SHARP REGULARITY THEORY OF SECOND ORDER HYPERBOLIC EQUATIONS WITH NEUMANN BOUNDARY CONTROL NON-SMOOTH IN SPACE

. The purpose of this paper is to complement available literature on sharp regularity theory of second order mixed hyperbolic problem of Neu- mann type [13, 15, 26] with a series of new results in the case–so far rather unexplored–where the Neumann boundary term (input, control) possesses a regularity below L 2 (Γ) in space on the boundary Γ. We concentrate on the cases H − 12 (Γ)), H − β (Γ)), H − 1 (Γ)), β being a distinguished parameter of the problem. Our present results are consistent with the sharp result of [13, 15, 26] (obtained through a pseudo-diﬀerential/micro-local analysis approach), whose philosophy is expressed by a gain of β in space regularity in going from the boundary control to the position in the interior. A number of physically relevant illustrations are given. Abstract result on time-lifting regularity [17, Theorem 7.3.1, p 651] Let X and U be reﬂexive Banach spaces with X ∗ and U ∗ their adjoints. For given 0 < T < ∞

1. Problem formulation, literature. Orientation The purpose of this paper is to complement available literature on sharp regularity theory of second order mixed hyperbolic problem of Neumann type [13,15,26] (see also [17, p 739]) with a series of new results in the case-so far rather unexplored-where the Neumann boundary term (input, control) possesses a regularity below L 2 in space on the boundary. We let throughout dim Ω ≥ 2. Though the starting point of our present analysis will be the sharp regularity theory available in [13,15,26], the results given in this paper are, except for Theorem 4.1, not explicitly contained in these references. Our present effort is prompted and stimulated by the recent papers [1], [2]. More precisely, [2] studies the shape differentiability and sensitivity analysis for the solution of the wave equation on a bounded domain with an active Neumann boundary term with respect to the shape of the geometric domain on which the wave equation is defined. In doing so, [2] needs a regularity result that guarantees that the wave equation will have H 1 -space interior regularity in the position. One such resultwith highly asymmetric assumptions on the Neumann boundary datum-is given in [1], with an ad-hoc proof which uses Galerkin techniques. Such regularity result of [1]-to be recalled as Theorem 1.3 below-is not sharp. It is however in line with the philosophy of the original treatment in [21, p 120] which yields a gain of regularity in space from the Neumann boundary term to the solution (position) in the interior of about 1 2 -space derivative. See the precise statement in (1.3) below. Why the result of [1] is within the philosophical setting of [21] can be explained if one factors-in that, for the wave equation, one derivative in time corresponds to one derivative 490 ROBERTO TRIGGIANI in space; see Remark 1.4 below Theorem 1.3. A completely different, soft proof of the (non-optimal) regularity result of [1] is given in the Appendix: starting from the original abstract model of the wave equation with Neumann boundary controlwhich was introduced originally in [27] and was extensively used by the authors of [9] to culminate in the treatment of the monograph [17]-it only employs basic elliptic theory and sine/cosine theory, in a purely functional analytic approach. It has been known since [13] that technical tools other than pseudo-differential microlocal analysis techniques are not suitable, and are not expected, to provide sharp regularity results for wave equation mixed problems in dimension greater than or equal to 2, with Neumann boundary control, a case where the Lopatinski condition is not satisfied. In contrast, in the case of second order hyperbolic mixed problems with Dirichlet boundary control-where the Lopatinski condition is satisfied-differential multipliers suffice to obtain sharp/optimal interior and boundary regularity [8].
[Such reference provided a cleaner proof by duality of the basic level regularity result of [9], originally obtained for an L 2 (Σ)-Dirichlet boundary control through a more complicated analysis of the non-homogeneous problem.] Problem formulation. Let Ω be an open bounded domain in R n , n ≥ 2, with sufficiently smooth boundary Γ. We consider a second order hyperbolic problem defined on Ω with Neumann boundary control g acting on Γ: y t=0 = y 0 , y t t=0 = y 1 in Ω (1.1b) In (1.1a), A(ξ, ∂) is a (time-independent) partial differential operator of order two in Ω, with smooth real coefficients: a ij (ξ) ∂ 2 ∂ξ i ∂ξ j + n j=1 b j (ξ) ∂ ∂ξ j + c 0 (ξ) (1.2a) and with principal part uniformly elliptic in Ω: a ij (ξ)η i η j ≥ c n j=1 η 2 i , a ij = a ji , c > 0, (1.2b) whereby then g is applied to the corresponding co-normal derivative, denoted by ∂ ∂ν . As noted, this is the generality of references [13,15,26], whose results will be the starting point of our present analysis. Goal The purpose of this paper is to provide sharp regularity theory for problem (1.1a)-(1.1c)-fully consistent with the philosophical setting of references [13,15,26] noted below in Theorem 1.1 in the new and presently considered case where the Neumann boundary control is assumed to possess a space regularity (on Γ) below the L 2 (Γ)-level; for instance, H − 1 2 (Γ), or H −1 (Γ) being relevant cases. These are treated in Section 4 and 5, respectively. This is the key feature and novelty of the present paper over available literature. Literature To the best of our knowledge, the relevant references on interior and boundary sharp regularity of second order hyperbolic equations with Neumann boundary control g of at least L 2 (Σ)-regularity are [13,15,26]. [For the purpose of the present discussion, we may take throughout y 0 = 0, y 1 = 0, h ≡ 0, for the standard data.] The 1 2 -gain in space [21,22] References [13,15,26] (in chronological order) followed by at least twenty years the original contribution of [21]. More precisely, [21, Vol. II, p 120] shows the following result on the wave equation problem corresponding to (1.1a)-(1.1c) [−A(ξ, ∂) = ∆]: continuously, with two unites less in space for y tt = ∆y. Result (1.3) embodies the 'about 1 2 -gain in space' -philosophy from Neumann boundary term g to solution y in the interior, which was referred to above.
Reference [22] studies likewise the regularity of second order hyperbolic problem (1.1a)-(1.1c) with Neumann boundary control, where however the coefficients of the operator A(·, ·) in (1.2a) are also time-dependent. Fourier/pseudo-differential operator techniques, in the style of [23] for the Dirichlet-problem, are used in [22]. The main result of [22] shares with [21] the philosophy of an improvement of " 1 2 " in Sobolev space regularity, from H 1 2 (Γ) for g to H 1 (Ω) for the interior solution (position) y.
Lifting of time regularity With regard to result (1.3) for y ∈ L 2 (0, T ; H 1 2 (Ω)), we note that it was proved in [10,11] (originally, in the context of second order hyperbolic equations with Dirichlet control) that the L 2 -regularity in time such as the one in (1.3) can be lifted to C([0, T ]; ·) regularity in time, while preserving the regularity in space. More precisely, and in particular in the present case, with reference to problem (1.1a)-(1.1c), as soon as one has L : g −→ Lg = y : continuous L 2 (0, T ; L 2 (Γ)) −→ L 2 (0, T ; H This result was originally proved in [10], lifting to C-the original L 2 -time regularity of [9], in the context of the wave equation with Dirichlet boundary control, where L 2 (Γ) is the boundary space, and L 2 (Ω) is the interior space for the position. Its proof relied on the property that, given the abstract model for a wave equation with either Dirichlet or Neumann boundary control which was introduced in [27], the corresponding variation of parameter formula for the map: boundary controlg → solution Lg, d dt (Lg) can be expressed in terms of a strongly continuous group, precisely the group generated by the free-dynamics operator. An abstract version of the time-lifting property was next given in [11], [17,Chapter 8,p 651], assuming more generally initial L p (0, T ; ·)-regularity, for any 1 ≤ p < ∞, not only p = 2. It is reproduced below for further use. The implication (1.4)-(1.5), combined with the result in (1.3), makes now precise the notion of 1 2 -gain in space regularity for the mixed problem (1.1a)-(1.1c) referred to before in going from the Neumann boundary datum g to the corresponding hyperbolic solution Lg = y (when y 0 = y 1 = 0, h ≡ 0). It thus improves by " " the original result of [21] corresponding to the mild solution of the abstract equatioṅ subject to the following standing assumptions: (i) A : X ⊃ D(A) → X is a linear operator, which is the infinitesimal generator of a strongly continuous (s.c.) group e At on X; (ii) B is a linear, continuous operator U → [D(A * )] , where A * is the X-adjoint of A, and [D(A * )] is the dual of D(A * ) with respect to the pivot space X, or equivalently A −1 B ∈ L(U : X).
(1.9) [W.l.o.g. for consideration of the operator L below over a finite interval, we may assume A −1 ∈ L(X) for otherwise we replace A with a suitable translation]. so that, equivalently by duality, the following (abstract trace regularity) property holds true Thus, the closable operator B * e A * t admits a continuous extension (denoted by the same symbol) satisfying B * e A * t : continuous X * −→ L q (0, T ; U * ).
From a 1 2 -gain to a α=β-gain, where α = β = 2 3 for a general domain Ω, and α = β = 3 4 for parallelepipeds: from the boundary datum to the interior solution y. For ready connection with the results in the references [13,15,26] to be critically invoked throughout this paper, we shall henceforth set for a general sufficiently smooth domain Ω in R n (1.14a) for a parallelepiped Ω in R n . (1.14b) as well as the independent boundary (trace) regularity (1. 16) In (1.15a)-(1.15b), A is the operator defined in (3.1) below.
Theorem 1.1 shows that, for a general smooth domain Ω ⊂ R n (respectively, for a parallelepiped in R n ), n ≥ 2, the actual gain in space regularity from g to y is 2 3 − 1 2 = 1 6 (respectively, 3 4 − 1 2 = 1 4 ) higher than the original result (1.3) in [21], as improved by (1.4)-(1.5). Corresponding improved regularity results for the adjoint problem (i.e. the operator L * to be defined below) apply.
Consider the problem where −A * (ξ, ∂) is the formal adjoint, which together with the zero corresponding co-normal derivative realize the adjoint operator A * . The adjoint operator L * with respect to Lg = y (for y 0 = y 1 = 0, h ≡ 0 in (1.1a)-(1.1c)) is the trace operator continuously [the compatibility condition (C.C.) f (T ) = 0 is not recognized for θ ≤ 1 2 ].
Remark 1.2. Throughout the paper we are maintaining the distinction between the parameter "α" (which pertains to interior regularity) and the parameter "β" (which pertains to boundary regularity), even though at the end of the investigations in [13,15,26] it turned out that they are numerically equal, "α = β", as noted in (1.14). This strategy will make it easier to follow the arguments of the present paper when they make direct reference to, typically, results of [15], where they were stated in terms of "α" or "β" separately, as it was not yet determined at that time that in all cases "α = β" The result of [1] We close this section by stating the regularity result of [1], which, under asymmetric assumptions on the Neumann datum g, yields the H 1 (Ω)-space regularity in the interior for Lg = y, which is needed in the shape differentiability and sensitivity analysis of [2]. By asymmetric assumptions, we mean that the Neumann control g is assumed smooth in time (in fact, with two time derivatives) and non-smooth in space (in fact, with space regularity H − 1 2 (Γ)).
Then, continuously, We note that the regularity assumptions on g andġ in (1.21) imply [20, Theorem 3.1 for j = 0, page 19] that g ∈ C([0, T ]; H − 1 2 (Γ))), whereby then the extra condition g(0) = 0 makes sense in H − 1 2 (Γ)). The proof in [1] with C 2 -domains is obtained by Galerkin technique. It is further noted in [2] that the same proof can be exended to Lipschitz domains as well, by a density argument. In the Appendix, we shall provide an altogether different proof yielding a slightly stronger result than Theorem 1.3 by means, essentially, of 'soft' arguments.
Remark 1.4. Theorem 1.3 is not sharp. Qualitatively, for second order hyperbolic equations, one time derivative corresponds to one space derivative. Thus, philosophically, Theorem 1.3 corresponds to taking an extra space derivative for g; i.e. g ∈ H 1 2 (Γ). It then delivers a solution Lg in H 1 (Ω), with a gain of " 1 2 " from the Neumann boundary datum g to the interior solution Lg. In this case, Theorem 1.3 (as well as its counterpart in Theorem A.1, in the Appendix) is in the " 1 2 -gain" setting of [21], as explained in (1.4). The definite improvement of Theorem 1.3 will be provided instead in Theorem 4.5 in Section 4 (same as Theorem B in Section 2). See also Theorem 4.3 in Section 4 (same as Theorem A in Section 2). The same conclusion (1.22) with time regularity actually boosted to C([0, T ]; ·) will be obtained essentially only with g ∈ H 1 (0, T ; H −β (Γ)), g(0) = 0. This improvement from 1 2 to β in the gain of regularity will rest on the sharp regularity theory of [13,15,26], of which Theorem 1.1 (where α = β) is a canonical illustration. Remark 1.5. After the sharp/optimal regularity theory in [9,10,8] was obtained for the Dirichlet boundary control case of wave equations, it was conjectured that in going from Dirichlet to Neumann boundary datum, the regularity will improve by one unit, as in the case of elliptic or parabolic equations. However, a counterexample for dim Ω ≥ 2 was given in [13], showing that with an L 2 (0, T ; L 2 (Γ))-Neumann boundary control, the interior regularity of the position y = Lg cannot exceed H  (Ω) is achieved in the case of Ω being a parallelepiped. On the other hand, it was also shown that the conjectured 1-gain of regularity from Dirichlet to Neumann is possible if the data are compactly supported [14,25]. Finally, of course, in the one dimensional case, an elementary treatment gives the desired improvement of one unit [17, p 758, or 859, or 882, or 962].
2. Some main results. While we refer to the subsequent Sections 4 and 5 for a list of results, and their proofs, in this section we single out a selection of them, with particular reference to the motivating Theorem 1.3.
Then, continuously, Remark 2.1. This is Theorem 4.3 in Section 4. Under a slightly stronger hypothesis than in Theorem 1.3, it yields a solution Lg which is slightly smoother in time regularity, and more importantly is smoother in space regularity over Theorem 1.3, Eq. (1.22), by (β + 1 2 ) − 1 = β − 1 2 , which is equal to 1 6 for a general domain Ω, or is equal to 1 4 for parallelepipeds.
Then, continuously, Remark 2.2. This is Theorem 4.5. It is the definite improvement of Theorem 1.3.
Theorem D. Let Then, continuously, Remark 2.4. This is Theorem 5.4. It shows a gain of β (from −1 to (β − 1)) in the space regularity, in going from g to Lg, again consistently with (1.15a) of Theorem 1.1, where α = β.
3. Priliminary background. We return to problem (1.1a)-(1.1c) and its adjoint (1.17a)-(1.17c) and introduce a number of relevant objects: Similarly, −A * generates a s.c. cosine operator on L 2 (Ω), given precisely by C * (t). For general background on cosine operators, we refer to [24,3], and bibliography cited therein. If A is self-adjoint, so are C(t) and S(t), and the standard self-adjoint calculus applies. 3. Without loss of generality for the regularity problem here considered, we may assume that the null space of A is trivial: N (A) = {0}, for replacement of the original A with a suitable translation (A + k 2 I) does not change the regularity over [0, T ], T < ∞. Thus, we may take fractional powers A θ of A, 0 < θ < 1, to be well defined [Recall that −A (canonically the Laplacian ∆ with Neumann homogeneous B.C.) a fortiori generates a s.c. analytic semigroup on L 2 (Ω), t > 0]. 4. As in [17,28], define the Neumann map N (elliptic extension of a Neumann boundary datum) by We have [20] as well as [4,5,7] the identification in (3.3) being set-theoretical and topological with 5. As introduced in [27,9], the abstract model of the operator L : in appropriate topologies based on L 2 (Ω) to be described below, where A in 6. The following are standard properties of the cosine/sine operators: Moreover with reference to (1.17a)-(1.17c) and (1.19) Proofs. Throughout this section, without further mention, we shall provide results for the operator L : g → y for problem (1.1a)-(1.1c) with y 0 = y 1 = 0, h ≡ 0; as well as for its adjoint We begin with a result which is actually contained in [15, Theorem 8.1, for θ = 0, p 161].
Remark 5.1. Theorem 5.1 is sharp in both time and space. Nevertheless, for our purposes, we prefer a result that start with g smoother in time, i.e. assumed in L 2 (0, T ; ·) in time. This is provided by the analysis below.
Remark 5.2. Theorem 5.2 shows an improvement of the sharp gain β in time (from 0 to β) as well as in space (from 1 to 1 + β). In the next result we deliberately accept a loss of time regularity in order to have an attractive result expressed in terms of L 2 (0, T ; ·) regularity, consistently with a gain β in space regularity from the boundary to the interior.
as 1 − β < 1 2 , so that the compatibility condition f (T ) = 0 is void. Eq. (6.5) is nothing but (5.3). We now consider the duality pairings in (5.4) with g = ∂y ∂ν Σi from (6.2c); that is: . (6.6) We now notice that in addition to the regularity noted in (6.5), it also holds that by (3.12) Moreover, under the hypotheses in (6.3), we have that the solution of the y-problem . Consequently, we have that the RHS duality pairings in (6.6) is well-defined. Thus the LHS duality pairings in (6.6) is likewise well-defined and then we then conclude that Theorem 6.1 is proved.
Remark 6.1. The above illustration deals with an irregular set of data in (6.3) which produce a highly irregular normal derivative on Γ i given by (6.11). This very irregular 'input' then produces the corresponding solution w in Q i , as in (6.13).
Illustration #2: An uncontrolled Dirichlet-wave equation feeding in a serial connection another wave equation via the tangential gradient of the Neumann trace.

ROBERTO TRIGGIANI
With such Neumann input as in (6.18) applied in (6.15b), Theorem 5.4 yields and ( feeding through time integration of the Neumann trace on Γ i = ∂Ω i a second w-wave mixed problem defined on the external 3-dimensional domain Ω e : in Ω e (6.22b)
Illustration #4: A point-control thermoelastic system feeding in a serial connection a wave equation via the tangential gradient of the trace of the "moment" of the elastic displacement; n = 2.
Illustration #5: An uncontrolled Kirchoff Equation feeding in a serial connection a wave equation.
The setting of the internal domain Ω i immersed in an external domain Ω e is exactly the same as in Illustration #1, #2, #3,and #4, with same symbols Γ i = ∂Ω i and ∂Ω e = Γ e ∪ Γ i . Illustration #6: An uncontrolled Euler-Bernoulli feeding in a serial connection a wave equation.
Appendix: A soft proof of the non-optimal Theorem 1.3 (" 1 2 -gain in space"). In this Appendix we provide a result which is a slightly stronger version of Theorem 1.3. Refer also to Remark 1.4. The proof is in spirit of the radically different re-proof, given in [15, p 122