRATING AT THE BOUNDARY

. In this paper we show how solutions of a wave equation with distributed damping near the boundary converge to solutions of a wave equation with boundary feedback damping. Suﬃcient conditions are given for the convergence of solutions to occur in the natural energy space.

1. Introduction. In this paper we consider some singular perturbation of a forced wave equation with localized damping. The singular perturbation consists in the damping region to be concentrated in a neighborhood of the boundary. This neighborhood is of width ε and shrinks to the boundary as ε → 0. Also we consider that some part of the forcing is also concentrated in that region. Our goal is then to describe the limit equation and to analyze the convergence of solutions.
To be more precise, we consider the following family of hyperbolic problems for λ > 0 and T > 0 fixed, in Ω (1) where Ω is an open bounded smooth set in IR N with a C 2 boundary Γ = ∂Ω and X ωε is the characteristic function of the set ω ε defined as for sufficiently small ε, say 0 ≤ ε ≤ ε 0 , where n(x) denotes the outward normal vector at a point x ∈ Γ. Hence, the set ω ε is a neighborhood of Γ in Ω, that collapses to the boundary when the parameter ε goes to zero. The given functions f ε and g ε are defined, respectively, in Ω × (0, T ) and ω ε × (0, T ).
As ω ε shrinks to the boundary as ε → 0, the goal in this work is to show that negative feedback boundary conditions can be obtained as a result of this limiting process. More precisely, the main result in this work is to prove that the family of solutions, u ε , converges in some sense, when the parameter ε goes to zero, to a limit function u 0 , which is given by the solution of the following hyperbolic problem with boundary feedback damping term where u 0 , v 0 , f are obtained as the weak limits of initial data u ε 0 , v ε 0 and f ε , while g is obtained as the limit of the concentrating terms in some sense that we make precise below. In particular, we will obtain that the time derivative of the solution concentrates to the time derivative of the restriction to the boundary, as ε → 0. Notice that all concentrating terms in (1) are transferred, in the limit, to the boundary condition in (2). Different problems with concentrating terms near the boundary have been considered before. For example linear elliptic problems have been studied in [6]. Nonlinear elliptic problems, some including oscillations in the boundary, have been considered in [7], [5], [2]; see also [19]. Linear parabolic problems can be found in [27] while nonlinear ones where considered in [14], [26], [3]. Other type of problems have also been studied. For example, delay nonlinear parabolic problems can be found in [4], while parabolic dynamic boundary conditions can be found in [15]. Also, asymptotic behavior of non-autonomous damped wave equations have been studied in [1].
Finally, similar eigenvalue problems to the ones associated to (1) appear in homogenization of vibration problems with inclusions near the boundary, see [25,12,13,10] and references therein.
In all these examples a common feature is that concentrating terms near the boundary give rise, in the limit, to a boundary term. The form of the boundary term depends on the problem under consideration. Also, this influences the way the solutions of the approximate problem converge to those of the limit one. Notice that one source of difficulties is that in (3) one term is defined in ω ε ⊂ Ω while the other is defined on Γ = ∂Ω so that convergence has to be seen in a dual space of regular test functions.
In the present paper, we build up a suitable functional setting for both problems (1) and (2) in Section 2 and we construct mild and strict solutions for both problems. Mild solutions are constructed in such a way that they satisfy the natural energy estimates that control the norm of the solution in the natural energy space E = H 1 (Ω) × L 2 (Ω) and shows the energy dissipative effect of the damping term of both (1) and (2), given by the term 1 ε X ωε u ε t and u 0 t on the boundary respectively, see Theorem 2.1, Proposition 2 and Theorem 2.2. Strict solutions require stronger regularity of the data and have additional regularity. Then in Section 3 we rely on energy estimates and classical compactness arguments, e.g. [21], to obtain mild solutions of (2) as weak limits of the mild solutions of (1), see Theorem 3.1. Then under stronger regularity assumptions on the data we prove in Theorem 3.4 and Proposition 5 that strict solutions of (1) actually converge in the energy space L 2 ((0, T ), E) and also that Some required results on the convergence of concentrating terms as in (3) are collected in Appendix A.

2.
On the well-posedness of the approximating and limit problems. In this section we prove the well-posedness results for both the approximating and limit problems (1) and (2), respectively. For this we will make use of minor variations of the results in [28,29].
Here and below H s (Ω) denote, for s ≥ 0, the standard Sobolev spaces and for s > 0 we denote their duals as Also, H −1/2 (Γ) will denote the dual space of H 1/2 (Γ).
Finally, we will consider below traces on Γ of functions defined in Ω. Hence, we will denote by γ(u) the trace of a function u and denote by γ the trace operator on H s (Ω) → H s− 1 2 (Γ) for s > 1 2 , and we use the embeddings H 2.1. Well-posedness of approximating damped hyperbolic problem. Here we consider (1) for 0 < ε ≤ ε 0 which we write as This, in turn, can be written as Note that in [29] a very similar problem to (1) was considered but with Dirichlet boundary conditions on Γ instead of Neumann ones as in this paper. Then in a similar fashion as in Theorem 5.1, Theorem 5.2 in [29], we have the following result that states the well-posedness of (5). Notice that no proofs are given, as the change in the boundary conditions does not introduce significant differences in the proofs. Theorem 2.1. i) (Existence of solutions). If h ∈ L 1 ((0, T ), L 2 (Ω)) and U 0 = (u 0 , v 0 ) ⊥ ∈ E = H 1 (Ω) × L 2 (Ω) then there exists a unique mild solution, U (t) = (u, v) ⊥ of (5) satisfying U (0) = U 0 , which is given by the variation of constants formula where S ε (t) is a C 0 semigroup of contractions generated by the operator −A ε in E and H(t) = (0, h(t)) ⊥ . In this case, U ∈ C([0, T ], E) and U (0) = U 0 or equivalently Moreover, the mapping (U 0 , h) → U is Lipschitz between E × L 1 ((0, T ), L 2 (Ω)) and C([0, T ], E).
Moreover, in the first case for h, U t = (u t , u tt ) ⊥ with u t (0) = v 0 and u tt (0) = − 1 ε X ωε v 0 + ∆u 0 − λu 0 + h(0), is a mild solution of (5) in E, with right hand side H t , that is Observe that in case ii) of Theorem 2.1, u satisfies the PDE (4) in Ω and the boundary condition in Γ in a point-wise sense.
We now show that mild solutions in Theorem 2.1 also satisfy the energy equality.
for 0 < τ < T , where E 0 is the energy functional given by Proof. As usual, we argue by density. First, assume the solution is smooth enough such that u ∈ H 2 N (Ω) and u t ∈ H 1 (Ω). Part ii) in Theorem 2.1 gives sufficient conditions on the data for this assumption to hold true.
, E) and, as above U n satisfies (10) for every n.
By the Lipschitz dependence of mild solutions in part i) of Theorem 2.1 we have and passing to the limit in (10) we obtain the energy estimate for the mild solution U (·, U 0 , h).

2.2.
Well-posedness of the limit problem. Now we consider the hyperbolic problem (2), that is Note that a very similar problem was considered [28] where the boundary Γ was assumed to be split into two regular subsets Γ = Γ 1 ∪ Γ 0 . Then Dirichlet boundary conditions were assumed on Γ 0 and dynamic boundary conditions on Γ 1 . Therefore we adapt here the results in [28] to our setting. We will often find below some elements in H −1 (Ω) for which we will employ the notation where f and g are functions defined in Ω and on Γ respectively. This will denote the functional defined by for all sufficiently smooth function φ in Ω. Thus, we define the normal derivative operator as follows: for every . This can be recast as with L as in (9).
In what follows we will denote by U = (u, v) a generic element of E, while U * = (u, w) will denote a generic element in E .
Then if g = 0 problem (13) can be written as To handle the case g = 0, following [28], we proceed as follows. By transposition, −A * generates in E the C 0 semigroup S * (t), i.e. the transposed semigroup of S(t), and is given by see Lemma 2.1 in [28]. In this way the solution of the limit problem (13) are given by the following result which relates them with the mild solutions in E of the dual equation with which can be written as and is a weak formulation of (13) in the sense that for every φ ∈ H 1 (Ω), a.e. t ∈ (0, T ). Indeed, as in Theorem 2.3 in [28] we get the following. Again no proofs are given since the arguments follow line by line those in [28].
Remark 1. We observe that from (16) despite H(s) is not in D(A * ) (unless g = 0), nor is regular in time we still have U * (s) is in D(A * ) for 0 < s < T . This is due to the particular form of H and a subtle smoothing effect of the semigroup. Also note that u tt ∈ L 1 ((0, T ), H −1 (Ω)) and from the energy estimates (17) we get γ(u) t ∈ L 2 ((0, T ) × Γ) and therefore in (18) we have where the derivative is to be understood as weak derivative, i.e. as d dt for every φ ∈ H 1 (Ω).
Observe that Theorem 2.2 suggest that when going from E into E we employ the following linear injective (not onto) "change of variables", see [28] for more details, From the above theorem we can make the following definition.
In particular, this mild solution is unique and satisfies the energy equality (17).
Proof. First, note that if U * (t) and U (t) = (u, u t ) ⊥ are given as in Theorem 2.2, then u is a mild solution of (13) as Definition 2.3.
Conversely if u is a mild solution as in Definition 2.3, we consider U * (t) = (u(t), w(t)) with w(t) = u t (t) + γ(u(t)) and we will prove below that U * is the mild solution of the dual equation (15) and satisfies (16) ). For this we will use the characterization in [8] of the variation of constants formula for mild solutions.

5134ÁNGELA JIMÉNEZ-CASAS AND ANÍBAL RODRÍGUEZ-BERNAL
In particular, the mild solution is unique and since Theorem 2.2 applies, then the energy estimate (17) holds true.
Concerning further regularity, as in Theorem 2.4 in [28] we have the following result that allows to construct strict solutions of (13).

Then
Moreover, u t is a mild solution of (13) in E, with right hand sides f t and g t in Ω and Γ, respectively.

Remark 2. i) Note that (20) is a weak formulation, in H −1 (Ω), of the condition
ii) Under the hypotheses of Proposition 4 we have u(t) ∈ Y 0 and u tt (t) ∈ L 2 (Ω) then u satisfies (13) in the sense that 3. Passing to the limit as ε → 0. We analyze here the limit of the solutions of the hyperbolic problems (1), with 0 < ε ≤ ε 0 .
Proof. We proceed in several steps. Below we will use K a generic constant that does not depend on ε.
In the remaining steps, we will pass to the limit as ε → 0.
Step 4. In this part, we study the convergence of u ε t to u 0 t and prove (30). i) Now, we will prove First, from (38) and taking another subsequence if necessary, there exists v * ∈ L ∞ ((0, T ), L 2 (Ω)) such that (Ω)) as ε → 0. Second, we will prove that v * = u 0 t , and we get (43). In effect, for every φ(t, ·) smooth such that φ(T, ·) = φ t (T, ·) = 0 and integrating by parts we have From the convergence of u ε 0 , u ε and u ε t we get Thus, we have v * = u 0 t ∈ L ∞ ((0, T ), L 2 (Ω)) and we conclude (43) and (30). ii) In what follows we will prove First, from (41), we obtain that for t i ∈ [0, T ] and we get {u ε t } ε is equicontinuous with values in H −1 (Ω). Next, we also note that for every t ∈ [0, T ] fixed, we have u ε t (t, ·) is uniformly bounded in L 2 (Ω) ⊂ H −1 (Ω) with compact embedding. Therefore from Ascoli-Arzela's Theorem there exists a subsequence which converge to a limit function in C([0, T ], H −1 (Ω)). Finally, we note that this limit function must be u 0 t and we conclude.
In particular, for Step 5. Now we study the convergence of u ε tt to u 0 tt and prove (31). In fact from (41) we obtain a subsequence that u ε tt → v * weakly in L p ((0, T ), H −1 (Ω)), with 1 < p < 2 as in (25). Analogously to (44) we have v * = u 0 tt . Step 6. We will prove now u 0 is a mild solution of (13) as in Definition 2.3.
Step 7. Notice that we actually proved above that any sequence of the family u ε (t) has a subsequence that converges to u 0 (t), the unique solution of (13) as Definition 2.3. Hence all the family converges.

3.2.
Convergence of strict solutions. Now we impose stronger assumptions than (21)-(23) on the data and obtain stronger convergence of solutions than in Theorem 3.1. In particular, we obtain convergence of strict solutions.
We will also assume the compatibility condition of the initial data Then by taking subsequences if necessary, we can assume (24) and moreover see Lemma A.1 iii).
The following result shows that the mild solutions u ε (t) and u 0 (t) of (1) and (2), respectively, constructed in Sections 2.1 and 2.2, see in (27) and (28), are actually strict solutions. Lemma 3.3. With the assumptions above, the function u ε (t) in Theorem 3.1 is a strict solution of (1) as in part ii) in Theorem 2.1.
Also, the function u 0 (t) in Theorem 3.1 is a strict solution of (2) as in Proposition 4.
Hence, we have the following result that improves the convergence in Theorem 3.1.