ON HYPERBOLIC MIXED PROBLEMS WITH DYNAMIC AND WENTZELL BOUNDARY CONDITIONS

. We study mixed hyperbolic systems with dynamic and Wentzell boundary conditions. The boundary condition contains a tangential operator which is strongly elliptic on the boundary. We prove results of generation of strongly continuous groups and well-posedness.


1.
Introduction. The aim of this paper is the study of a problem in the form Roughly speaking (precise assumptions will be given in the following), A(x, D x ) is a strongly elliptic differential operator in divergence form in the bounded domain Ω with smooth boundary ∂Ω; γf is the trace of f on ∂Ω; ∇ τ is the tangential gradient in ∂Ω, ∇ τ · is the divergence operator in ∂Ω, B(x ) is a positive definite symmetric operator in the tangent space T x (∂Ω), with x ∈ ∂Ω, F (x , D x ) is a linear differential operator of order not exceeding one (not necessarily tangential) and coefficients defined in ∂Ω.
(1) is strictly connected with the problem formally obtained replacing in (1) D 2 t γu(t, x ) in the second equation with the trace of the second term in the first equation. In case (2) one usually speaks of Wentzell boundary conditions. A physical interpretation of (2) is given in [10], Chapter 6.
In our knowledge, problems (1) and (2) have been always considered in the particular case that where we indicate with ∂ ∂ν A the conormal derivative associated with A(x, D x ). See, for example, [1], [9], [12], often connected with problems of control.
The most general results are contained in [2], where F (x , D x ) is in the form (3) with β(x ) > 0 which is allowed (to some extent) to be unbounded and with infimum equal to 0. The authors do not even assume that the coefficients of A(x, D x ) and B(x ) are continuous; they need to be just measurable and bounded. They work in the basic space L 2 (Ω, dµ) := L 2 (Ω) × L 2 (∂Ω, dS/β). with F (x , D x ) as in (3). They show that a certain operator connected with (1) and (2) is self-adjoint and upper bounded. This allows to formulate theorems of well-posedness in a certain generalized sense. They consider also the case when D 2 t is replaced by D 2 t + aD t (this is the telegraph equation).
Roughly speaking, in this paper we want to show that, at least in case of "regular coefficients" for A(x, D x ) and B(x ), (1) and (2) are well posed whenever the operator F (x , D x ) has bounded and measurable coefficients in ∂Ω.
This is the plan of this paper: Section 2 is dedicated to the proof of Theorem 2.1. We begin by considering a particular case, with In this situation the result is essentially known (see for this also [3]), but we have decided to give a complete proof in order to make the paper more or less self-contained. The general statement is obtained by combining an estimate of the conormal derivative of the solution to a hyperbolic Cauchy-Dirichlet system (see Theorem 2.5) with a perturbation theorem of Miyadera type (Theorem 2.6). The estimate is inspired by a nice result due to I. Lasiecka, J. L. Lions, R. Triggiani (see [8]) The final Section 3 contains developments and applications of Theorem 2.1 to a generalization of (1), and to (2).
To conclude this preliminary section, we describe some notations we are going to use.
If Ω is a domain with smooth boundary and x ∈ ∂Ω, we shall indicate with ν(x ) the unit normal vector to ∂Ω in x , pointing outside Ω, with ∂ ∂ν the corresponding normal derivative. T x (∂Ω) will be the tangent space to ∂Ω in x and T (∂Ω) the tangent bundle. If A is the differential operator C will indicate a positive constant we are not interested to precise. In a sequence of formulas we shall write C 1 , C 2 , . . . . If the constants depend on T , we shall write C(T ), C 1 (T ), . . . . If X and Y are normed spaces, we shall indicate with L(X, Y ) the space of linear bounded operators from X to Y . If X = Y , we shall write L(X). If V is a Hilbert space, we shall indicate with V * the space of antilinear bounded functionals in V .
2. The main theorem. As we said, in this section we shall study a simplified version of (1). We begin by stating our assumptions.
(A1) Ω is an open bounded subset of R n lying on one side of its boundary ∂Ω, which is a submanifold of R n of dimension n − 1 and class C 2 . ( for any x ∈ Ω, ξ ∈ R n , for some ν positive. (A3) ∀x ∈ ∂Ω B(x ) is is symmetric and positive definite element of L(T x (∂Ω)).
Of course, H is a Hilbert space with the usual scalar product where σ is the standard Riemannian measure in ∂Ω. We set also We equip V with the scalar product We introduce the following operator The main result if this section is the following We begin the proof of Theorem 2.1 by recalling the well known procedure of identifying the element (f, h) of H with the element J(f, h) of V * defined as The following result is well known (for a proof, see [11], Chapter 2.2).
Proof. We consider the operator A 1 : W → H, It is immediately seen, employing Green's formula, that On the other hand, as A 0 is self adjoint and positive, −A 0 is the infinitesimal generator of an analytic semigroup in H. By Theorem 4.1 in [6], the same happens for A 1 . So ρ(A 0 ) ∩ ρ(A 1 ) = ∅. This, together with (5), implies the conclusion.
. Now we are able to employ the following result (see [7], Theorem 7.2): Theorem 2.4. Let B be the infinitesimal generator of a strongly continuous group in the Banach space X. Assume that 0 ∈ ρ(B). Define Then M 0 is the infinitesimal generator of a strongly continuous group in the Banach space D(B) × X.
Corollary 1. Suppose that (A1)-(A4) hold. We introduce the following operator Then M 0 is the infinitesimal generator of a strongly continuous group in V × H.
Proof. We set B := iA We shall indicate with (e tM0 ) t∈R the group generated by M 0 in V × H.
Remark 3. If (φ, ψ, α, β) belongs to V ×H and its components are real valued, then the components of e tM0 (φ, ψ, α, β) are real valued. In case (φ, ψ, α, β) ∈ W × V , This can be easily deduced from the uniqueness of the solution of the problem which follows from Corollary 1. The general case follows by continuity. Then So φ is also the solution of the mixed Cauchy-Dirichlet problem Now we want to replace − ∂· ∂ν A − γ with an essentially arbitrary differential operator of order not exceeding one. To this aim, the key fact is the following result, following the idea of [8], Theorem 2.1. For a slightly different situation, see also [5].
Proof. We continue to employ the notation (7). Concerning the proof, it suffices to consider the case that the components of (u 0 , γu 0 , u 1 , γu 1 ) are real valued, so that, by Remark 3, φ is real valued. We set We introduce a function h ∈ C 1 (Ω; R n ) such that, for each j ∈ {1, . . . , n}, if x ∈ ∂Ω, If b ∈ H 1 (Ω), we have, by Green's formula, with We have also that in (a, b) × ∂Ω, for j = 1, . . . , n, employing the notation (7), with T j differential operator of order one in ∂Ω, with coefficients in C 1 (∂Ω). Then, by Remark 4, Now, We have Moreover, by (9), So, by (11), Moreover, We deduce that We have By (10), we have , with S 1 differential operator of order one in ∂Ω, while with S 2 differential operator of order one in ∂Ω. From (12) we deduce and, as A(x ) 2 is lower bounded by a positive constant, for some C 0 positive independent of u 0 , u 1 , Then there exists C(T ) positive, independent of u 0 and u 1 , such that Proof. Let g(x ) = (g 1 (x ), . . . , g n (x )).
So the conclusion follows from Theorem 2.5.
Now we recall the following perturbation result of Miyadera type (see [4], Corollary 3.16): Theorem 2.6. Let A be the infinitesimal generator of a strongly continuous semigroup (T (t)) t≥0 on a Banach space X and let B ∈ L(D(A), X) satisfy, for some Then A + B, with domain D(A), is the infinitesimal generator of a strongly continuous semigroup in X.