PDE PROBLEMS WITH CONCENTRATING TERMS NEAR THE BOUNDARY

. In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modiﬁcation in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions.

1. Introduction. In this paper we consider several types of PDE problems in which certain terms in the equations concentrate, as a parameter ε → 0. Typically, near the boundary of the domain. This implies that the problem under consideration are subjected to singular perturbations that drastically change the nature of the problem, when passing to the limit.
In all the problems considered the goal is then to identify the form of the limit problem and to describe the process of convergence of solutions, as ε → 0.
To illustrate the type of concentrated terms we will consider, assume we have a family of functions {j ε } ε such that 1 ε ωε |j ε | r ≤ C, for some 1 ≤ r < ∞ (for r = ∞ we assume j ε L ∞ (ωε) is bounded uniformly in ε). Then we prove that, taking subsequences if necessary, there exists a function j 0 ∈ L r (Γ) (or a bounded Radon measure on Γ, j 0 ∈ M(Γ) if r = 1) such that for any smooth function ϕ, we have lim ε→0 1 ε ωε j ε ϕ = Γ j 0 ϕ. (1.2) Thus say that we have a "L r -concentrated convergent subsequence" and write 1 ε X ωε j ε → j 0 cc − L r (1. 3) and say {j ε } ε is an "L r -concentrated (sequentially) compact family". See Section 2.1. We will consider several types of PDEs with concentrating terms as in (1.2), (1.3). These terms could be non-homogeneous data, linear potentials or even linear or nonlinear terms depending on the unknowns themselves. For example, in Section 2.2, we will consider general elliptic problems in divergence form of the type −div(a(x)∇u ε ) + c(x)u ε + λu ε + 1 ε X ωε V ε (x)u ε = 1 ε X ωε f ε + g ε in Ω, a(x) ∂u ε ∂n + b(x)u ε = h ε on Γ, (1.4) where λ ∈ IR. Observe that here a non-homogeneous data, f ε and a linear potential, V ε , concentrate as in (1.3). We will find conditions on the concentrating terms to prove that assuming that the terms g ε and h ε converge in certain weak sense to g 0 and h 0 , respectively, then the solutions of (1.4) converge to the unique solution of −div(a(x)∇u) + c(x)u + λu = g 0 in Ω, a(x) ∂u ∂n + (b(x) + V 0 (x))u = f 0 + h 0 on Γ.
( 1.5) Notice that the singular concentrating terms in (1.4) transfer to boundary terms in the limit problem (1.5). Similar type of singular, concentrating, terms will be considered in further sections. For example, in Section 2.3 we will considers some elliptic nonlinear eigenvalue problems related to optimal constants in Sobolev embeddings. In Section 3 we will analyze the natural linear parabolic problems associated to (1.4) and (1.5). For these problems we will prove strong results on the convergence of the associated linear semigroups, in optimal families of Bessel-type spaces. In Section 4 we will consider nonlinear parabolic problems in which the concentrating term is the nonlinear one. Hence in this case the limit problem is a parabolic problem with nonlinear boundary conditions. Under natural dissipativity assumptions we show that the asymptotic behavior of solutions are close as ε → 0, by showing the upper semicontinuity of the family of global attractors. Then, in Section 5 we turn back to linear parabolic problems with now concentrating linear potentials and where the time derivative of the unknown also concentrates. This implies that the approximating problem is a coupled elliptic-parabolic transmission problem while the limit one has dynamic boundary conditions; that is, an evolution problem on the boundary Γ which has a nonlocal coupling with an elliptic equation in Ω. Our goal is then to give sufficient conditions on the data to prove convergence and to describe the limit process. The last two sections are devoted to damped type wave equations. As we lose the parabolic structure that implies regularization of solutions, we are bound then to rely on energy estimates and compactness arguments. First, in Section 6, the main feature is that the damping is concentrated and, therefore, the limit problem is a wave equation with boundary feedback boundary condition; in particular, the damping acts only on the boundary in the limit. In such a situation the type of approximating problems we consider, appear naturally in control theory/stabilization of waves, see [15,44,59,47,48], or in homogenization of vibration problems with inclusions near the boundary, see [51,28,29,17] and references therein. On the other hand, the limit problem appears in the boundary control theory, see [40,41,39,67,50,19] and references therein. Second, in Section 7 we explore the situation in which the damping and linear potential concentrate around a smooth, compact orientable hypersurface without boundary, M, contained in Ω, that is not touching the boundary Γ. In that case we show that the limit problem is then a wave-wave transmission problem on both sides of M coupled with an evolution problem on M. The limit problem can also be considered as a wave equation in Ω with damping concentrated only on M, see [32] for a two dimensional case. The results in this section appear for the first time in print.
Problems with concentrating terms near the boundary have been considered in the literature, some of which we are reporting about in this paper. For example linear elliptic problems have been studied in [10]. Nonlinear elliptic problems, some including oscillations in the boundary, have been considered in [12], [7], [4], [11]; see also [42]. Linear parabolic problems can be found in [55] while nonlinear ones where considered in [34], [53], [5]. Delay nonlinear parabolic problems can be found in [6], while parabolic dynamic boundary conditions can be found in [35]. Also, asymptotic behavior of non-autonomous damped wave equations have been studied in [3].
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 (1.2) one term is defined in ω ε ⊂ Ω while the limit one is defined on Γ = ∂Ω so that convergence has to be seen in a dual space of regular test functions.
2. Stationary problems. In this section we present the basic results and tools for stationary problems before approaching evolution ones.
2.1. Concentrating integrals. In this section we prove several results that describe how different concentrated integrals converge to surface integrals. Most of the results are taken from [10] where full details can be found.
i.e, s = 1 = q and r = 1 below. Then for sufficiently small ε 0 , we ii) There exist a positive constant C independent of ε and v such that for any ε ≤ ε 0 , we have In particular and lim ε→0 1 ε ωε |v| r = Γ |v| r .

2152ÁNGELA JIMÉNEZ-CASAS AND ANÍBAL RODRÍGUEZ-BERNAL
We can now analyze how concentrating integrals converge for certain families of functions which vary with ε and have weak regularity properties.
Lemma 2.2. Assume that a given family f ε defined on ω ε is an "L r -concentrated bounded family" near Γ, that is, for some 1 ≤ r < ∞ and a constant C independent of ε, . ii) For every sequence converging to zero (that we still denote ε → 0) there exist a subsequence (that we still denote the same) and a function f 0 ∈ L r (Γ) (or a bounded Radon measure on Γ, f 0 ∈ M(Γ) if r = 1) such that a) For any smooth function ϕ, defined in Ω, we have or strongly in case of equal sign in (2.2), then In other words 1 ε X ωε f ε → f 0 in H −s,q (Ω).
Also the following consequence will be used further below.
Corollary 1. i) Assume ϕ ∈ H σ (Ω) with σ > 1 2 , and denote ϕ 0 the trace of ϕ on Γ. Then for any s such that s > 1 2 and Then, by taking subsequences if necessary, there exists u 0 ∈ H 1 (Ω) such that, as ε → 0, We can also prove, Proposition 1. Assume we have a family of functions V ε , 0 ≤ ε ≤ ε 0 , satisfying the hypotheses of Lemma 2.2. Moreover, assume that (taking subsequences if necessary) there exits a function V 0 ∈ L r (Γ) (or a bounded Radon measure on Γ, V 0 ∈ M(Γ) if r = 1) such that for any smooth function ϕ, we have

2.2.
Elliptic problems and resolvent estimates. In this section we analyze the behavior, as ε → 0, of the solutions of the elliptic problem , with suitable nonhomogeneous given terms g ε , j ε and concentrating potentials V ε and concentrating non-homogeneous terms h ε . We present here results from [10] and [55].
We show below that the corresponding limit problem is the elliptic problem where the concentrating terms in (2.5) turn into boundary terms in (2.6). For the setting of problems (2.5) and (2.6) we define the elliptic operator A 0 by regarded as an unbounded operator in L q (Ω), for 1 < q < ∞, with domain given by D(A 0 ) = H 2,q bc (Ω) := {u ∈ W 2,q (Ω) : a(x) ∂u ∂n + b(x)u = 0 on Γ}. Using the complex interpolation-extrapolation procedure in [2], for which the reader is referred for further details, one can construct the scale of Banach spaces associated to this operator, which will be denoted H 2α,q bc (Ω) for α ∈ [−1, 1], which are closed subspaces of H 2α,q (Ω) incorporating some boundary conditions. In particular, we have H 0,q bc (Ω) = L q (Ω), and H 1,q bc (Ω) = H 1,q (Ω). Note that the scale with negative exponents satisfies H −2α,q bc (Ω) = (H 2α,q bc (Ω)) , for 0 < α < 1 H −2α,q (Ω) → H −2α,q bc (Ω). See [2] for details. Therefore we consider now nonsmooth perturbations of the operator A 0 . More precisely we consider a nonsmooth potential m(x) in Ω, a nonsmooth perturbation, m 0 (x) of the boundary coefficient b(x) in Γ as well as a family of concentrated perturbations near Γ.
In order to treat all perturbations in a unified form, we define for 0 < ε ≤ ε 0 , for suitable u and ϕ.
2154ÁNGELA JIMÉNEZ-CASAS AND ANÍBAL RODRÍGUEZ-BERNAL Theorem 2.3. Assume that m lies in a bounded set in L p (Ω), with p > N/2, m 0 lies in a bounded set in L r (Γ) and also that the family of potentials V ε is a L r -concentrated bounded family, for r > N − 1, that is 1 ε ωε |V ε | r ≤ C, r > N − 1.
We have then the following consequences.
Assume moreover that 1 ε X ωε h ε → h 0 cc − L q for some q > 1.
The convergence above implies also that the spectrum of the operators are close. See Corollary 4.2 and Remark 4.3 in [10] or [37] for a precise statement. In particular, we have the following Corollary 3. Assume (2.9) and denote by λ ε 1 the first eigenvalue of the eigenvalue problem Then, as ε → 0, which is the first eigenvalue of the limit eigenvalue problem 3. Sharp embeddings and nonlinear eigenvalue problems. The existence of a trace H 1 (Ω) → L q (Γ) for 1 ≤ q ≤ 2 * = 2(N − 1)/(N − 2), implies that we have the Sobolev trace inequality: there exists a constant C such that for all v ∈ H 1 (Ω). The best Sobolev trace constant is the largest C such that the above inequality holds, that is, For subcritical exponents, 1 ≤ q < 2 * , the embedding is compact, so we have existence of extremals, i.e. functions where the infimum is attained. These extremals can be taken strictly positive in Ω and smooth up to the boundary. If we normalize the extremals with Γ |u| q dS = 1, (2.11) it follows that they are weak solutions of the following problem −∆u + u = 0 in Ω, ∂u ∂ n = T q |u| q−2 u on Γ. (2.12) In the special case q = 2 (2.12) is a linear eigenvalue problem of Steklov type, see [62]. In the rest of this section we will assume that the extremals are normalized according to (2.11).
Let us consider the usual Sobolev embedding associated to the set ω ε , that is, which is continuous for exponents q such that 1 ≤ q ≤ 2 * = 2N/(N −2), see Lemma 2.1. Note that 2 * = 2N/(N − 2) is larger than 2 * = 2(N − 1)/(N − 2). The best constant associated to this embedding is given by and for q < 2 * , by compactness, the infimum is attained. The extremals, normalized by 1 ε ωε |u| q dx = 1, (2.14) are weak solutions of where χ ωε denotes the characteristic function. Therefore we are bound to study the convergence of the solutions of (2.15), (2.14) to those of (2.12), (2.11) and so the convergence of the optimal constant (2.10) to (2.13). This was analyzed in [12].
Theorem 2.4. Let Ω be a bounded, C 2 domain and let T q and S q (ε) be the best Sobolev constants given by (2.10) and (2.13).
In the critical case, q = 2 * = 2(N − 1)/(N − 2), the extremals of S q (ε) converge weakly (along subsequences) in H 1 (Ω) to a limit, u 0 , that is a weak solution of (2.12). This convergence is strong in H 1 (Ω) if and only if the limit verifies Γ u q 0 = 1 and in this case u 0 is an extremal for T 2 * .
Remark 1. Observe that in the critical case, using a sequence of minimizers and subsequences if necessary we have u ε → u 0 weakly in H 1 (Ω) and S ε (q) → T q . Also, we have Hence if u 0 is a minimizer, then Γ |u 0 | q dS ≤ 1. Conversely, if Γ |u 0 | q dS ≥ 1 then the argument above shows that this integral is actually equal to 1 and u 0 is a minimizer. Moreover in such a case, we get the convergence of the H 1 (Ω) norms and hence the strong convergence in this space.
Thus, u 0 is a minimizer if and only if Γ |u 0 | q dS = 1 which in turn is equivalent to the strong convergence.
Also, in the critical case it may happen then that one has (2.14) and Γ |u 0 | q dS < 1.
In [12] the question of radial symmetry of minimizers was also discussed and the following result was proved.
(1) For 1 ≤ q ≤ 2 and for every R, ε > 0, the extremals of (2.13) in a ball are radial functions that do not change sign. In particular, there exists a unique non negative extremal of (2.13) satisfying (2.14).
Remark 2. As a consequence of our results we get that extremals for the Sobolev trace embedding in small balls are radial. For symmetry results of extremals of Sobolev inequalities see for example, [22], [38] and references therein.
3. Linear parabolic problems. In this section we are interested in the behavior, for small ε, of the solutions of the linear parabolic problem where a ∈ C 1 (Ω) with a(x) ≥ a 0 > 0 in Ω, and b(x) a C 1 (∂Ω) function, with m ∈ L p (Ω), p > N/2 and m 0 ∈ L r (Γ), r > N − 1 and X ωε denotes the characteristic function of the set ω ε . Following [55], we will show in this section that the "limit problem" for the singularly perturbed problem above is given by Since from the results in Section 2, the solutions of the elliptic problems (2.5) converge to the unique solution of the elliptic limit problem (2.6), see Corollary 2, then it is enough to consider here the linear homogeneous problems with m ∈ L p (Ω), p > N/2 and m 0 ∈ L r (Γ), r > N − 1.
Note that if V 0 ∈ L r (Γ), for r > N − 1, with the choice V ε = 0 and m 0 + V 0 replacing m 0 , the result above allows to define the semigroup S m,m0+V0 (t) such that for every u 0 ∈ H 2γ,q bc (Ω), with γ as above, the function u(t; u 0 ) := S m,m0+V0 (t)u 0 is a weak solution of (3.3) in the sense that for all sufficiently smooth ϕ. With these notations we have Theorem 3.2. Assume that as ε → 0, (2.9) holds true and for any 1 < q < ∞, consider the semigroups S mε,m0,ε,ε (t) and S m,m0+V0 (t) as above.

Remark 3. i) The constant C(ε) in Theorem 3.2 can be estimated in terms of
where the last norm is a suitable norm in a space L(H s,q (Ω), H −σ,q (Ω)) for suitable s, σ see e.g. Proposition 1. ii) From Theorem 3.2 and Corollary 3 we have that for sufficiently small ε, in Theorem 3.1 we can take any µ > −λ 0 1 .

4.
Nonlinear parabolic problems. We analyze now the behavior, for small ε, of the solutions of the nonlinear parabolic problem where a ∈ C 1 (Ω) with a(x) ≥ a 0 > 0 in Ω and b(x) a C 1 (∂Ω) function and X ωε denotes the characteristic function of the set ω ε . Note that without loss of generality we can assume that g ε is defined on Ω × IR.
We will show below that the "limit problem" for the singularly perturbed problem (4.1) is given by where g 0 is obtained as the limit of the concentrating terms 1 ε X ωε g ε (·, u) → g 0 (·, u) as we now explain. To be more precise, observe that the nonlinear terms in (4.1) may contain zero and first order terms in u, so they can be written as with g 0 ε (x, 0) = 0, ∂ ∂u g 0 ε (x, 0) = 0, with certain regularity properties that will be made precise below.
Analogously for (4.2) we will assume where h 0 , V 0 and g 0 0 (x, u) are obtained as the limits of the concentrating terms Our goal is to prove that under assumptions (4.7) and (4.6), plus some growth and dissipativity conditions on the nonlinear terms, problems (4.1) and (4.2) have globally defined solutions for certain classes of initial data. Moreover, we are going to show that the solutions of both problems have enough compactness so that they are attracted to the global attractors, A ε , 0 ≤ ε ≤ ε 0 respectively. The global attractor for each problem contains all information about the asymptotic behavior of all solutions.
Furthermore, we are going to show that the asymptotic dynamics of (4.1) and (4.2) are close in the sense that the family of attractors A ε is upper semicontinuous at ε = 0. That is, in a suitable and strong norm which here implies, among others, uniform convergence in Ω for the functions and convergence of the derivatives in Lebesgue spaces. The results in this section are taken from [33,34].
Observe that the approach for upper semicontinuity has grounds in, e.g. Section 2.5. in [30], see also [61], and requires the following ingredients. First, we must prove that all problems have attractors and that they are uniformly bounded with respect to the parameter 0 ≤ ε ≤ ε 0 . Then we must prove that the nonlinear semigroups defined by (4.1) converge as ε → 0 to the one defined by (4.2). This in turn, will be obtained from the convergence of solutions for the corresponding linear equations, see [55].

4.1.
Well posedness for nonlinear problems. In this section we give some results on the well posedness for both problems (4.1) and (4.2). For these we use the results in [8] adapted to the particularities of problems (4.1) and (4.2) mentioned above. Also note that we will make use of the semigroups described in Section 3 with boundary potential m 0 = 0.
Hence we consider (4.1) and (4.2) in the space X = L q (Ω) or X = H 1,q bc (Ω) = H 1,q (Ω), for 1 < q < ∞. For either choice of X there exist suitable growth restrictions on the nonlinearities, such that problems (4.1) and (4.2) are locally well posed in X. For this we consider the following class of nonlinear terms N X Definition 4.1. The class N X is formed up with functions j(x, u) such that i) j(x, ·) : IR → IR is locally Lipschitz, uniformly on x ∈ Ω or x ∈ Γ ii) If X = L q (Ω), assume that c) if q > N , no further conditions are assumed.
Then the techniques from [8] applied here give the following result.
In order to ensure that the local solutions constructed above are globally defined, following [9], we will assume the following sign conditions on the nonlinear terms and either i) for (4.1), Remark 4. Observe that comparing (4.3) with (4.10), (4.4) with (4.11) and (4.5) with (4.12), we get for either X = L q (Ω) or X = H 1,q bc (Ω).

Existence of attractors and uniform bounds.
In this section we give conditions that allow to prove that the nonlinear semigroups defined by problems (4.1) and (4.2) in Theorem 4.3 have global attractors A ε and A 0 respectively and to obtain suitable uniform bounds on A ε independent of ε. For this, by Lemma 2.2, we will assume that in (4.11) (4.14) Therefore we will also assume the following dissipativity condition.
Then there exist a constant K ∞ and a function R ∞ (M, t), for M, t > 0, independent of ε such that for each fixed M > 0, R ∞ (M, t), is monotonically decreasing and converges to zero, as t → ∞ and such that for sufficiently small 0 ≤ ε ≤ ε 0 , the global solutions of problems (4.1) and (4.2) in Theorem 4.3, satisfy that for initial data such that In particular, for any M > 0, With this and the smoothing effect of the equations we get for all x ∈ Ω, and |u| ≤ R. Then, for any 1 < ρ < ∞ and γ < 1 2 + 1 2ρ there exists a constant K ρ,γ and a function R ρ,γ (M, t), for M, t > 0, independent of ε such that for each fixed M > 0, R ρ,γ (M, t), is monotonically decreasing and converges to zero, as t → ∞ and such that for sufficiently small 0 ≤ ε ≤ ε 0 , the global solutions of problems (4.1) and (4.2) in Theorem 4.3, satisfy that for initial data such that In particular, Therefore, the global semigroups defined by problems (4.1) and (4.2) in Theorem In particular the attractors are uniformly bounded in H 1,ρ bc (Ω) and C ν (Ω) for any 1 < ρ < ∞ and for any 0 < ν < 1.
Remark 5. From here on we can assume that the nonlinear terms are globally Lipschitz and the semigroups T ε (t) and T 0 (t) are defined on L ρ (Ω) for any 1 < ρ < ∞. In particular, the attractors A ε attract solutions in the norm of H 2γ ,ρ bc (Ω) for any 1 < ρ < ∞ and γ < 1 2 + 1 2ρ . Now since the nonlinear semigroups T ε (t) and T 0 (t) are order preserving and the estimates above, from Theorem 3.2 in [56], see also [18], we get the existence of extremal equilibria for problems (4.1) and (4.2) which are the caps of the attractors Proposition 2. Under the above notations and hypotheses, uniformly in x ∈ Ω and for initial data u 0 such that u 0 L q (Ω) ≤ M .

Concentrated nonlinear terms.
In this section, we prove two technical results that will allow to pass to the limit in nonlinear terms which are concentrating near the boundary as ε → 0. For this, we consider a family of functions g 0 ε : Ω × IR −→ IR, for 0 ≤ ε ≤ ε 0 , satisfying the following conditions i) {g 0 ε (x, u)} ε is uniformly bounded in Ω on bounded sets of IR, i.e. for any R > 0 there exists a positive constant C(R) independent of ε such that ii) {g 0 ε (x, u)} ε is uniformly continuous in Ω, uniformly on bounded sets of IR and also uniformly Lipschitz on bounded sets of IR, i.e. for any R > 0 there exists a positive constant L(R) independent of ε such that ε (x, u) converges to g 0 0 (x, u) uniformly on Γ and on bounded sets of IR, i.e. for any R > 0 g 0 ε (x, u) → g 0 0 (x, u) as ε → 0, uniformly on x ∈ Γ and |u| ≤ R (4.18) Then we have the following result. Note that here p and q are not meant to be the same as in previous sections. Also, the result below applies in the case g 0 ε = g 0 0 , that is, when the family does not depend on ε. for 0 ≤ ε ≤ ε 0 . Also, consider a family of functions, C, in Ω such that, for some , then there exists a positive constant, M (R), independent of ε such that for every 1 < q < ∞ and any ϕ ∈ H s,q (Ω) with s > 1 q and every v ∈ C we have In particular sup v∈C .17) and (4.18), then there exists M (ε, R) → 0 as ε → 0 such that for every ϕ ∈ H 1,q (Ω) and v ∈ C (4.20) In particular 1 ε X ωε g 0 ε (·, v) → g 0 0 (·, v) in H −1,q (Ω), uniformly in v ∈ C.
In particular, we get and Finally, consider a family C as in Lemma 4.6, that is satisfying (4.19). Then we have that for any 1 < q < ∞ and 1 q < s ≤ 1 i) There exists C > 0 independent of ε > 0 such that

4.4.
Upper semicontinuity of attractors. With all the above we can then obtain the convergence of the nonlinear semigroups. Note that although the nonlinear problems (4.1) and (4.2) are set in the space X = L q (Ω) or X = H 1,q bc (Ω) as in Section 4.1, depending on the growth of the nonlinear term, the convergence results below always take place in H 1,ρ bc (Ω) for any 1 < ρ < ∞. Lemma 4.7. Fix any M > 0 and t 0 > 0 and consider any initial data such that u 0 L q (Ω) ≤ M and denote u ε = T ε (t 0 )u 0 .
Then, for any 1 < ρ < ∞ and any T > 0, there exists a constant C(M, T, ε) → 0 if ε → 0, such that for ε ∈ (0, ε 0 ), We are now in a position to prove the upper semicontinuity of the family of attractors.
Theorem 4.8. Under the above assumptions, for any 1 < ρ < ∞, the family of global attractors of (4.1) and (4.2), A ε , is upper semicontinuous at ε = 0 in In particular, we get the upper semicontinuity of equilibria Corollary 5. i) For every sequence ε k with ε k → 0 as k → ∞ and for every sequence of equilibria ϕ ε k ∈ A ε k there exists a subsequence (that we denote the same) and a equilibrium point ϕ 0 ∈ A 0 such that ϕ ε k → ϕ 0 , k → ∞ in H 1,ρ bc (Ω) for any 1 < ρ < ∞. ii) In particular, considering the extremal equilibria in Proposition 2, we obtain that Dynamic boundary conditions. Dynamic boundary conditions have the main characteristic of involving the time derivative of the unknown. They have been used, among others, as a model of "boundary feedback" in stabilization and control problems of membranes and plates, [15,40,41,39,44,66], in phase transition problems, [65,24,25,26,49,16], in some hydrodynamic problems, [27,63] or in population dynamics, [21]. They have also been considered in the context of elliptic-parabolic problems, [20,57]. Also several of so called "transmission problems" have been described and analyzed in [60], some of which lead, under some singular perturbation limits, to problems with dynamical boundary conditions.
In this section our goal is to prove that dynamic boundary conditions can be obtained as the singular limit of elliptic/parabolic problems in which the time derivative concentrates in a narrow region close to the boundary.
Hence we consider the following family of parabolic problems where X ωε is the characteristic function of the set ω ε and λ ∈ IR. Then, following [35], we show that the limit problem is the following parabolic problem with dynamic boundary conditions where v 0 , V and g are obtained as the limits of the concentrating terms Notice that all concentrating terms in (5.1) are transferred, in the limit, to the boundary condition in (5.2). Note that (5.1) is formally equivalent to solving and that in (5.3) boundary conditions are missing on Γ ε = ∂ω ε \ Γ = ∂(Ω \ω ε ).
Since there would be several ways of connecting the solutions of the elliptic and the parabolic equations in (5.3) along that boundary, we consider the boundary conditions on Γ ε that ensure maximal smoothness of solutions. This is achieved by imposing the classical transmissions conditions on Γ ε , that is, no jump of the u ε and its normal derivate across Γ ε , see [58], Hence, (5.3) and (5.4) is a formulation of an elliptic-parabolic transmission problem, see [46], Chapter 1, Section 9, for related problems.
On the other hand, (5.2) must be understood as an evolution problem on the boundary Γ, such that, for each time t > 0, the solution must be lifted to the interior of Ω by means of the elliptic equation in (5.2). In this way the term ∂u 0 ∂ n , which is the so called Dirichlet Neumann operator, becomes a linear nonlocal operator for functions defined on Γ.
Here and below H s (Ω) denote, for s ≥ 0, the standard Sobolev spaces and for s > 0 we denote H −s (Ω) = H s (Ω) .
Also H −1 0 (Ω) will denote the dual space of H 1 0 (Ω). 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 H −1/2 (Γ) will denote the dual space of H 1/2 (Γ). We will also use the embeddings We will often find below some elements in H −1 (Ω) for which we will employ the notation h = f Ω + g Γ 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.

5.1.
The approximating parabolic problems. Note that in [58] a very similar problem to (5.1) was considered. In fact in [58] Dirichlet boundary conditions were assumed on Γ instead as Neumann ones and also V ε = 0. Therefore, we modify the arguments in [58] to apply them to (5.1). See Theorem 1.1, Theorem 4.9 and Proposition 4.10 in [58]. Also, temporarily, we remove the dependence on ε.
Then in a similar fashion as in Theorems 1.1 and 4.9 in [58], we have the following result that states the well-posedness of (5.6).

Time dependent concentrating integrals.
In this section we show several results that describe how different concentrated integrals converge to surface integrals. Hereafter we denote by C > 0 any positive constant such that C is independent of ε and t. This constant may change from line to line.
The results in Section 2.1 can now be extended to handle concentrating integrals including a time dependence. B) Consider a family g ε defined on (0, T ) × ω ε , such that for some 1 < q < ∞, 1 ≤ r < ∞ and a positive constant C independent of ε, x)| q dt ≤ C for the case r = ∞. Then, for every s satisfying s − N 2 > − N −1 r , and for every sequence converging to zero (that we still denote ε → 0) there exists a subsequence (that we still denote the same) and a function g ∈ L q ((0, T ), L r (Γ)) (or a bounded Radon measure on Γ, g ∈ L q ((0, T ), M(Γ)) if r = 1) such that where X ωε is the characteristic function of the set ω ε . In particular, for any smooth function ϕ, defined in [0, T ] ×Ω, we have Also, if u ε → u 0 strongly in L q ((0, T ), H s (Ω)) then 2170ÁNGELA JIMÉNEZ-CASAS AND ANÍBAL RODRÍGUEZ-BERNAL C) Consider a family g ε defined on (0, T ) × ω ε , and assume that for some 1 < r, q < ∞, there exist h ∈ L q (0, T ), and g ∈ L q ((0, T ), L r (Γ)) such that for σ, s as in (2.4). If ϕ ∈ C([0, T ] ×Ω), (5.9) holds for any q > 1 and s > 1 2 .
Now we prove the following result that will be used below in the analysis of (5.1) and (5.2). Note that the assumption on the potentials below is, not only uniform in ε, but more restrictive in ρ than the one needed for fixed ε, as in (5.5), i.e. ρ > N/2.
for any smooth function ϕ defined inΩ and for some function V ∈ L ρ (Γ), see Lemma 2.2. Then i) There exists some λ 0 ∈ IR, independent of ε > 0, such that for λ > λ 0 the elliptic operator, associated to the parabolic problems (5.1) and (5.2), are positive.
In particular, for every ϕ ∈ L 2 ((0, T ), H 1 (Ω)) we have We will finally make use of the following result.
Lemma 5.5. Assume the family of potentials V ε is as in Lemma 5.3. Also, assume u ε is as in Lemma 5.4, that is, satisfies (5.10) and (5.11), and let u 0 be as in the conclusion of Lemma 5.4. Then if s is such that

5.4.
Passing to the limit. We analyze the limit of the solutions of the parabolic problems (5.1), with 0 ≤ ε ≤ ε 0 . For this we will assume that the data of the problem satisfy, for each ε > 0 the assumptions in the first part of Theorem 5.1 with h ε = f ε + 1 ε X ωε g ε and the following uniform bounds in ε > 0: for some constant C independent of ε.
Hence, we have the following result that improves the convergence in Theorem 5.6. Then if u ε and u 0 are as in Theorem 5.6, we have that in addition to the convergence in Theorem 5.6 we have now that u ε converges to u 0 , weak * in L ∞ ((0, T ), H 1 (Ω)) and weakly in H 1 ((0, T ), H −1 (Ω)) and strongly in C([0, T ], H −1 (Ω)). Also If additionally ρ > 2(N − 1) then u ε converges to u 0 also in L 2 ((0, T ), H 1 (Ω)). 6. Damped wave equations. Now we consider some singular perturbation of a forced wave equation where the damping region to be concentrated in a neighborhood of the boundary that shrinks to the boundary as ε → 0. To be more precise, we consider the following family of damped wave equations for λ > 0 and T > 0 fixed. Then, we are going to show that the limit problem is the following damped wave equation with boundary feedback damping 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 Notice that again all concentrating terms in (6.1) are transferred, in the limit, to the boundary condition in (6.2). The results here are taken from [36].
6.1. The approximating damped wave equations. Here we consider (6.1) for 0 < ε ≤ ε 0 which we write as x). This, in turn, can be written as Note that in [58] a very similar problem to (6.1) was considered but with Dirichlet boundary conditions on Γ instead of Neumann ones as in this section. Then in a similar fashion as in Theorem 5.1, Theorem 5.2 in [58], we have the following result that states the well-posedness of (6.4).
Also, the mild solutions of (6.4) in part i) of Theorem 6.1 possess the following properties.

2176ÁNGELA JIMÉNEZ-CASAS AND ANÍBAL RODRÍGUEZ-BERNAL
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 (6.9) can be written as To handle the case g = 0, following [57], 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 [57]. In this way the solution of the limit problem (6.9) are given by the following result which relates them with the mild solutions in E of the dual equation Observe that a strict solution U * = (u, w) ⊥ of this equation satisfies which can be written as and is a weak formulation of (6.9) in the sense that for every φ ∈ H 1 (Ω) and a.e. t ∈ (0, T ), Indeed, as in Theorem 2.3 in [57] we get the following.
Observe that Theorem 6.2 suggest that when going from E into E we employ the following linear injective (not onto) "change of variables", see [57] for more details, From the above theorem we can make the following definition. H 1 (Ω)) such that u t ∈ C([0, T ], H −1 (Ω)) and γ(u) ∈ C([0, T ], H −1 (Ω)) is a mild solution of (6.9) if u t + γ(u) has a weak derivative in H −1 (Ω) and satisfies a.e. t ∈ [0, T ] Now, we will show that mild solutions as in Definition 6.3 are given by Theorem 6.2.
In particular, this mild solution is unique and satisfies the energy equality (6.12).
Concerning further regularity, as in Theorem 2.4 in [57] we have the following result that allows to construct strict solutions of (6.9).
In particular, i) (Further regularity) ii) (Energy estimate for U t ). U t satisfies the energy equality for 0 < τ < T where E 0 is the energy functional given by (6.8) Remark 6. i) Note that (6.13) is a weak formulation, in H −1 (Ω), of the condition ii) Under the hypotheses of Proposition 7 we have u(t) ∈ Y 0 and u tt (t) ∈ L 2 (Ω) then u satisfies (6.9) in the sense that 6.3. Passing to the limit. We analyze here the limit of the solutions of the hyperbolic problems (6.1), with 0 < ε ≤ ε 0 .

6.3.2.
Convergence of strict solutions. Now we impose stronger assumptions than (6.14)-(6.16) on the data and obtain stronger convergence of solutions than in Theorem 6.4. In particular, we obtain convergence of strict solutions. With the notations in Theorem 6.4, we consider the initial data u ε 0 ∈ H 2 N (Ω), v ε 0 ∈ H 1 (Ω) satisfying the following uniform bounds We also assume that the nonhomogenous terms satisfy f ε W 1,1 ((0,T ),L 2 (Ω)) ≤ C and f ε W 1,p ((0,T ),H −1 (Ω)) ≤ C, where 1 < p < 2 and g ε ∈ H 1 ((0, T ), L 2 (ω ε )) with where C is a positive constant independent of ε. We will also assume the compatibility condition of the initial data Then by taking subsequences if necessary, we can assume (6.17) and moreover strongly in L 2 (Ω) and see Corollary 1.
On the other hand, on f ε and g ε by taking subsequences if necessary, we can assume (6.18) and (6.19) and moreover the convergence given in the following result.
The following result shows that the mild solutions u ε (t) and u 0 (t) of (6.1) and (6.2), respectively, constructed in Sections 6.1 and 6.2, are actually strict solutions. Lemma 6.6. With the assumptions above, the function u ε (t) in Theorem 6.4 is a strict solution of (6.1) as in part ii) in Theorem 6.1.
Also, the function u 0 (t) in Theorem 6.4 is a strict solution of (6.2) as in Proposition 7.
Hence, we have the following result that improves the convergence in Theorem 6.4. Theorem 6.7. With the notations above, as ε → 0, .
In particular, with E 0 the energy functional defined by (6.8).
Now we show that strong convergence of the initial data in the energy space, E = H 1 (Ω) × L 2 (Ω) implies convergence of the solution (u ε , u ε t ) → (u 0 , u 0 t ) in L 2 ((0, T ), E). From the convergence in Theorem 6.7, it is enough to show the convergence u ε → u 0 in L 2 ((0, T ), H 1 (Ω)).

7.
Concentrating terms away from the boundary. In this section we explore the potential use of the techniques in previous sections to analyze problems with singular terms concentrating away from the boundary. These cases will reflect a different nature in the limit problem which does not influence the boundary conditions.
For this we will consider in an open bounded smooth set in IR N , Ω, with a C 2 boundary, Γ = ∂Ω, an embedded smooth, compact, orientable hypersurface M ⊂ Ω such that Ω \ M has two components, an interior one Ω 1 that does not touch Γ and ∂Ω 1 = M and an exterior one Ω 2 such that ∂Ω 2 = Γ ∪ M.
Next, we define, for sufficiently small ε > 0, 0 < ε ≤ ε 0 , a neighborhood of M where n(x) denotes the normal vector at a point x ∈ M outwards from Ω 1 . As before X ωε denotes the characteristic function of the set ω ε . We will also denote by γ(u) the trace on M of a regular enough function defined in either Ω 1 or Ω 2 .
Hence, we consider the following family of damped wave equations in Ω (7.1) with λ > 0 and T > 0 fixed.
We will show that the corresponding limit problem, as ε → 0 is now given by the wave equation with damping on M where [F ] denotes the jump of the function F across M, 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 1 2ε This last convergence will be achieved in a similar manner as in Sections 2.1 and 5.3. We start with the following technical lemma that will be used below. Recall that we assume λ > 0.
. Notice the similarities with the operator considered in Section 6.1.
Our first result now concerns the homogeneous problem (7.6) with h ε = 0.
Proof. Observe that for U = (u, v) ⊥ ∈ D(A ε ) using the scalar product associated to the norm (7.3), we have Since D(A ε ) is clearly dense in E then to conclude the proof it is enough to show that R(A ε + I) = E. For this A ε U + U = (j, k) ⊥ ∈ E is equivalent to u − v = j and −∆u + λu + 1 2ε Hence −∆u + (λ + 1)u + 1 2ε X ωε u = k + j − 1 2ε X ωε j which has a unique solution u ∈ H 2 (Ω) ∩ H 1 0 (Ω)). Hence v ∈ H 1 0 (Ω) and we get the result.
We also get the following characterization of the mild solutions of (7.6) in part i) of Theorem 7.2. This results uses the characterization of the functions given by the variations of constant formula (7.7) in [14] and is similar to Proposition 5.3 in [58].
We now show that mild solutions in Theorem 7.2 also satisfy a natural energy equality.

2184ÁNGELA JIMÉNEZ-CASAS AND ANÍBAL RODRÍGUEZ-BERNAL
Then U = (u, u t ) ⊥ satisfies the energy equality for 0 < τ < T , where E ε 0 is given by (7.3). Proof. As usual, we argue by density. First, assume the solution is smooth enough such that u ∈ H 2 (Ω) ∩ H 1 0 (Ω) and u t ∈ H 1 0 (Ω). Part ii) in Theorem 7.2 gives sufficient conditions on the data for this assumption to hold true.
By the Lipschitz dependence of mild solutions in part i) of Theorem 7.2 we have which implies in particular U n (t) E → U (t) E , and taking into account that E ε 0 (u, v) is an equivalent norm in E, see Lemma 7.1, we get E ε 0 (u n , u n t )(t) → E ε 0 (u, u t )(t) as n → ∞. Also, 1 2ε X ωε u n t → 1 2ε X ωε u t in C([0, T ], L 2 (Ω)) as n → ∞ and 1 2ε Finally, since u n t → u t in C([0, T ], L 2 (Ω)) and h n → h in L 1 ((0, T ), L 2 (Ω)) we get τ 0 Ω h n u n t → τ 0 Ω hu t , and passing to the limit as n → ∞ in (7.10) we obtain the energy equality for the mild solution U (·, U 0 , h).

7.2.
Convergence of mild solutions. We analyze here the limit of the solutions of the hyperbolic problems (7.1), with 0 < ε ≤ ε 0 . For this we will obtain uniform energy estimates and, by compactness, study the limt as ε → 0. For this we will assume that the data of the problem satisfy the following assumptions: (i) u ε 0 H 1 (Ω) ≤ C, v ε 0 L 2 (Ω) ≤ C (7.12) and, by taking subsequences if necessary, as ε → 0,
Proposition 13. The operator −A generates a C 0 semigroup in E = H 1 (Ω) × L 2 (Ω), denoted S(t), which is a semigroup of contractions for the norm E 0 in E given in (7.4).
Hence the mild solutions of the limit problem (7.37) are given by the strict solutions in E of the dual equation.
Then U * = (u, w) ⊥ ∈ C([0, T ], E ), w = u t + γ(u), and U (t) = (u, u t ) ⊥ satisfies i) (Regularity). U = (u, u t ) ⊥ ∈ C([0, T ], E), γ(u) ∈ C([0, T ], H 1 2 (M)) ∩ H 1 ((0, T ), L 2 (M)) ∩ L ∞ ((0, T ), L 2 (M)) and u tt ∈ L 1 ((0, T ), H −1 (Ω)). ii) (Energy equality). U satisfies the energy equality E 0 (u(τ ), u t (τ )) + 2 8. Some further research. The problems presented here suggest some further research along the following lines. First, rates of convergence of solutions have been obtained for the resolvent estimates and the linear semigroups in Sections 2 and 3. It seems plausible to use some general techniques to derive rates of convergence for nonlinear problems and their attractors, see e.g. [13]. Also lower semicontinuity of attractors of parabolic problems, under generic hyperbolicity conditions of equilibria seem within reach; see [31] for an approach that has been successfully applied in different instances. For problems in Sections 5, 6 and 7 it is interesting to study the spectral continuity (or stability) as ε → 0. Observe that in the case of Section 5, the limit eigenvalues are given by the Steklov eigenvalue problem, see [62,42]. In case of Section 6 a related important problem is that of the uniform stabilization of waves by the localized/boundary damping, see [32] and references therein. For Section 7 stronger convergence of solutions needs to be explored. Also, the convergence for the associated nonlinear problems seems relevant. Finally, problems in which also the second order time derivative concentrates may deserve some attention.