Appl Math Optim DOI 10.1007/s00245-016-9354-4

Hyperbolic Equations with Mixed Boundary Conditions: Shape Differentiability Analysis Lorena Bociu1 · Jean-Paul Zolésio2

© Springer Science+Business Media New York 2016

Abstract We consider the wave equation with Dirichlet–Neumann boundary conditions on a family of perturbed domains s . We discuss the shape differentiability analysis associated with the above mentioned problem, namely the existence of strong material and shape derivatives of the solution, and the rendering of the new wave problem whose solution is given by the shape derivative. The study shows that the Neumann boundary conditions completely change the focus and strategy involved in the shape differentiability analysis, in comparison to the case of the wave equation with purely Dirichlet boundary conditions. In this paper we show that for the existence of weak material derivative, the classical sensitivity analysis of the state can be bypassed by using parameter differentiability of a functional expressed in the form of Min–Max of a convex–concave Lagrangian with saddle point. Then we analyze the strong material derivative via a brute force estimate on the differential quotient, using known regularity results on the solution of the wave problem.

1 Introduction and Main Results We are interested in the shape differentiability and sensitivity analysis for the solution of the wave equation on a bounded domain with mixed boundary conditions of Dirichlet–Neumann type with respect to the shape of the geometric domain on which the PDE is defined. We build the variable domains using the well-known “velocity

B

Lorena Bociu [email protected] Jean-Paul Zolésio [email protected]

1

Department of Mathematics, NC State University, Raleigh, NC 27695, USA

2

CNRS and INRIA, CNRS-INLN, 1136 route des Lucioles, 06902 Sophia Antipolis, France

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method”, i.e. through a family of diffeomorphisms which preserve the smoothness of the images of a fixed domain. The velocity method has been widely used in various applications, and a complete shape calculus, which makes intensive use of tangential differential calculus, is now available (for details see [17,28]). The problem under consideration is a fundamental question in shape optimization and control problems for the linear wave equation and coupled systems where the hyperbolic equation is coupled with other dynamics, and the matching conditions at the boundary are of Neumann type. These are the natural conditions often present in interactive systems where the wave equation is coupled with other dynamics, like in the case of structural acoustic problems and fluid structure interactions. In more details, (i) For shape optimization problems in general [6,17,28], the cost function takes the form of an integral over a domain or its boundary, where the integrand depends smoothly on the solution of the boundary value problem (BVP). The goal is to minimize the functional with respect to the geometrical domain (which must belong to an admissible family of domains). This involves performing shape sensitivity analysis for the solution of the BVP; (ii) In control problems for coupled systems with hyperbolic components, especially when the wave is coupled with nonlinear dynamics (for example in fluidelasticity interactions, where the wave is coupled with the Navier-Stokes equation [2]), the optimality conditions that characterize the optimal control must be derived from differentiability arguments on the cost functional with respect to the control. This involves computing a directional derivative of the cost functional in the direction of a perturbation. Thus one has to introduce a small parameter s, and then calculate the derivative w.r.t. s at s = 0 of the perturbed functional (which now depends on a perturbed domain). The derivative of the cost will depend on the derivatives of the states with respect to s, and one needs to have a characterization for these “pseudoshape” derivatives [3]. This brings us again to the question of sensitivity for the solution of the hyperbolic component of the coupled system with respect to the shape of the geometrical domain. The shape differentiability analysis has received a lot of attention in the literature and has been solved for various classical linear and nonlinear boundary value problems on manifolds ([1,7,8,22,23,28,33] and references within). In particular, a full analysis of the shape differentiability for the solution to the second order hyperbolic equation with Dirichlet boundary conditions was provided in [8]. However, the hyperbolic case with Neumann boundary conditions is more delicate, due to the complications arising from the boundary regularity of the solutions, as described in details in the next section. Therefore, in this paper we focus on the second order hyperbolic equation with Dirichlet–Neumann boundary conditions, which brings in new challenges and implicitly requires new techniques.

1.1 The PDE Model Below we present the PDE model for the wave problem under consideration. The wave problem on . Let D ⊂ Rn , potentially included in a C 2 -manifold. Let ¯ S¯ = ∅,  ⊂ D be an open, bounded, C 2 domain, with boundary ∂ = ∪S, where ∩ and meas(S)= 0.

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Consider the following wave equation on  × I = (0, T ), with mixed boundary conditions ⎧ ⎪ ⎨ ytt − y = f  × I (1.1) y=0 S×I ⎪ ⎩ ∂y  × I, ∂n = g and initial conditions given by y(0) = yt (0) = 0. Since we are interested in the shape differentiability for the solution of the boundary value problem introduced above with respect to the shape of the geometric domain , we build the family of perturbed domains using the “velocity method”, as described below. The moving domain Let Q = (0, T ) ×  be the cylindrical evolution domain, while  = (0, T ) ×  be the lateral boundary of Q. For s ∈ [0, s ∗ ], let V be a smooth vector field (knows as the “speed”), V ∈ C 0 ([0, s ∗ ), C 2 (D, Rn )) with V · n = 0 on ∂ D. The flow transformation associated to V is given by: Ts : D¯ → D¯ such that at the point x, V (s)(x) =



 ∂ Ts ◦ Ts−1 (x). ∂s

From [13] and [14], we know that for each s ∈ [0, s ∗ ), the transformation map Ts is injective from D to D, and it satisfies the following properties: (1) T0 = I , (2) Ts ∈ C 1 ([0, s ∗ ), C 2 (D, D)), and Ts (∂ D) = ∂ D, (3) T2−1 ∈ C([0, s ∗ ), C 2 (D, D)). Using Ts (V ), we build the family of perturbed domains {s }s by s (V ) = Ts (V )(). We call Q s = (0, T )×s (V ) the perturbed cylinder, and s = (0, T )×s the perturbed lateral boundary, where s = ∂s . The wave problem on s : Let ys be the solution to the wave equation in the cylinder Q s , with homogeneous Dirichlet condition on the fixed part S of the boundary, and verifying a non-homogeneous Neumann condition on the moving part s : ⎧ ⎪ ⎨(ys )tt − ys = 0 I × s = Q s ys = 0 I×S ⎪ ⎩ ∂ ys = g I × s =  s , s ∂n s

(1.2)

with ys (0) = ∂∂tys (0) = 0. The assumptions on the regularity of the boundary data gs are discussed in the next section.

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1.2 Goals and Assumptions The goal of this paper is to provide complete details for the results announced  in [4],  ∂ and namely the existence of strong material derivative y˙ (; V ) = ∂s [ys ◦ Ts ] {s=0}

strong shape derivative y  (; V ) = y˙ (; V ) − ∇ y · V (0) in suitable spaces, plus a careful analysis of the new wave problem whose solution is the shape derivative y  (; V ). The shape differentiability analysis under consideration is of course very much dependent on the well-posedness results available for system (1.1), and implicitly on the regularity assumptions posed for the Neumann boundary data g. It is well known in the literature that for L 2 (I × )- Neumann boundary data, the maximal amount of regularity that one obtains for the solution to the linear wave equation is in general H 2/3 × H −1/3 [26,29]. As our goal is to provide shape differentiability analysis for finite energy solutions to the wave equation, we will consider appropriate regularity for the Neumann data g, i.e., one that guarantees that the wave solution will have H 1 -regularity. Specifically, we assume that g(0, x) = 0 and g ∈ W 1,1 (I, H −1/2 ()), along with f (0, x) = 0 and [ f ∈ L 2 (I × ) or f ∈ W 1,1 (I, H −1 ())]. Then we know that there exists unique solution y of (1.1) which verifies y ∈ E() := L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()),

(1.3)

where

H∗1 ()

= {φ ∈ H (), φ = 0 on S} 1

with φ2H 1 () ∗

=

Moreover, we have the following estimate: ∃ c > 0 such that



|∇φ|2 d x.

y2E() = yt 2L ∞ (I,L 2 ()) + y2L ∞ (I,H 1 ()) ≤ c( f, g)2 , ∗

(1.4)

where ( f, g)2 = g2W 1,1 (I,H −1/2 ()) +  f 2 , and  f 2 =  f 2W 1,1 (I,H −1 ()) , or  f 2L 2 (I ×)) . The above existence and regularity result is provided in [5] for the case of smooth (C 2 ) domains using Galerkin techniques. We would like to point out that the same result can be obtained for Lipschitz domains as well, and this is valuable in applications, due to the fact that non-smooth boundary data is usually associated with non-smooth domains. The details of the proof in this case follow along the same lines as in [5], with the following observation: under the Lipschitz continuity assumption, the Hilbert space H 1 () is separable as there exists a continuous extension operator from H 1 () in H 1 (D). Since D is smooth, there exists a dense countable family E i in H 1 (D) and therefore the restrictions ei to the open set  are also dense in H 1 (). Moreover, we recall that in such domains, Stokes’ formula holds, as a special case of the result provided in [21], where it was obtained for open sets with bounded perimeter, i.e. with characteristic function in BV (D).

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Remark 1.1 The existence and regularity result mentioned above in (1.3)–(1.4), even for C 2 -domains, has not apparently appeared explicitly in the literature, prior to [5]. However, it is in line with the philosophy of [27], which provided a gain of regularity in space from the Neumann boundary term g to the solution y in the interior of about 1/2. More specifically, [27] assumes the Neumann boundary datum g ∈ L 2 (0, T ; L 2 ()) and obtains the solution y ∈ C([0, T ]; H 1/2−ε ()) ∩ L 2 (0, T ; H 1/2 ()). To see this qualitative philosophical correspondence with our result (1.3)–(1.4), one recalls that for the wave equation one derivative in time corresponds to one derivative in space. Remark 1.2 Optimal regularity results for hyperbolic equations with nonhomogeneous Neumann boundary condition are provided in [25,26], and are based on pseudo-differential calculus and sine/cosine operator techniques. In particular [30], one can consider Neumann data of the type g ∈ H 1 (0, T ; H −2/3 ()) ∩ C([0, T ]; H −1/2 ()), with g(0, x) = 0, and obtain the desired regularity for the wave solution (y, yt ) ∈ C([0, T ]; H 1 ()) × C([0, T ]; L 2 ()). In this paper, we will use the regularity for the solution of the wave system (1.3)– (1.4) in a brute force estimate on the differential quotient that will allow us to obtain the desired shape differentiability results. 1.3 Material and Shape Derivatives In this section we introduce the notation and recall the definition of shape differentiability, and the relation between the material and shape derivatives. Definition 1 (Shape Differentiability) The map [ → y()] associated with problem (1.1) is shape differentiable in L 2 (I, H m (D)) if ∃Y ∈ C 1 ([0, s ∗ ], L 2 (I, H m (D))) such that Y | Q s = y(s ) and Y(s) − Y(0) − ∂s Y(0) → 0 in L 2 (I, H m (D)) as s → 0. s

(1.5)

Similarly, the map  → y() is weakly shape differentiable in L 2 (I, H m (D)) if the convergence in (1.5) is weak. Definition 2 (Shape Derivative) The shape derivative of y is the unique element

  y  (, V ) = ∂s Y 

s=0,(t,x)∈Q

∈ L 2 (Q).

It is known [33] that ∂s Y(0)| Q does not depend on the choice of the function Y introduced above.

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Definition 3 (Material Derivative) The element y˙ (, V ) is the material derivative of y in L 2 (I, H m (D)) if 1 (ys ◦ Ts − y()) − y˙ (, V ) → 0 in L 2 (I, H m (D)) as s → 0. s 1.4 Main Results Before stating our main results, we introduce our assumptions on the boundary data g. First, recall the definition of a transported distribution [28]. For any gs ∈ H −1/2 (s ), gs Ts (V ) ∈ H −1/2 (s ) is the distribution defined by



gs Ts (V ) d =

s

gs

 ωs

◦ Ts−1 ds ,

where ∈ D is any test function, and the density ωs is given by ws = det(DTs )|(DTs )−∗ n|. Assumption 1.1 (Boundary Data g) 1. Regularity: gs ∈ W 1,1 (I, H −1/2 (s )), ∀ s ∈ [0, s ∗ ). 2. Differentiability: ∃ g˙ ∈ W 1,1 (I, H −1/2 ()) such that gs Ts − g˙ −→ 0 in W 1,1 (I, H −1/2 ()). s→0 s

(1.6)

Our first result provides the existence and regularity of strong material derivative for ys . Theorem 1.1 (Strong Material Derivative) Let ys ∈ H = L ∞ (I, H∗1 (s )) ∩ W 1,∞ (I, L 2 (s )) be the unique solution of (1.2). Under Assumption 1.1, ys has a strong material derivative in the following topology: ys ◦ Ts (V ) − y − (Y˙t − divV (0) y) → 0 in L ∞ (I, L 2 ()) ∩ W −1,∞ (I, H∗1 ()), s where Y˙ is the solution to the variational problem (2.27) described in Sect. 3.2., and V and Ts are the the vector field and the flow map that build the family of perturbed domains s , introduced in Sect. 1.1. As an immediate consequence, we obtain the following corollary: Corollary 1.1 (Strong Boundary Material Derivative) In the context of Theorem 1.1, assume that gs ∈ W 2,1 (I, H −1/2 (s )) and there exists g˙ ∈ W 1,1 (I, H −1/2 ()) verifying gs Ts (V ) (1.7) − g(V ˙ ) → 0 str ongly in W 2,1 (I, H −1/2 ()). s

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Then the solution ys has a strong boundary material derivative in the following topology: ys ◦ Ts (V ) − y − (Y˙t − divV (0) y) → 0 in L ∞ (I, H 1/2 ()). (1.8) s The next result provides shape differentiability for the wave-solution, as well as a characterization for the shape derivative. As expected, the shape derivative solves a linear wave equation with a Neumann boundary condition that involves the “speed” and curvature of the boundary. Theorem 1.2 (Shape Differentiability) Under the same assumptions as Theorem 1.1, the map  → y() is shape differentiable (in the direction V ) in L ∞ (I, L 2 ()) ∩ W −1,∞ (I, H∗1 ()). Moreover, assuming that  is at least C 1,1 domain, the shape derivative y  (; V ) = y˙ (; V ) − ∇ y · V (0) ∈ L ∞ (I, H −1 ()) ∩ W −1,∞ (I, L 2 ()) solves the following wave problem: ⎧   ⎪ ⎨(y )tt − y = 0 in  

∂y = div (y ∇ v ) − v ⎪ ⎩ ∂n y (0) = yt (0) = 0,

∂2 y ∂t 2

+ H g v + g˙ on 

(1.9)

where g˙ is defined in (1.6). The last result is concerned with weak material differentiability for the solution ys . We want to point out that the result does not follow from Theorem 1.1 (the 2 results provide topologies that are not comparable), and its proof requires completely different techniques. Theorem 1.3 (Weak Material Derivative) Under Assumption 1.1, the solution ys has weak material derivative in [W 1,1 (I, H −1 ())] , and weak boundary material derivative in [W 1,1 (I, H −1/2 ())] (X  stands for the dual notation).

2 Proof of Theorem 1.1 Our goal is to show that ys ◦Ts is differentiable at s = 0 in a suitable (to be determined) space and find its derivative w.r.t s at s = 0. First, we will check continuity for the map [s → ys ◦ Ts ] at 0. Using (1.3)–(1.4) from the preceding section, we know that for each s ∈ [0, s ∗ ) there exists unique solution to system (1.2) ys ∈ H = L ∞ (I, H∗1 (s )) ∩ W 1,∞ (I, L 2 (s )),

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which solves the variational problem: ∀ϕs ∈ H with ϕ(T ) = 0, we have

s

I

(∇ ys · ∇ϕs − (ys )t (ϕs )t )ds dt =

s

I

gs ϕs ds dt,

(2.1)

and we have the following estimate for the H-norm of the solution: ∃ks > 0 such that ys 2H ≤ ks gs 2W 1,1 (I,H −1/2 ( )) .

(2.2)

s

where ks is continuous at s = 0. 2.1 [Continuity]

t

For G s (t) =

gs (σ )dσ ∈ W 2,1 (I, H −1/2 (s )), consider

0

Ys (t) =

t

ys (σ )dσ

0

solution to the following problem: ∀ϕs ∈ H,

I

s

(∇Ys · ∇ϕs − (Ys )t (ϕs )t )ds dt = I

s

G s ϕs ds dt.

(2.3)

Again, by (1.3)-(1.4) and our definition of Ys as the primitive of ys , we have that for each s ∈ [0, s ∗ ), Ys has the following regularity: Ys ∈ H1 (s ) = W 1,∞ (I, H∗1 (s )) ∩ W 2,∞ (I, L 2 (s )), and the H1 (s )-norm of Ys satisfies the following inequality: ∃ks > 0 (continuous at s = 0) such that (2.4) Ys 2H1 ( ) ≤ ks G s 2W 2,1 (I,H −1/2 ( )) . s

s

Our first goal is to show that s → Ys ◦ Ts is continuous at 0. With that scope in mind, in (2.3) we change variables using the flow map Ts and bring all the integrals on fixed domains: Js (DTs−1 )(DTs−∗ )∇(Ys ◦ Ts ) · ∇(ϕs ◦ Ts )ddt I  = Js (Ys ◦ Ts )t (ϕs ◦ Ts )t )ddt I  + ωs (G s Ts )(ϕs ◦ Ts )ddt, (2.5) I



where Js = det(DTs ) is the Jacobian of the transformation Ts , and ωs = Js |DTs−∗ n| is called the density [17].

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For the sake of exposition, we introduce the following notation. Let Y s = Js (Ys ◦ Ts ) ∈ H1 (),

(2.6)

As = Js (DTs−1 )(DTs−∗ ) ∈ L ∞ (D),

(2.7)

ϕ = ϕs ◦ Ts ∈ H, and G s = G s Ts ∈ W 2,1 (I, H −1/2 ()). Therefore we can rewrite (2.5) as As ∇(Js−1 Y s ) · ∇ϕddt − Yts ϕt ddt = G s ωs ϕddt. (2.8) I



I



I



Lemma 2.1 The map π : [0, s ∗ ) → H1 () given by π(s) = Y s is continuous at s = 0. Proof Consider the difference Z s = Y s − Y . Note that Z s solves: ∀ϕ ∈ H, [As ∇(Js−1 Y s ) − ∇Y ] · ∇ϕddt I  − (Yts − Yt )ϕt ddt I  = (G s ωs − G)ϕddt I



(2.9)

At this point, we want to recover the variational form of the wave equation for Z s . Note that we can rewrite As ∇(Js−1 Y s ) − ∇Y

= As (∇ Js−1 )Y s + As Js−1 ∇Y s − ∇Y = As (∇ Js−1 )Y s + As Js−1 ∇Y s − ∇Y s + ∇Y s − ∇Y = As (∇ Js−1 )Y s + (As Js−1 − I )∇Y s + ∇ Z s

(2.10)

Using (2.10) in (2.9), we obtain that Z s solves the following problem: ∀ϕ ∈ H, (∇ Z s · ∇ϕ − Z ts ϕt )ddt = − m(s) · ∇ϕ ddt + Gˆ s ϕ ddt I



I



I



(2.11) where the RHS terms are given by m(s) = As (∇ Js−1 )Y s + (As Js−1 − I )∇Y s , and

(2.12)

Gˆ s = G s ωs − G.

(2.13)

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Appl Math Optim

In the first term on the RHS of (2.11), we integrate by parts in space and obtain that Z s satisfies: ∀ϕ ∈ H, (∇ Z s · ∇ϕ − Z ts ϕt )ddt I  = div[m(s)]ϕ ddt I  + (Gˆ s − m(s) · n)ϕ ddt. I



(2.14)

Since Y s ∈ W 1,∞ (I, H∗1 ()), then we know that ∇Y s ∈ W 1,∞ (I, L 2 ()), and implicitly div[m(s)] ∈ W 1,∞ (I, H −1 ()). On the boundary , we can rewrite m(s) as follows: m(s) = As (∇ Js−1 )Y s | + (As Js−1 − I )∇Y s | = As (∇ Js−1 )Y s | + (As Js−1 − I )∇ Y s + (As Js−1 − I )∇Y s , nn

(2.15)  

Lemma 2.2 The boundary term m(s), n = As (∇ Js−1 )Y s , n + (As Js−1 − I )∇Y s , n ∈ W 1,∞ (I, H −1/2 ()). Proof This follows from (2.8), and the fact that the normal derivative of Y s on the boundary can be recovered in terms of its tangential gradient and the co-normal derivative. In particular, we know that Y s satisfies the following boundary condition As ∇(Js−1 Y s ), n = G s ws on  This is equivalent to As ∇(Js−1 )Y s , n + As Js−1 ∇ (Y s ), n + ∇Y s , nAs Js−1 n, n = G s ws on . (2.16) Solving for ∇Y s · n in (2.16), we obtain that ∇Y s · n = As Js−1 n, n−1 [G s ws − As ∇(Js−1 )Y s , n − As Js−1 ∇ (Y s ), n] (2.17) In (2.17) we used the fact that As Js−1 = DTs−1 DTs−∗ is coercive (for details, see [8], p.189). Since Y s ∈ W 1,∞ (I, H∗1 ()), we know that Y s | ∈ W 1,∞ (I, H 1/2 ()) and ∇ Y s | ∈ W 1,∞ (I, H −1/2 ()). Combining these with (2.17), we obtain that m(s), n = G s ws − ∇Y s , n ∈ W 1,∞ (I, H −1/2 ()).

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Now we apply the existence and regularity results of Section 1.2 with fˆ(s) = div[m(s)] ∈ W 1,∞ (I, H −1 ()) and g(s) ˆ = Gˆ s − m(s), n ∈ W 1,∞ (I, H −1/2 ()), and obtain that for each s ∈ [0, s ∗ ), Z s ∈ L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()). Moreover, we have the following estimate for our solution Z s : ∃ks > 0 (continuous at s = 0) such that Z ts 2L ∞ (I,L 2 ()) +Z s 2L ∞ (I,H 1 ()) ∗

≤ ks [ fˆ(s)2W 1,1 (I,H −1 ()) 2 +g(s) ˆ ] W 1,1 (I,H −1/2 ())

(2.18)

What is left to show is that the RHS of (2.18) goes to 0 as s → 0. First of all, we know from classical estimates [28] that ∃ C V (depending on V (0)) such that (2.19) As Js−1 − I [L ∞ (D)]n2 + ||A(s)∇ Js−1 || L ∞ (D) ≤ s C V . Recalling the definitions (2.12) and (2.6) of m(s) and Y s , respectively, and using the above mentioned estimate, we obtain that there exists C˜ V > 0 such that m(s)W 1,∞ (I,L 2 ()) ≤ s C˜ V G s W 2,1 (I,H −1/2 (s )) .

(2.20)

Since G s is differentiable at s = 0 in W 2,1 (I, H −1/2 (s )) by its definition and the Assumption 1.1 on gs , we conclude that fˆ(s) −→ 0 in W 1,1 (I, H −1 ()).

(2.21)

s→0

Now we deal with the boundary term g(s). ˆ Recall that on  we have g(s) ˆ = Gˆ s − m(s), n = ∇Y s , n − G =As Js−1 n, n−1 [G s ws − As ∇(Js−1 )Y s , n − As Js−1 ∇ (Y s ), n] − G =As Js−1 n, n−1 G s ws − G− − As Js−1 n, n−1 [As ∇(Js−1 )Y s , n + (As Js−1 − I )∇ (Y s ), n] (2.22) Since Y s ∈ W 1,∞ (I, H∗1 ()), then we have that Y s | ∈ W 1,∞ (I, H 1/2 ()) and ∇ Y s ∈ W 1,∞ (I, H −1/2 ()), with their respective norms bounded by the W 1,∞ (I, H∗1 ())-norm of Y s (due to the continuity of the trace map), which in turn is bounded above by G s W 2,1 (I,H −1/2 (s )) . Combining this with (2.19), we know that As ∇(Js−1 )Y s , n + (As Js−1 − I )∇ (Y s ), nW 1,1 (I,H −1/2 ()) −→ 0. s→0

(2.23)

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Appl Math Optim

We also have that Gˆ s = G s ωs − G is differentiable (and thus continuous) at s = 0 in W 2,1 (I, H −1/2 (s )). More specifically, we know that As Js−1 n, n−1 G s ws − GW 1,1 (I,H −1/2 ()) −→ 0. s→0

(2.24)

Combining (2.24) with (2.23), we obtain that g(s) ˆ −→ 0 in W 1,1 (I, H −1/2 ()). s→0

(2.25)

From (2.18), combined with (2.25) and (2.21), we obtain that Z ts 2L ∞ (I,L 2 ()) + Z s 2L ∞ (I,H 1 ()) −→ 0, ∗

s→0

(2.26)

which then allows us to conclude that ys ◦ Ts −→ y in L ∞ (I, L 2 ()) ∩ W −1,∞ (I, H∗1 ()). s→0

This finishes the proof of continuity of the map s ∈ [0, s ∗ ) → ys ◦ Ts ∈   L ∞ (I, L 2 ()) ∩ W −1,∞ (I, H∗1 ()) at s = 0. 2.2 Differentiability Let Y˙ ∈ H = L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()) be the solution to the following problem: ∀ψ ∈ H ,

T 0

 

−

 T T ∂ ˙ ∂ gψ, ¯ (2.27) f¯ψ + Y ψ + ∇ Y˙ · ∇ψ d xdt = ∂t ∂t   0 0

with f¯ = div[(A − divV (0)I )∇Y − Y ∇(divV (0))] ∈ W 1,∞ (I, H −1 ()), and

g¯ = [∇(divV (0)Y ) − A ∇Y ]n + G˙ + G H v ∈ W 1,∞ (I, H −1/2 ()), where the following notation was used above: ∂ As  • A = is the shape derivative of As w.r.t. s at s = 0,  ∂s s s=0 ∂G  • G˙ = is the material derivative of G s w.r.t. s at = 0 (this exists by the  ∂s s=0 definition of G s and Assumption 1.1), • H represents the additive curvature of the boundary , and • v = V (0) · n.

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Appl Math Optim

Note that problem (2.27) is obtained by formally differentiating (2.8) with respect to s at s = 0. The existence and regularity of the solution Y˙ are guaranteed by (1.3)–(1.4). From [28], we know that A = divV (0)I −2ε(V (0)), and therefore we can simplify ¯ f and g¯ accordingly: f¯ = div[−2ε(V (0)∇Y − Y ∇(divV (0))] ∈ W 1,∞ (I, H −1 ()), and g¯ = 2ε(V (0))∇Y · n + Y

∂ [div(V (0))] + G˙ + G H v ∈ W 1,∞ (I, H −1/2 ()) ∂n

Next we consider the element: Zs − Y˙ , s

δ(s) =

which satisfies the following problem: ∀ϕ ∈ H ,

T 0





∇δ(s) · ∇ϕ − δt (s)ϕt = 0

T





div(M(s))ϕ +

T

0



G(s)ϕ. (2.28)

The terms on the RHS are given by M(s) =

m(s) + 2ε(V (0)∇Y + Y ∇(divV (0)), s

(2.29)

and G(s) =

Gˆ s − m(s) · n ∂ − 2ε(V (0))∇Y · n − Y [div(V (0))] − G˙ − G H v. s ∂n (2.30)

Using the definition (2.12) of m(s), we can rewrite M(s) and G(s) in the following forms: M(s) =

A J −1 − I S s + 2ε(V (0)) ∇Y s − 2ε(V (0))∇ Z s s

A ∇ J −1 s s + ∇[div(V (0))] Y s − ∇[div(V (0)]Z s , + s

(2.31)

and G(s) =

Gˆ s − M(s) · n − G˙ − G H v. s

(2.32)

Now it is clear that the time regularity of Z s (Z s ∈ L ∞ (I, H∗1 ())) is not sufficient to allow us to use the existence and regularity results for (2.28). To overcome this obstacle, we invoke the primitive again. Let

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Appl Math Optim



t

P(s) = 0

δ(s)(σ ) dσ ∈ W 1,∞ (I, H∗1 ()) ∩ W 2,∞ (I, L 2 ()),

t corresponding to Y˜ = 0 Y s (σ )dσ ∈ W 2,∞ (I, H∗1 ()) ∩ W 3,∞ (I, L 2 ()), and Z˜ = t s 1 1,∞ (I, H 1 ()) ∩ W 2,∞ (I, L 2 ()). ∗ 0 Z (σ )dσ ∈ H () = W The primitive P(s) is then solution to the following problem: ∀ψ ∈ H 1 (),

  ∂ ∂ − P(s) ψ + ∇ P · ∇ψ d xdt ∂t ∂t  0 T T = div(M P (s))ψd xdt + G P (s) ψd dt, T

0



0



(2.33)

where M P (s) =

A J −1 − I s s + 2ε(V (0)) ∇ Y˜ s − 2ε(V (0))∇ Z˜ s s

A ∇ J −1 s s + ∇[div(V (0))] Y˜ s − ∇[div(V (0)] Z˜ s , + s

(2.34)

and t G P (s) =

0

t t Gˆ s (σ )dσ ˙ )dσ − G(σ )H vdσ. (2.35) G(σ − M P (s) · n − s 0 0

By well-posedness theory for (2.28) we have the following estimate on the norm of P(s): ∃ks > 0 (continuous at s = 0) such that P(s)W 1,∞ (I,H∗1 ()) +P(s)W 2,∞ (I,L 2 ()) ≤ ks [div(M P (s))W 1,1 (I,H −1 ()) +G P (s)W 1,1 (I,H −1/2 ()) ]

(2.36)

Lemma 2.3 For the term M P (s) defined above in (2.34), we have that M P (s)W 1,1 (I,L 2 ()) −→ 0. s→0

Proof First, we have the following limits as s → 0 (their proofs can be found in [28] and [8]): As Js−1 − I 2 + 2ε(V (0)) −→ 0 in [L ∞ (D)]n (2.37) s→0 s As ∇ Js−1 + ∇(div(V (0))) −→ 0 in [L ∞ (D)] (2.38) s→0 s Moreover, we know that • Y˜ s W 2,∞ (I,H∗1 ()) ≤ ks G˜ s W 3,1 (I,H −1/2 (s ) , where G˜ s is the primitive of G s (implicitly shape differentiable), and

123

Appl Math Optim

•  Z˜ s H1 () −→ 0. s→0

Combining all the above limits and estimates, we obtain the desired result.

 

As an immediate consequence, we obtain that div(M P (s)) −→ 0 in W 1,1 (I, H −1 ()). s→0

Lemma 2.4 For the boundary term G P (s), we have the following limit as s → 0: G P (s)W 1,1 (I,H −1/2 ()) −→ 0. s→0

Proof Recall that G P (s) is given by: 1 G P (s) = −M P (s) · n + s



t

Gˆ s (σ )dσ −

0



t

˙ )dσ − G(σ

0





t

Gˆ s (σ )dσ −



0

t

˙ )dσ − G(σ

0



t

G(σ )H vdσ.

0

Using the definition of Gˆ s , the assumption 1.1, and the fact that [28], we obtain that 1 s

t



∂ ∂s ωs



|s=0 = H v

G(σ )H vdσ −→ 0 in W 1,1 (I, H −1/2 ()). s→0

0

Lastly, we deal with the term M P (s) · n, which on the boundary  takes the following form: M P (s) · n =

A J −1 − I s s + 2ε(V (0)) ∇ Y˜ s · n − 2ε(V (0))∇ Z˜ s · n s

A ∇ J −1 s s + ∇[div(V (0))] Y˜ s · n − ∇[div(V (0)] Z˜ s · n, + s (2.39)

Since Z s H1 () −→ 0, we immediately obtain that s→0

∇[div(V (0)] Z˜ s −→ 0 in W 1,1 (I, H −1/2 ()). s→0

ˆ and we already showed Moreover, we know that ∇ Z s · n = Gˆ s − m(s) · n = g(s), (2.25) that g(s) ˆ −→ 0 in W 1,1 (I, H −1/2 ()). s→0

From Lemma (2.3), we know that Y˜ s ∈ W 2,∞ (I, H∗1 ()) and ∃k(s) > 0 such that Y˜ s W 2,∞ (I,H∗1 ()) ≤ k(s)G˜ s W 3,1 (I,H −1/2 (s ) . Therefore, by the continuity of the trace operator H 1 → H 1/2 , we have a similar bound on the norm of trace Y˜ s | upper −1 A J −I in W 2,∞ (I, H 1/2 ()). To deal with the term s s + 2ε(V (0)) ∇ Y˜ s · n, we repeat s

the strategy used in Lemma 2.2 (namely, recover the normal derivative in terms of the

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Appl Math Optim

tangential gradient and the co-normal derivatives, then use their known regularities to bound the norm of the normal derivative on the boundary by the norm of the given boundary data). Combining these estimates with (2.37) and (2.38), we obtain that  A J −1 − I

 s s + 2ε(V (0)) ∇ Y˜ s · n  s

A ∇ J −1 s s + ∇[div(V (0))] Y˜ s · nW 1,1 (I,H −1/2 ()) → 0 as s → 0. (2.40) + s   In conclusion, we showed that P(s) −→ 0 in W 1,∞ (I, H∗1 ()) ∩ W 2,∞ (I, L 2 ()), s→0

which implies that δ(s) −→ 0 in L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()). s→0

Recalling the definition of δ(s), i.e. δ(s) =

Js Ys ◦ Ts − Y Zs − Y˙ = − Y˙ , s s

we obtain the desired result, i.e. the solution ys of (1.1) has strong material derivative given by Y˙t − div(V (0))y in W −1,∞ (I, H∗1 ()) ∩ L ∞ (I, L 2 ()).

3 Proof of Theorem 1.2 From Theorem 1.1, we know that the solution ys has strong material derivative y˙ = Y˙ − div(V (0))y in W −1,∞ (I, H∗1 ()) ∩ L ∞ (I, L 2 ()). We define Y(s) = [P(ys ◦ Ts )] ◦ Ts−1 , where P : H 1 () → H 1 (D) is a linear extension operator. Note that Y(s) is differentiable as ys ◦ Ts is differentiable, and moreover, ∀s, Y(s, .) = ys (.) on s . From [28], we know that the term y  (; V ) := [

∂ Y(0, x)]{x∈} ∈ L ∞ (I, H −1 ()) ∩ W −1,∞ (I, L 2 ()) ∂s

is independent on the choice of the function Y and is given by y  (; V ) = y˙ (; V ) − ∇ y · V (0).

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Appl Math Optim

Now we want to find the equation that characterizes y  . Let  ∈ D(D \ S) with (T ) = t (T ) = 0. We know that Y(s) solves

T 0



s

(Y(s)tt − Y(s))d xdt +

T



0

s

∂ Ys ds dt = ∂n s



T



0

s

gs  ds dt (3.1)

Taking derivative w.r.t s at s = 0 in (3.1) we obtain:



T



0

(y  tt − y  )d xdt



T

+



0

(ytt − y)v ddt



∂n  s + ∇ · |s=0 y + (∇ · n)y  ∂s  0 +[∇(y ∇ · n) · n + H (y ∇ · n)]v ddt T = g ˙ + g∇ · V (0) + H gv ddt T



0

(3.2)

Recall that we can represent the outer normal vector to  in terms of the oriented distance function b = b as ∇b| = n, and that we have the following identities [17]: (D 2 b)∇b = 0, H = b (the additive curvature), ∇(∇ · ∇b) = (D 2 b)∇ + (D 2 )∗ ∇b. Therefore if we select the test function  such that it satisfies the following properties at the boundary: ∇ · n = 0 and (D 2 )n, n = 0 on , we can simplify (3.5) to



T



0

(y  tt − y  )d xdt



+ = 0

T



 0 T 

[(ytt − y )v + (∇ · ∇b )y] ddt

g ˙ + g∇ · V (0) + H gv ddt

(3.3)

From [17] (p.360) we know that ∇b | = (DV (0)n · n)n − (DV (0))∗ n − (D 2 b)V (0),

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Appl Math Optim

and therefore ∇b · n = 0 on .

(3.4)

Using (3.4) back in (3.2), we obtain the following identity:



T



0

(y  tt − y  )d xdt



+



T

0  T

=



0

[(ytt − y )v + (∇  · ∇ b )y] ddt

g ˙ + g∇ · V (0) + H gv ddt,

(3.5)

Integrating by parts in (3.2) and using the fact that b = −v on  ([12], Theorem 4.2, p. 2298) and the initial conditions y  (0) = yt (0) = 0, we obtain  ∂ 2 y  − y  d xdt ∂t 2  0  T   ∂y ∂2 y + −  (y v)  ddt + ∂n ∂t 2 0 τ  div (y ∇ v ) ddt =−



T



+

0 T

0







[g ˙ + g∇ · V (0) + H gv] ddt

Now we choose V (0) normal, i.e., V (0) = v n, and we have that the final variational identity for y  :  ∂ 2 y  − y  d xdt ∂t 2  0  T   ∂y ∂2 y + + −  (y v)  ddt  ∂n ∂t 2  0 τ div (y ∇ v ) ddt =−



T



+

0 T

0







[g ˙ + H gv] ddt

In conclusion, we obtained that y  satisfies the linear wave equation ytt − y  = 0 on ,

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Appl Math Optim

with the following boundary condition on : ∂ y ∂2 y = −div (v ∇ y) − v 2 +  (y v) + H g v + g˙ ∂n ∂t ∂2 y = div (y ∇ v) − v 2 + H g v + g˙ ∂t

(3.6)

4 Proof of Theorem 1.3 To prove existence of weak material derivative, we will take advantage of the regularity of solution for the linear wave equation and use the parameter differentiability technique for any functional expressed in form of a MinMax of a convex–concave Lagrangian with saddle points [11,15,17,32]. In fact, weak material derivative existence is immediate from any regularity result for a linear equation. Here is a summary of the details. Let Ps ∈ L(A, B) be a linear operator depending on a perturbation parameter s (A, B are Banach spaces). We assume that for any f (s) ∈ B the problem Ps ys = f (s) is well-posed. Now for any θ ∈ A , we consider the functional J (s) = y s , θ  A×A , where y s = ys ◦ Ts . Note that J can be written as J (s) = I n f φ∈A Supψ∈B L(s, φ, ψ), where the Langrangian L is given by L(s, φ, ψ) = φ, θ  A×A + Ps φ − f (s), ψ B×B  . Now we consider the following problem: find the pair (y s , p s ) ∈ A × B that solves the linear system:  ∂φ L(s, y s , p s ; φ) = 0 ∂ψ L(s, y s , p s ; ψ) = 0

 ⇔

Ps∗ p s = −θ Ps y s = f

Since the state and adjoint equations are well-posed, the pair (y s , p s ) exists and is unique in A × B. Since the Lagrangian L(s, ·, ·) is convex–concave on A × B, this pair (y s , p s ) is the unique saddle point of L(s, ·, ·) over A × B. Therefore we know from [11] that the functional J can be written in Min Max form as follows: J (s) = Min φ∈A Maxψ∈B L(s, φ, ψ). Now we assume that for any p ∈ B  the map s →  f (s), p is differentiable. Moreover, we assume that the maps s → y s and s → p s are weakly continuous. Then

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Appl Math Optim

we obtain that J is differentiable at 0 and the derivative J  (0) is given by J  (0) =

∂ L(0, y 0 , p 0 ). ∂s

Lastly, we can write J  (0) = P  y 0 , p 0  −  f  , p 0 , 

d Ps |{s=0} , and where p 0 is the solution to the adjoint problem P0∗ p 0 = with P  = ds −θ . Now we apply this general framework to our case. Recall that the non-cylindrical wave problem that we are considering is ⎧ 2 ∂ ys ⎪ ⎨ ∂t 2 − ys = 0 I × s = Q s ys = 0 I×S ⎪ ⎩ ∂ ys = g(s) I × s =  s , ∂n s

(4.1)

with initial conditions ys (0, ·) = (ys )t (0, ·) = 0 on s . From (1.3)–(1.4), we know that for each s ∈ [0, s ∗ ), (4.1) has unique solution ys ∈ H = L ∞ (I, H∗1 (s )) ∩ W 1,∞ (I, L 2 (s )), and we have the following estimate for the H-norm of the solution: ∃ks > 0 (continuous at s = 0) such that ys 2H ≤ ks gs 2W 1,1 (I,H −1/2 ( )) . s

(4.2)

Let y s = ys ◦ Ts ∈ L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()) be the transported solution in the non perturbed geometry. Note that y s satisfies the following wave system ⎧ s −1 s ⎪ ⎨ ytt − Js div(As ∇ y ) = 0 I ×  = Q I×S ys = 0 ⎪ ⎩ s s I ×  = , A s ∇ y · n = g ws

(4.3)

where Js = det(DTs ) is the Jacobian of the transformation Ts (V ), the matrix As is defined as As = J (s)(DTs )−1 (DTs )−∗ , g s = gs ◦ Ts , and the density ωs is given as ωs = Js |(DTs )−∗ n|. The initial conditions associated with (4.3) are y s (0, ·) = yts (0, ·) = 0 on .

123

(4.4)

Appl Math Optim

For R ∈ W 1,1 (I, H −1 ()) and q ∈ W 1,1 (I, H −1/2 ()), we define the following functional



T

J (s) =



0



y Rd x + s

s



y q d

dt.

Following a classical tool developed in structural mechanics, we can write J as J (s) = inf sup L(s, φ, ψ), φ∈E ψ∈E 

where the Lagrangian L is given by

τ

L(s, φ, ψ) = 0





τ

+ 0





 

φ Rd x + (−J (s)



φ q d

∂φ ∂ψ ∂t ∂t



τ

+A(s)∇φ, ∇ψ)d xdt − 0

dt



ω(s) g s ψ ddt.

and E and E  are the following Banach spaces E = {φ ∈ L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()) | φ(0, ·) = 0 in }, and E  = {ψ ∈ L ∞ (I, H∗1 ()) ∩ W 1,∞ (I, L 2 ()), ψ(T, ·) = 0 in }. Clearly, the map φ → L(s, φ, ψ) is convex on E and the map ψ → L(s, φ, ψ) is concave on E  . Moreover, the state equations (4.3) are equivalent to ∂ψ L(s, y s , p s ; ψ) = 0, ∀ψ ∈ E  . The adjoint state equations are given by ∂φ L(s, y s , p s ; φ) = 0, ∀φ ∈ E ⇔ ⎧ s −1 s ⎪ ⎨ ptt − Js div(As ∇ p ) = −R I ×  = Q I×S ps = 0 ⎪ ⎩ I ×  = , As ∇ p s · n = −q

(4.5)

From the regularity of R and q, we know what Eq. (4.5) has unique solution in E  . Therefore (y s , p s ) is the unique saddle point of L(s, ·, ·) on E × E  .

123

Appl Math Optim

Moreover, the Lagrangian is Gateaux differentiable with respect to s, and the map s → (y s , p s ) is weakly continuous. Therefore we have that J  (0) =

∂ L(0, y 0 , p 0 ). ∂s

At s = 0 the unique saddle point of L is (y, p) = (y 0 , p 0 ), where the co-state p is the solution to the following problem: ⎧ ptt − p = −R on Q ⎪ ⎪ ⎪ ⎨ ∂ p = −q on  ∂n ⎪ p=0 on I × S ⎪ ⎪ ⎩ p(T ) = 0

(4.6)

Since R ∈ W 1,1 (H −1 ()) and q ∈ W 1,1 (H −1/2 ()), we know that the previous problem has unique solution p ∈ L ∞ (H∗1 ()) ∩ W 1,∞ (L 2 ()). Lastly, the functional L is differentiable w.r.t s at s = 0 and we obtain the explicit expression for its derivative w.r.t s at s = 0: ∂ L(0, y, p) ∂s  τ  ∂y ∂p = −divV (0) + [divV (0) − 2(V (0)) ]∇ y, ∇ p d xdt ∂t ∂t 0 τ  − p (g(V ˙ ) + H g v )ddt

J  (0) =

0



d where 2(V ) = DV + (DV )∗ , g(V ˙ ) = [ ds (gs Ts (V ))]s=0 , v = V (0), n, and H is the mean curvature of the boundary. In conclusion, we obtained the existence of the weak material derivative of the map ys in [W 1,1 (H −1 ())] , and the weak differentiability of the trace mapping: s → y s | in [W 1,1 (H −1/2 ())] .

Acknowledgments The authors would like to thank the anonymous referees for their insightful comments and corrections. The research of first author was partially supported by NSF grant DMS-1312801.

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