On the Linear Water Wave Problem in the Presence of a Critically Submerged Body

On the Linear Water Wave Problem in the

Presence of a Critically Submerged Body

I. V. Kamotski and Vladimir Maz´ya

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

I. V. Kamotski and Vladimir Maz´ya, On the Linear Water Wave Problem in the Presence of

a Critically Submerged Body, 2012, SIAM Journal on Mathematical Analysis, (44), 6,

4222-4249.

http://dx.doi.org/10.1137/120868074

Copyright: Society for Industrial and Applied Mathematics

http://www.siam.org/

Postprint available at: Linköping University Electronic Press

ON THE LINEAR WATER WAVE PROBLEM IN THE PRESENCE OF A CRITICALLY SUBMERGED BODY∗

I. V. KAMOTSKI† ANDV. G. MAZ’YA‡

Abstract. We study the two-dimensional problem of propagation of linear water waves in deep

water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Maz’ya in 1978, we prove an important property of the scattering matrix, which may be inter-preted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g.,H2

loc∩ L∞) without use of the radiation

conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

Key words. water waves, limiting absorption principle, radiation conditions, uniqueness,

do-mains with cusps

AMS subject classifications. 35J05, 35C20, 35J25, 35P25, 76B15 DOI. 10.1137/120868074

1. Introduction. We study the problem of propagation of linear water waves

in a domain Ω, which represents water of infinite depth in the presence of a critically submerged body Ω. Let us describe the domain Ω. We fix a Cartesian system x = (x₁, x₂) with the origin O and consider a bounded domain Ω ⊂ R2

+ = {(x1, x2) ∈

R2_{: x}

2> 0} (notice that the axis x2 points downward). We assume that S := ∂ Ω is

smooth and touches the water surface Γ :={x₂= 0} only at the origin O. Further we define Ω :=R2₊\ Ω and set Ω_τ:= Ω∩ {|x₁| < τ, x₂< τ}, where τ is a small positive

number. We assume that Ω_τ coincides with the set (1.1) {x : |x₁| < τ, 0 < x₂< φ(x₁)}, where φ is a function from C2_{[−τ, τ], such that}

(1.2) φ(0) = φ(0) = 0 and

(1.3) κ := φ(0) > 0.

Moreover, let φ be strongly decreasing on (−τ, 0) and strongly increasing on (0, τ). The governing equations are the following:

(1.4) Δu = f in Ω,

∗_{Received by the editors February 29, 2012; accepted for publication (in revised form)}

Septem-ber 13, 2012; published electronically DecemSeptem-ber 13, 2012. http://www.siam.org/journals/sima/44-6/86807.html

†_{Department of Mathematics, University College London, Gower Street, London, WC1E 6BT,}

UK (i.kamotski@ucl.ac.uk).

‡_{Department of Mathematics, Link¨oping University, SE-581 83 Link¨oping, Sweden (vlmaz@mai.}

liu.se).

4222

ON THE LINEAR WATER WAVE PROBLEM 4223 (1.5) ∂_nu = g₁ on S\ O,

(1.6) ∂_nu− νu = g₂ on Γ\ O,

where n is the external normal to Ω, ν > 0 is a fixed spectral parameter, and f, g₁, g₂

are given functions.

The linear water wave problems for fully submerged and semisubmerged bodies in deep water has been studied extensively; see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. The presence of a critically submerged body implies that the domain Ω contains two external cusps. The problems in domains with cusps were studied from various points of view in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] (see also [45], where more references can be found).

Our main condition on Ω is as follows.

Condition 1. The homogeneous problem (1.4)–(1.6) does not have nontrivial solutions in the energy space V (Ω) ={u :

Ω|∇u|2dx +



∂Ω|u|2ds < +∞}.

This condition may hold for many fully submerged bodies. For example, it is well known (see [3, 10]) that the following geometric condition implies the uniqueness for fully submerged bodies.

Condition 2. Let n(x) = (n₁(x), n₂(x)) be the unit normal to S, external to Ω. Then we have

(1.7) x₁(x2₁− x2₂)n₁(x) + 2x2₁x₂n₂(x)≥ 0, x ∈ S.

One of the results of this paper is that Condition 2 still implies uniqueness for the case of critically submerged bodies (see also [46], where similar problem without cusps had been considered); in fact we can say more—see Theorems 4.4, 4.6, and 4.7. We are interested in the existence of solutions which satisfy an outgoing radiation

condition at infinity (see (2.5) below for the precise definition):

(1.8) u∼ d+e− iνx1−νx2 _{as x}

1→ +∞ and u ∼ d−eiνx1−νx2 as x1→ −∞,

where d+ _{and d}− _{are some constants.}

If Ω is completely submerged (i.e., there is no cusp, Ω∈ R2

+), the existence of a

solution to (1.4)–(1.6) satisfying radiation conditions at infinity follows immediately under Condition 1; see [10] and the references therein. Our situation is more subtle, for two reasons: the first reason is purely technical, namely, that we cannot directly apply the method of [10], which was based on integral equations, due to the presence of the cusps.

The other reason is that, depending on the parameter ν, the solutions might

not be in H_loc1 (Ω). The situation is in fact even more complicated: there may be many “reasonable” solutions, and so we need to select only one. The latter implies that we need to additionally employ new conditions at the cusp. We will refer to them as radiation conditions at the cusp since they play similar role to radiation conditions (1.8). To be more precise, we prove, under suitable conditions on f, g₁, g₂

and assuming Condition 1, that there is a unique solution to (1.4)–(1.6) satisfying radiation conditions at infinity and such that, provided ν > κ/8,

(1.9) u∼ c₁x−1/2+i √_2ν κ−14 1 , x1→ +0, u∼ c2|x1|−1/2+i √_2ν κ−14, x₁→ −0.

In the case ν < κ/8 we have (1.10) u∼ c₁x−1/2+ √₁ 4−2νκ 1 , x1→ +0, u∼ c2|x1|−1/2+ √₁ 4−2νκ, x₁→ −0.

In the above formulae c₁and c₂are some constants. For the case ν = κ/8 we have the same expressions as in (1.9), but Condition 1 needs to be modified; see Condition 1 in section 3.

Let us mention that radiation conditions for the water wave problems in finite geometry have been studied in [18] and [42].

The presence of radiation conditions both at infinity and at the origin presents new challenges. In particular, we need to employ to this end a nonstandard version of the limiting absorption principle; cf., e.g., [47].

Asymptotic representations (1.9) and (1.10) show, in particular, that if ν < κ/8, then the solution is in the space H_loc1 (Ω). In the case ν ≥ κ/8 the solution does not belong to H_loc1 (Ω), in general. Moreover, in the latter case there are many solutions with a similar type of behavior; we show, however, that the condition (1.9) fixes the unique one.

Expressions (1.8) and (1.9) can be interpreted as “outgoing waves” and their complex conjugates as “incoming waves.” This introduces, for ν > κ/8, a 4× 4

scattering matrix describing the relation between the incoming and outgoing solutions

at both infinities but also at two cusps.

We study properties of this scattering matrix and show that, apart from the standard ones of unitarity and symmetry, it has more subtle “block properties”; see Theorem 4.3. The latter ensures in particular that any combination of waves coming in from the infinities will at least partially “scatter in the cusps” (and vice versa). This may be interpreted as the absence of an analogue of total internal reflection (i.e., of “infinity to infinity” or of “cusp to cusp” scattering).

The crucial ingredient for establishing the above properties of the scattering ma-trix is the uniqueness theorem, Theorem 4.4, roughly in the class of arbitrary com-binations of the cusp incoming and outgoing waves (in fact in the class of functions which can grow as|x₁|−M for any finite M , as|x| → 0). We prove this by showing that, under Condition 2, the method of multipliers (see, e.g., [3, 10]) surprisingly works also in the presence of functions singular at the cusp.

Moreover, the properties of the scattering matrix allow us to establish the unique-ness and existence results for problem (1.4)–(1.6) in various functional spaces. We prove that if ν > κ/8, and f, g are regular enough and have a compact support, then there exists a unique solution of problem (1.4)–(1.6) in the space H2

loc(Ω)∩ L∞(Ω).

Under the same conditions we also establish the existence and uniqueness results in the space H2

loc(Ω\ O) ∩ Lp(Ω), p∈ (2, 6) (the result is not true for other values of p;

see (1.8) and (1.9)). The former may be interpreted as a solution with no waves either incoming or outgoing to the cusps (hence bounded), and the latter with no similar waves either from or to infinity (hence a localized solution in some sense). In particular these spaces of functions do not differentiate between the incoming and the outgoing waves and the radiation conditions are no longer employed.

The paper is organized as follows: In section 2 we consider the problem without a submerged body and derive some useful estimates which are employed in section 3. There we prove the existence of the solution of (1.4)–(1.6) in the space of functions with the radiation conditions, using the limiting absorption principle. In the last section we introduce the scattering matrix for problem (1.4)–(1.6), prove some of its properties, and establish the uniqueness and existence results for (1.4)–(1.6) in various spaces of functions without radiation conditions.

ON THE LINEAR WATER WAVE PROBLEM 4225

2. Problem in R2₊. Here we consider an auxiliary problem in the entire

half-space:

(2.1) Δu = f inR2₊,

(2.2) ∂_nu− νu = 0 on Γ.

(Here f is not necessarily the same as before; the precise assumption on f are given below.) We are interested in solutions which satisfy the following radiation condition

at infinity: u can be represented as a sum of two outgoing waves and of a function

decaying at infinity. To make this more precise we define the outgoing waves at

infinity, e.g.,

(2.3) u−₁(x) = χ(x₁)e−iνx1−νx2_{, u}−

2(x) = χ(−x1)eiνx1−νx2,

where χ is the cut-off function, such that

(2.4) χ∈ C∞(R), χ(t) = 0 for t < N, χ(t) = 1 for t > 2N,

and N is a fixed positive number. (Physically, function u−₁ represents an outgoing wave moving to the right, respectively, the outgoing wave u−₂ moving to the left.) Then we say that u satisfies radiation condition at infinity (see, e.g., [10]) if

(2.5) u = c₁u−₁ + c₂u₂−+ ˜u, in Ω\ B_N, |˜u| + |x||∇˜u| = O(|x|−1) as|x| → ∞, where c₁and c₂ are some constants and B_N ={x : x < N}.

The existence of a solution which satisfies radiation conditions (under certain assumptions on f ) is well known; see, e.g., [10]. Below we discuss the relation of this solution to the limiting absorption principle and derive some useful estimates which we will apply in the next section.

Consider now the problem with a small absorption described by ε > 0: (2.6) Δu_ε− iεu_ε= f inR2₊,

(2.7) ∂_nu_ε− νu_ε= g₂on Γ.

In order to describe precisely a solution of (2.6), (2.7) we introduce the following spaces. Denote x_j = (1 + x2

j)1/2, j = 1, 2, and for real β, γ and l = 0, 1, . . . , for

relevant domain Θ, let (2.8) W_β,γl (Θ) = ⎧ ⎨ ⎩u :  |δ|≤l  Θ e2βx1_x 1 2γx2 2|∇δu|2dx <∞ ⎫ ⎬ ⎭, (2.9) H˙l(Θ) = ⎧ ⎨ ⎩u :  1<|δ|≤l  Θ |∇δ_u|2_{dx +}  Θ x2 −2|u|2dx <∞ ⎫ ⎬ ⎭, l≥ 1 , with the corresponding definitions of the norm and of the trace spaces. For the case Θ =R2

+ we omit the dependence on the domain in the notation.

Application of the Fourier transform with respect to x₁ and shift of the contour of integration (see, e.g., [20, 48]) yields the following result.

Lemma 2.1. Let β > 0 and let ε be such that β >−Im(ν2− iε)1/2 and ε√1/2

2 >

−Im(ν2_{− iε)}1/2_{. Suppose further that f∈ W}0

β,γ and g2∈ Wβ,γ1/2(Γ), and γ∈ R. Then

there exists a unique solution u_ε ∈ W_0,γ2 of problem (2.6)–(2.7), and the following representation holds: (2.10) u_ε= bε₁U₁ε+ bε₂U₂ε+ ˜u_ε, where ˜u_ε∈ W_β2∗_,γ, β∗< min  β,ε 1/2 √ 2 . Here (2.11) U₁ε(x) = χ(x₁)e−i(ν2−iε)1/2x1−νx2_, (2.12) U₂ε(x) = χ(−x₁)ei(ν2−iε)1/2x1−νx2_,

bε₁, bε₂ are constants, and

(2.13) |bε₁| + |bε₂| + ˜u_εW2 β∗,γ ≤ c  fW0 β,γ +g2Wβ,γ1/2  .

Remark 2.1. Clearly we have U0

j = u−j, j = 1, 2.

The constant c appearing in (2.13) depends on ε. The estimate which appears in the next lemma overcomes this disadvantage.

Theorem 2.2. _{Suppose that the conditions of Lemma 2.1 hold, and additionally}

let us assume that γ = 1 and g₂= 0. Then the following estimate holds: (2.14) |bε₁| + |bε₂| + ˜u_εH˙2 ≤ cf_W0

β,1,

where c does not depend on ε.

(Henceforth c is a constant whose value may change from line to line.) Theorem 2.2 is proved in the appendix.

The above statement allows us to pass to the limit in (2.6), (2.7), and we have

bε₁→ b₁, bε₂→ b₂, and ˜u_εconverges to ˜u weakly in the space ˙H2as ε→ 0. As a result we obtain a solution u to problem (2.1), (2.2), which can be represented in the form

u = b₁u−₁ + b₂u−₂ + ˜u.

3. Critically submerged body. Consider now the original problem (1.4)–(1.6)

with critically submerged body Ω. Let us associate with this problem an “energy space”: V ={u :_Ω|∇u|2_{dx +}

∂Ω|u|2dS <∞}. Let us notice that

(3.1)  Ω (x2₂+ 1)−1|v|2  Ω |∇v|2₊  ∂Ω|v| 2_dS.

(From now on,  denotes ≤ c with a constant c.) This inequality follows from two obvious inequalities, (3.2)  Ω∩{|x1|>N} |u|2 x2 2+ 1 dx  Ω∩{|x1|>N}  _∂x∂u₂2dx +  {x2=0,|x1|>N} |u|2_dx 1,  Ω\BN∩{|x1|<N} |u|2 x2 2+ x21 dx  Ω\BN |u|2 x2 2+ x21 dx (3.3)   Ω\BN 1 r  ∂u_∂θ2dθdr +  {x2=0,|x1|>N} |u|2_dx 1,

ON THE LINEAR WATER WAVE PROBLEM 4227 and the Friedrichs inequality

(3.4)  Ω∩BN |u|2_dx  Ω∩BN |∇u|2_{dx +}  ∂(Ω∩BN) |u|2_dS,

which is valid for any bounded domain; see [49, section 6.11.1]. Here we assume that constant N from the previous section is such that Ω⊂ B_N :={x : x < N}.

We are planning to find a solution to problem (1.4)–(1.6) by employing the princi-ple of limiting absorption. In fact we need an absorption in (1.4) and in the boundary conditions (1.5), (1.6), locally in the neighborhood of the origin.

Let us fix a cut-off function μ ∈ C₀∞(Γ), such that μ(x₁) = 1, |x₁| < N, and

μ(x₁) = 0, |x₁| > 2N. Consider now the following problem with a small absorption

ε≥ 0,

(3.5) Δv_ε− iεv_ε= f in Ω, (3.6) ∂_nv_ε+ iεv_ε= g₁ on S, (3.7) ∂_nv_ε− (ν − iεμ) v_ε= g₂ on Γ, and the corresponding energy space:

V :=  u :  Ω |∇u|2_{dx +}  ∂Ω|u| 2_{dS +}  Ω |u|2_{dx <∞} .

Lemma 3.1. Let g₁ ∈ L₂(S), g₂ ∈ L₂(Γ), and f ∈ L₂(Ω). Then for any ε > 0

there exists a unique solution v_ε∈ V of problem (3.5)–(3.7).

Proof. Let us associate with (3.5)–(3.7) a variational problem: Find v_εsuch that (3.8) a_ε(v_ε, ϕ) := F (ϕ)∀ϕ ∈ V. Here a_ε(v, ϕ) :=  Ω ∇v · ∇ϕ dx −  Γ (ν− iεμ(x₁)) vϕ dx₁ (3.9) + iε  S vϕ dS + iε  Ω v_εϕ dx and (3.10) F (ϕ) :=−  Ω f ϕ dx +  Γ g₂ϕ dx₁+  S g₁ϕ dS.

The sesquilinear form a_ε(·, ·) is clearly continuous and coercive on V, and F is an anti-linear continuous functional on V, and the application of the Lax–Milgram lemma (see, e.g., [50]) gives us a unique solution v_εfrom energy space V. Due to ellipticity, the local estimates give us v_ε∈ H2(Ω\ B_σ) for any positive σ.

We aim to pass to the limit in (3.8) as ε→ 0. The main difficulty is the absence of compactness of embedding of H1(Ω) into L₂(∂Ω) and L₂(Ω) due to the fact that Ω is unbounded, and lack of compactness of embedding of H1_{(Ω) into L}

2(∂Ω) in the

neighborhood of the origin due to the presence of the external quadratic cusps. To overcome this problem, we need to employ more detailed information about properties of the solutions.

We start from the description of v_εin right and left neighborhoods of the origin: (3.11) Ω+_τ := Ω_τ∩ {0 < x₁}, Ω−_τ := Ω_τ∩ {x₁< 0}, Υ±_τ := ∂Ω±_τ ∩ ∂Ω . In order to describe precisely a solution of (3.8), we need to introduce the following weighted Sobolev spaces: Let Ξ be a domain and let γ be real, l = 0, 1, . . .; then we define W_γl(Ξ) and V_γl(Ξ) as the closures of the set C₀∞(Ξ\ O) with respect to the norms (3.12) u2_Wl γ(Ξ):=  |δ|≤l  Ξ |x1|4(γ−l+|δ|)|∂xδu|2dx, (3.13) u2_Vl γ(Ξ):=  |δ|≤l  Ξ |x1|2(γ−l+|δ|)|∂xδu|2dx, respectively, where δ∈ Z2

+ is the usual multi-index. Furthermore, for l≥ 1 we define

Wγl−1/2(∂Ξ) and V_γl−1/2(∂Ξ) as the trace space forW_γl(Ξ) and V_γl(Ξ) on the boundary

∂Ξ.

Finally we define the space

V2

γ(Ω+τ) ={u ∈ V2γ2(Ω+τ) : P2u∈ Wγ2(Ω+τ)},

with the norm

(3.14) u_V2

γ(Ω+τ)=uV2γ2(Ω+τ)+P2uWγ2(Ω+τ).

Here the projection operator P₂ is defined as follows. We represent u∈ V_2γ2(Ω+_τ) as (3.15) u(x₁, x₂) = u₁(x₁) + u₂(x₁, x₂), 0 < x₁< τ, 0 < x₂< φ(x₁), where u₁(x₁) = φ(x₁)−1  _φ(x1) 0 u(x₁, x₂)dx₂, and we define P₁u := u₁, P₂u := u− u₁:= u₂.

We also define the fully analogous spaceV2

γ(Ω−τ).

One of characteristic properties of the above scale of spaces is the following. If

u∈ V2 γ(Ω+τ), then for (3.16) Lu := (Δ − iε)u and (3.17) Bu :=  (∂_n+ iε)u|_S∩Υ+ τ, (∂n− ν + iε)u|Γ∩Υ+τ 

we have{Lu, Bu} ∈ W_γ0(Ω+_τ)× W_γ1/2(Υ+_τ). Denote (3.18) λ_ε:= 2(ν− 2iε)/κ − 1/4. The following theorem was proved in [42].

ON THE LINEAR WATER WAVE PROBLEM 4229 Theorem 3.2 (see [42, Theorem 4.3]). Let γ= 1/2 ± 1/2Im√λ_ε. Suppose that

u_ε∈ V2

γ(Ω+τ) is a solution of the problem

(3.19) Δu_ε− iεu_ε= f in Ω+_τ,

(3.20) ∂_nu_ε+ iεu_ε= g₁ on S∩ Υ+_τ,

(3.21) ∂_nu_ε− (ν − iε) u_ε= g₂ on Γ∩ Υ+_τ, where g := (g₁, g₂) and (f, g) ∈ W0

γ(Ω+τ)× Wγ1/2(Υ+

τ). Then for any ε, there exists

δ₀> 0 such that, for any 0 < δ < δ₀, solution u_εsatisfies the estimate

(3.22) u_V2 γ(Ω+δ/2)≤ c  f_W0 γ(Ω+δ)+gWγ1/2(Υ_δ+)+uL2(Ω+_δ\Ω+_δ/2)  .

Here constant c is independent of f, g, and u_ε.

The next theorem from [42] describes the structure of the solution.

Theorem 3.3. _{Let γ,γ1 be real numbers, and let γ, γ1} = 1/2 ± 1/2Im√_{λε and} (f, g)∈ W_γ0₁(Ω+_τ)× W_γ1/2₁ (Υ+_τ). Suppose that v_ε∈ W_γ2(Ω) is a solution of the problem (3.19)–(3.21). Then the solution v_ε admits the representation

(3.23) v_ε= c+Y_ε++ c−Y_ε−+ Y_ε in Ω+_δ for sufficiently small positive δ. Here Y_ε∈ V2

γ1(Ω

+

δ), Yε± are solutions of the

homoge-neous problem (3.19)–(3.21), and c± are constants.

Properties of the special solutions Y_ε± had been described in [42] (see Theo-rem 4.4), which can be reformulated in our context as follows.

Theorem 3.4. Suppose that in (3.18) λ_ε = 0. Then there exist solutions Y_ε+

and Y_ε− of the homogeneous problem (3.19)–(3.21) in Ω+_δ for small enough positive δ, such that (3.24) Y_ε±(x) = y₁±(x₁, ε) + y₂±(x, ε),  _φ(x1) 0 y₂±(x)dx₂= 0, x₁< δ, where (3.25) y₁±(x₁, ε) = x−Λ±ε 1 +y1±(x1, ε), y±1 ∈ V2γ2±(Ωδ)∀γ± >∓Im  λ_ε/2, y₂±(x₁, x₂, ε) = x2−Λ±ε 1 Q±ε(x2/φ(x1)) +y2±(x, ε), y2±∈ Wγ2±(Ωδ)∀γ±>∓Im  λ_ε/2. Here (3.26) Q±_ε(z) = κ 2(ν− iε)  (z− 1)2 2 − 1 6  −κ 2  κΛ±_ε + iε z 2 2 − 1 6  and Λ±_ε = 1/2± iλ_ε. Remark 3.1. Let us note that Y_ε±∈ V2

σ±(Ω+δ)∀σ±>∓Im

√

λ_ε/2 + 1/2. It will be

useful in what follows to use another representation for Y_ε±instead of (3.24), namely, (3.27) Y_ε±= v±_ε +v±_ε,

where (3.28) v±_ε(x) =|x₁|−Λε±₊|x₁|2−Λ±ε_Q± ε (x2/φ(x1)) and v±ε ∈ Vγ2±(Ω + δ) ∀γ±>∓Im  λ_ε/2.

Remark 3.2. In the case λ_ε= 0, i.e., ε = 0 and ν = κ/8 (see (3.18)), functions Y_ε± do still exist and belong toV2

σ±(Ω

+

δ)∀σ±> 1/2. We have the following representation

for them: (3.29) Y₀±= v±+v±, where (3.30) v−(x) =|x₁|−1/2+|x₁|3/2Q−₀ (x₂/φ(x₁)) , v+(x) =|x₁|−1/2ln|x₁| + |x₁|3/2ln|x₁|Q+₀ (x₂/φ(x₁)) (3.31) +|x₁|3/2Q (x₂/φ(x₁)) , and v±∈ V2 γ(Ωδ) ∀γ > 0.

Here Q±₀ is defined according to (3.26) (Q−₀ = Q+₀ in this case) and (3.32) Q(z) =−κ 2 2  z2 2 − 1 6  .

As we have seen, the structure of the solution crucially depends on the relation between ν and κ/8 (which determines the real part of λ_εaccording to (3.18)). Let us start from the most singular case, ν > κ/8.

Let us check the implications of the above theorems for the solution v_ε of the boundary value problem (3.5)–(3.7), under assumption that the pair

(f, g)∈ W₀0(Ω+_τ)× W₀1/2(Υ+_τ)

and has compact support. It follows from Theorems 3.3 and 3.4 and Remark 3.1 that

v_ε= d_εv+_ε + c_εv−_ε + w_ε in Ω+_τ,

where w_ε∈ V2

0(Ω+τ) and cε, dε are some constants. Moreover, Theorem 3.4 provides

information about functions v+

ε and v−ε, i.e., v+_ε(x) = O  x−1/2−i(λ₁ 0−4iεκ−1)1/2  , v−_ε(x) = O  x−1/2+i(λ₁ 0−4iεκ−1)1/2  as x₁→ 0, and consequently d_ε= 0 since vε∈ H1_(Ω).

It is easy to see that wε∈ V2

0(Ω+τ) solves the problem

(3.33) Δwε= f−Δ− iεc_εv−_ε + iεwεin Ω+_τ,

ON THE LINEAR WATER WAVE PROBLEM 4231 (3.34) ∂_nwε= g₁−∂_n+ iεc_εv−_ε − iεwε on S∩ Υ+_τ,

(3.35) ∂_n− νwε= g₂−∂_n− ν + iεc_εv−_ε − iεwεon Γ∩ Υ+_τ.

Clearly R_ε:=Δ− iεv−_ε,{∂_n+ iεv−_ε,∂_n− ν + iεv−_ε}∈ W0

γ(Ω+τ)× Wγ1/2(Υ+

τ)

∀γ > 2−1_Im√_λ

ε. Moreover, for any γ > 0, its norm is uniformly bounded with respect

to ε, due to explicit expression for v−_ε; see (3.28). Next, applying Theorem 3.2 (with

ε = 0), we obtain wεV2 γ(Ω+δ/2)≤ c  f_W0 γ(Ω+δ)+gWγ1/2(Υ+_δ)+wεL2(Ω+_δ\Ω+_δ/2)+|cε| + εwεWγ0(Ω+δ)  (3.36)

for any γ > 0 such that γ= 1/2 − 1/2Im√λ_ε. Consequently for any γ > 0, γ= 1/2, and for sufficiently small ε > 0, we have

(3.37) w_εV2 γ(Ω+δ/2)≤ c  f_W0 γ(Ω+δ)+gWγ1/2(Υ+_δ)+wεL2(Ω+_δ\Ω+_δ/2)+|cε|  .

Let us emphasize that the constant c in the above formula is independent of ε, f,

g, c_ε, and w_ε.

Following the same reasoning, for v_εin Ω−_τ we obtain (3.38) v_ε(x) = b_εv−_ε(x) + w_ε(x), x∈ Ω−_τ,

where b_εis some constant, w_ε∈ V2

γ(Ω−τ) for any γ > 0, and

(3.39) w_εV2 γ(Ω−δ/2)≤ c  f_W0 γ(Ω−δ)+gWγ1/2(Υ−_δ)+wεL2(Ω−_δ\Ω−_δ/2)+|bε|  ,

where the constant c is independent of ε, f, g, b_ε, and w_ε.

Now let us consider v_ε in Ω\ B_2N. Using Theorem 2.2 (we choose N such that supp g⊂ (−N, N)), we obtain (3.40) v_ε= bε₁U₁ε+ bε₂U₂ε+w_ε and (3.41) |bε₁| + |bε₂| + ˜u_εH˙2_(Ω\B2N)≤ c  f_W0 β,1+wεL2(Ω∩BN)  ,

where the constant c in the above formula is independent of ε, f, g, and w_ε.

In the intermediate region, say Ω∩B_2N\Ω_δ/2, we apply the usual elliptic estimates, yielding (3.42) vε H2_{(Ω∩B2N\Ωδ/2)}≤ c  fL2(Ω∩B4N\Ωδ/4)+gH1/2(∂Ω)+vεL2(Ω∩B4N\Ωδ/4)  ,

where c obviously does not depend on ε.

Now we are going to combine estimates (3.37), (3.39), (3.41), and (3.42). With this purpose we introduce the weighted space

(3.43) H2_γ(Ω) :={v ∈ H_loc2 (Ω\ O) : v ∈ V_γ2(Ω_τ),v ∈ ˙H2(Ω\ B_τ/2)}, v2 H2 γ(Ω):=v 2 V2 γ(Ωτ)+v2_H˙2_(Ω\Bτ/2),

and a space with “detached asymptotics”: (3.44) v ∈ H2_γ,ε(Ω)⇔ v = 4  j=1 c_jU_jε+v, v ∈ H2_γ(Ω), c_j∈ C, j = 1, . . . , 4. Here U₁ε and U₂εare as introduced in (2.11), and

(3.45) U₃ε(x) := ζ_τ+(x)v−_ε(x), U₄ε(x) := ζ_τ−(x)v−_ε(x),

ζ_τ±∈ C∞(Ω\O) are cut-off functions, such that ζ_τ±= 1 in Ω±_τ/2and ζ_τ±= 0 in Ω\Ω±_τ. We will refer toH2

γ,0(Ω) as space with radiation conditions at the infinity and at

the origin.

Finally, we obtain the following lemma. Lemma 3.5. Let {f, g} ∈ W0

0(Ω)× W 1/2

0 (∂Ω) and have compact support. Then

for any ε > 0 the unique solution v_ε∈ V of problem (3.5)–(3.7) ensured by Lemma 3.1 belongs to the space H2_γ,ε(Ω) for any γ > 0, in particular

(3.46) v_ε=

4



j=1

cε_jU_jε+v_ε.

Moreover, for any γ∈ (0, 1/2) and sufficiently small ε, the following estimate is valid: ˜vεV2 γ(Ωτ)+˜vε_H˙2_(Ω\Bτ/2) (3.47) ≤ c ⎛ ⎝fW0 γ(Ω)+gWγ1/2(∂Ω)+vεL2(Ω∩B4N\Ωτ)+ 4  j=1 |cε j| ⎞ ⎠ ,

where c does not depend on f, g, c_j, and ε.

In order to pass to the limit in (3.5)–(3.7), we need to demonstrate that the “extra” quantity which appears on the right-hand side of (3.47), namely,

b_ε:=v_{εL2(Ω∩B4N\Ωτ)}+ 4  j=1 |cε j| , is bounded.

Lemma 3.6. _{Under Condition 1, we have} (3.48) b_ε≤ c  fW0 γ(Ω)+gWγ1/2(∂Ω)  ,

where c does not depend on ε, f , and g.

Proof. Let us assume that b_εis not bounded; then there exists a subsequence ε_k, such that b_εk > kf_W0

γ(Ω)+gWγ1/2(∂Ω)



, k = 1, 2, . . . . Consider a “normalized”

subsequence of ˜vε, ˜uεk_:= ˜vεk

bεk (which we still denote ˜uε). The following representation

is now valid for this subsequence (cf. (3.46)):

u_ε= ˜u_ε+

4



j=1

αε_jU_jε

ON THE LINEAR WATER WAVE PROBLEM 4233 and (3.49) u_{εL2(Ω∩B4N\Ωτ)}+ 4  j=1 |αε j| = 1.

Then it follows from (3.47) that we can choose a subsequence (which we still denote u_ε) such that u_εrad u as ε→ 0. Here the “weak convergence with radiation

conditions,” denotedrad, is understood in the following sense:

1. u can be represented as u = ˜u + 4  j=1 α_jU_j0. 2. αε_j → α_j, ˜u_ε converges weakly to ˜u inH2

γ(Ω) for any γ > 0, and Ujε→ Uj0 in

H2_{(K), where K is any compact set not containing the singularity point O.}

Let us note that convergence of U_jε to U0

j follows from explicit expressions for

U₁ε, U₂ε (see (2.11), (2.12)) and explicit formulae for U₃ε, U₄ε (see formula (3.45) and Remark 3.1 to Theorem 3.4). This allows us to pass to the limit in (3.5)–(3.7) (with

u_ε instead of v_ε and _b1

ε{f, g} instead of {f, g}) and conclude that u ∈ H

2

γ,0(Ω) for

any γ > 0 and is a solution of the homogeneous problem. A standard trick with integration by parts (see Remark 4.1 below) shows that α_j = 0, j = 1, . . . , 4, and consequently u ∈ H_γ2(Ω) for any γ > 0 and in particular u ∈ V , which, due to Condition 1, implies u = 0. This contradicts (3.49), since, clearly, weak convergence of ˜u_ε in H2

γ(Ω) for any γ > 0 and convergence of αεj, j = 1, . . . , 4, imply the strong

convergence of u_ε in L₂ on compact sets.

Finally, combining Lemmas 3.1 and 3.5, and then treating v_εin the same way as

u_εin Lemma 3.6, we arrive at the following results.

Theorem 3.7. Suppose Condition 1 is satisfied and ν > κ/8. Assume further

that {f, g} ∈ W0

0(Ω)× W 1/2

0 (∂Ω) and have compact support. Then there exists the

unique solution of (1.4)–(1.6), v∈ H2

γ,0(Ω) for any γ > 0, in particular

(3.50) v =

4



j=1

c_jU_j0+v,

and for any γ∈ (0, 1/2), the following estimate is valid:

4  j=1 |cj| + ˜vV2 γ(Ωτ)+˜v_H˙2_(Ω\Bτ/2) (3.51) ≤ cfW_γ0_(Ω)+g_W1/2 γ (∂Ω)  ,

where c does not depend on f and g.

Moreover, this solution can be obtained as the result of the limiting absorption procedure, namely, vε rad v, where v_ε is a solution of (3.5)–(3.7).

If ν = κ/8, then we apply arguments as above, employing Remark 3.2 instead of Theorem 3.4. However, in order to obtain the result of Theorem 3.7, we need to employ a stronger condition.

Condition 1. The homogeneous problem (1.4)–(1.6) does not have nontrivial solutions in the space V(Ω) = {u : u = c+_x−12

1 ζτ++ c−|x1|− 1 2ζ_τ− +u, c± ∈ C,  Ω|∇u|2dx +  ∂Ω|u|2ds < +∞}.

The difference comes from the fact that we are able only to prove, in the analogue of Lemma 3.6, that the solution of the homogeneous problem from the space with radiation conditions is actually in energy space V(Ω). (For the case ν > κ/8 we were able to deduce that the solution is in V (Ω).) As a result, we conclude with the following corollary.

Corollary 3.8. If Condition 1 is satisfied, then condition ν > κ/8 in

Theo-rem 3.7 can be relaxed to ν ≥ κ/8.

If ν < κ/8, then there is no need to isolate waves in the cusp. Let us introduce the space with “detached asymptotics” at infinity only:

(3.52) v∈ F2_γ,ε(Ω)⇔ v = 2  j=1 c_jU_jε+v, v ∈ H2_γ(Ω), where c_j∈ C.

We have the following analogue of Lemma 3.5. Lemma 3.9. Let{f, g} ∈ W0

1/2(Ω)× W 1/2

1/2(∂Ω) and have compact support. Then

for any ε > 0 the unique solution v_ε of (3.5)–(3.7), delivered by Lemma 3.1, belongs to the space F2 1/2,ε(Ω), in particular (3.53) v_ε= 2  j=1 cε_jU_jε+v_ε.

Moreover, for sufficiently small ε, the following estimate is valid: ˜vεV2 1/2(Ωτ)+˜vεH˙2(Ω\Bτ/2) (3.54) ≤ c ⎛ ⎝fW0 1/2(Ω)+gW_1/21/2(∂Ω)+vεL2(Ω∩B4N\Ωτ)+ 2  j=1 |cε j| ⎞ ⎠ ,

where c does not depend on ε, f , and g.

Employing similar arguments to the above, we arrive at the following theorem. Theorem 3.10. Suppose Condition 1 is satisfied and ν < κ/8. Assume further

that {f, g} ∈ W0

1/2(Ω) × W 1/2

1/2(∂Ω) and have compact support. Then there is the

unique solution of (1.4)–(1.6), v∈ F2 1/2,0(Ω), in particular (3.55) v = 2  j=1 c_jU_j0+v,

and the following estimate is valid:

2  j=1 |cj| + ˜vV2 1/2(Ωτ)+˜vH˙2(Ω\Bτ/2) (3.56) ≤ c  f_W0 1/2(Ω)+gW_1/21/2(∂Ω)  ,

where c does not depend on f and g.

ON THE LINEAR WATER WAVE PROBLEM 4235

Moreover, this solution can be obtained as the result of the limiting absorption procedure, namely, vε rad v in the space F2_1/2,0, where v_ε is a solution of (3.5)–(3.7).

Remark 3.3. Now formula (1.10) follows from Theorems 3.3 and 3.4.

Finally let us comment on the applicability of Condition 1. It has been proved in [3] that in the case of a fully submerged body Condition 2 implies uniqueness. Various examples of bodies satisfying Condition 2 can be found in [10]. In particular, Condition 2 is satisfied by ellipses whose major axis is parallel to the x₂axis; see [51]. One can apply the method of [3] to the case of a critically submerged body. This method is based on multipliers techniques and integration by parts. So we only need to verify that integration by parts in the neighborhood of the origin is justified. For the case ν > κ/8, this can easily be seen, as a solution of the homogeneous problem (1.4)–(1.6), which is in V , belongs to the spaceH2

γ(Ω) for any γ; see Theorem 3.3. This

means that the solution decays quickly (in fact exponentially) in the neighborhood of the origin, and integration by parts is justified. In other words, for the case ν > κ/8 Condition 2 implies Condition 1. The same remains true if ν < κ/8. Then the solution of the homogeneous problem (1.4)–(1.6), which is in V , belongs to the space H2

γ(Ω)

for some γ < 1/2 (see Theorem 3.3), and analysis of possible singularities shows that we still can integrate by parts (see Theorem 4.4 below for details).

The critical case ν = κ/8, where we need to check Condition 1, is more subtle, but still one can prove that Condition 2 implies Condition 1; see Corollary 4.5.

4. Scattering matrix and its properties. Let us define the usual scattering

matrix for ν < κ/8. In this case we need to employ only waves at the infinity. First we need to introduce “incoming waves”:

(4.1) u+₁(x) = χ(x₁)eiνx1−νx2_{, u}+

2(x) = χ(−x1)e−iνx1−νx2;

compare the above with (2.3).

Theorem 4.1. Suppose that ν < κ/8 and Condition 1 is satisfied. Then

there exist two linearly independent solutions of the homogeneous problem (1.4)–(1.6), η_j, j = 1, 2, such that (4.2) η_j= u+_j + 2  n=1 s_jnu−_n + ˜η_j, where ˜η_j ∈ H2

1/2. Condition (4.2) determines ηj uniquely. The scattering matrix

s = (s_jn)2

j,n=1 is unitary and sjn= snj, j, n = 1, 2.

Proof. The arguments are standard; see, e.g., [48]. Consider, for example, case j = 1. We are looking for η₁ in the form

(4.3) η₁(x) = eiνx1−νx2_{+ ξ}

1(x),

where ξ₁(x) is a solution of (1.4)–(1.6) with f = 0, g₂= 0, and g₁=−∂_neiνx1−νx2|_S.

The solution to this problem exists in the space H2_1/2,0(Ω), due to Theorem 3.10. In particular (4.4) ξ₁= 2  j=1 c_jU_j0+v, where v ∈ H2

1/2(Ω). Since Uj0 = u−j (compare (2.11), (2.12) with (2.3)), we see that

η₁(x) = eiνx1−νx2_{+ ξ1(x) satisfies (4.2) with s11= c1 and s12= c2+ 1. Clearly this}

solution is unique. The same argument applies to η₂.

Next we verify the properties of scattering matrix s. We know that η_j solves the problem

(4.5) Δη_j = 0 in Ω, (4.6) ∂_nη_j = 0 on S, (4.7) ∂_nη_j= νη_j on Γ.

Let us multiply (4.5) by η_n, n = 1, 2, integrate over Ω∩{|x₁| < M}, and integrate by parts twice (which is justified since η_j ∈ H2

1/2(Ωτ) and due to conditions (2.5)).

We have 0 =  +∞ 0 η_n∂_x1η_j|_x1=Mdx₂−  +∞ 0 η_n∂_x1η_j|_x1=−Mdx₂ (4.8) −  _+∞ 0 η_j∂_x1η_j|_x1=Mdx₂+  _+∞ 0 η_j∂_x1η_j|_x1=−Mdx₂.

Next, using (4.2) and (4.1), we pass to the limit as M→ +∞ and obtain (4.9) 0 = δ_jn− 2  p=1 s_nps_jp. So s is indeed unitary.

Now the symmetry property s_jn= s_nj, j, n = 1, 2, follows easily, since (4.5)–(4.7) is a problem with real coefficients, s is unitary, and

(4.10) u−_j = u+_j, j = 1, 2.

In the case ν > κ/8, there are four linearly independent solutions to the homo-geneous problem (1.4)–(1.6), viewed as solutions of a scattering problem. First we renormalize functions U0 j, j = 3, 4 (see (3.28) and (3.45)): u−₃(x) := (ωκ)−12U₃0(x) = (ωκ)−12v−₀(x)ζ_τ+ (4.11) = (ωκ)−12|x₁|−12+ iω1 + x2₁Q−₀ (x₂/φ(x₁))ζ_τ+ and u−₄(x) := (ωκ)−12U₄0(x) = (ωκ)−12v−₀(x)ζ_τ− (4.12) = (ωκ)−12|x₁|−12+ iω1 + x₁2Q−₀ (x₂/φ(x₁))ζ_τ−, (4.13) ω :=  2ν κ − 1 4.

Similarly to (2.3), we will refer to these functions as outgoing waves in the cusps. Namely, u−₃ is the outgoing wave in the right cusp Ω+_τ and u−₄ is the outgoing wave in the left cusp Ω−_τ. In a similar way we introduce incoming waves in the cusps: (4.14) u+₃(x) := (ωκ)−12v+₀(x)ζ_τ+= (ωκ)−12x−12− iω

1



1 + x2₁Q+₀ (x₂/φ(x₁))ζ_τ+(x),

ON THE LINEAR WATER WAVE PROBLEM 4237 (4.15) u+₄(x) := (ωκ)−12v+₀(x)ζ_τ− = (ωκ)−12|x₁|−12− iω1 + x2₁Q+₀ (x₂/φ(x₁))ζ_τ−(x); see (3.28).

Theorem 4.2. _{Suppose that ν > κ/8 and Condition 1 is satisfied. Then there}

exist four linearly independent solutions of the homogeneous problem (1.4)–(1.6), η_j, j = 1, . . . , 4, such that (4.16) η_j= u+_j + 4  n=1 S_jnu−_n + ˜η_j, where ˜η_j ∈ H2

γ ∀γ > 0. Condition (4.16) determines ηj uniquely. The scattering

matrix S = (S_jn)4

j,n=1 is unitary and Sjn= Sjn, j, n = 1, . . . , 4.

Proof. The proof of the existence of η₁, η₂ follows the arguments of Theorem 4.1, with reference to Theorem 3.7 instead of Theorem 3.10. As for η₃and η₄, we need to take some more care. Consider, for example, η₃. Let us look for η₃ in the form (4.17) η₃= (ωκ)−12_Y+

0 ζδ++ ξ3,

where v+ _{is the function described in Theorem 3.4, δ is sufficiently small, and ξ} 3(x)

is a solution of (1.4)–(1.6) with f = (ωκ)−12ΔY₀+ζ_δ+and

g₁=−(ωκ)−12∂_nY₀+ζ_δ+|_S, g₂=−(ωκ)−12(∂_n− ν)Y₀+ζ_δ+|_Γ.

Since Y₀+is a solution of the homogeneous problem (3.19)–(3.21) (with ε = 0) for small enough δ, we conclude that (f, g)∈ W_γ0(Ω)× W_γ1/2(∂Ω) and Theorem 3.7 applies. As a result there is a solution of the problem for ξ₃ in the spaceH2_γ,0(Ω) for any γ > 0. In particular (4.18) ξ₃= 4  j=1 c_jU_j0+v, where v ∈ H2

γ(Ω) for any γ > 0. Since u−j = Uj0, j = 1, 2, and u−p = (ωκ)−12U0

p,

p = 3, 4, we see that η₃(x) = (ωκ)−12Y₀+ζ_δ++ ξ₃satisfies (4.16) with s_3j = c_j, j = 1, 2, and s_3p = (ωκ)−12c_p, p = 1, 2. Clearly this solution is unique. The same argument applies to η₄.

Next we verify the properties of scattering matrix S. We know that η_j solves the problem Δη_j= 0 in Ω, (4.19) ∂_nη_j= 0 on S, (4.20) ∂_nη_j= νη_j on Γ. (4.21)

Let us multiply (4.19) by η_n, n = 1, . . . , 4, integrate over Ω\ Ω_δ ∩ {|x₁| < M}, and integrate by parts twice (which is justified due to conditions (2.5)). As a result

we have 0 =  +∞ 0 η_n∂_x1η_j|_x1=Mdx₂−  +∞ 0 η_n∂_x1η_j|_x1=−Mdx₂ (4.22) −  _+∞ 0 η_j∂_x1η_j|_x1=Mdx₂+  _+∞ 0 η_j∂_x1η_j|_x1=−Mdx₂ −  _φ(δ) 0 η_n∂_x1η_j|_x1=δdx₂+  _φ(−δ) 0 η_n∂_x1η_j|_x1=−δdx₂ +  _φ(δ) 0 η_j∂_x1η_j|_x1=δdx₂−  _φ(−δ) 0 η_j∂_x1η_j|_x1=−δdx₂.

Passing to the limits as M→ +∞ and δ → 0, and using (4.16), (4.11), (4.12), (4.14), (4.15), (2.3), and (4.1), we obtain (4.23) 0 = iδ_jn− i 4  p=1 S_npS_jp,

so S is unitary. The property S_jn= S_jn, j, n = 1, . . . , 4, can be verified in the same way as in Theorem 4.1.

Remark 4.1. This type of argument with integration by parts implies the fact

(which we used in Lemma 3.6) that if v ∈ H2

γ,0 ∀γ > 0 and is a solution of the

homogeneous problem (1.4)–(1.6), then v∈ H2

γ ∀γ > 0.

Next we describe an important nonstandard property of the scattering matrix S. First we decompose S as follows:

(4.24) S =  S_(1,1) S_(1,2) S_(2,1) S_(2,2)  , where (4.25) S_(1,1)=  S_1,1 S_1,2 S_2,1 S_2,2  , S_(1,2)=  S_1,3 S_1,4 S_2,3 S_2,4  , and (4.26) S_(2,1)=  S_3,1 S_3,2 S_4,1 S_4,2  , S_(2,2)=  S_3,3 S_3,4 S_4,3 S_4,4  .

Theorem 4.3. Suppose Condition 2 is satisfied and ν > κ/8. Then det S_(1,2)= 0, det S_(2,1)= 0, and the absolute values of eigenvalues of the matrices S_(2,2) and S_(1,1) are strictly less than 1.

This theorem will follow from the following uniqueness result.

Theorem 4.4. Suppose Condition 2 is satisfied and ν > κ/8. If v is a solution

of the homogeneous problem (1.4)–(1.6) and v∈ H2

γ(Ω) for some γ, then v≡ 0.

Proof. Due to Theorems 3.3 and 3.4 we have

(4.27) v = 4  j=3 c+_ju+_j + 4  j=3 c−_ju−_j +v, where c±_j, j = 3, 4, are some constants andv ∈ H2

γ(Ω) for any γ > 0. Let us consider

the real part of v, which we denote by u. It is a solution of the homogeneous problem (1.4)–(1.6) and has the same structure as (4.27).

ON THE LINEAR WATER WAVE PROBLEM 4239 We start by recalling the method of multipliers of [3, 10], where it was applied for the case of fully submerged bodies. Let Z = (Z₁, Z₂) be a real vector field in Ω with at most linear growth as |x| → ∞ and Z₂(x₁, 0) = 0 ∀x₁, and let H be a constant. The following identity, which can be found in [10, p. 71] can be verified directly:

2{(Z · ∇u + Hu)Δu} = 2∇ · {(Z · ∇u + Hu)∇u} + (Q∇u) · ∇u − ∇ ·|∇u|2Z.

(4.28)

Here Q is a 2×2 matrix with components Q_ij = (∇·Z −2H)δ_ij−(∂_iZ_j+ ∂_jZ_i), i, j = 1, 2. Let us choose small positive δ < τ , integrate (4.28) over Ω\ Ω_δ, and integrate by parts:

0 = 2 

∂Ω\∂Ωδ

(Z· ∇u + Hu)∂_nuds +

 Ω\Ωδ (Q∇u) · ∇udx (4.29) −  ∂Ω\∂Ωδ |∇u|2_{(Z· n)ds + A}+ δ + A−δ , where (4.30) A±_δ =∓2  φ(±δ) 0 (Z· ∇u + Hu)∂_x1u|_x1=±δdx₂±  φ(±δ) 0 |∇u|2_Z 1|x1=±δdx2. Hence, 0 = 2  Γ\∂Ωδ

(Z· ∇u + Hu)(∂_n− ν)udx₁+ 2ν  Γ\∂Ωδ (Z· ∇u + Hu)udx₁ + 2  S\∂Ωδ

(Z· ∇u + Hu)∂_nuds

+  Ω\Ωδ (Q∇u) · ∇udx −  S\∂Ωδ |∇u|2_{(Z· n)ds + A}+ δ + A−δ = 2ν  Γ\∂Ωδ (Z₁∂_x1u + Hu)udx₁ +  Ω\Ωδ (Q∇u) · ∇udx −  S\∂Ωδ |∇u|2_{(Z· n)ds + A}+ δ + A−δ = ν  Γ\∂Ωδ (2H− ∂_x1Z₁)|u|2dx₁ +  Ω\Ωδ (Q∇u) · ∇udx −  S\∂Ωδ |∇u|2_{(Z· n)ds + A}+ δ + A−δ + Bδ++ Bδ−. (4.31) Here (4.32) B_δ±=∓νZ₁(±δ, 0)u2(±δ, 0). Following [10] (see p. 76), we choose

(4.33) Z(x₁, x₂) =  x₁x 2 1− x22 x2 1+ x22 , 2x 2 1x2 x2 1+ x22  and H = 1/2.

Then, in particular, the first term on the right-hand side of (4.31) is equal to zero. Moreover it was also verified in [10] (see p. 76) that the quadratic form (Q∇u) · ∇u is

nonpositive. In fact it has been shown in [10], and can be verified by direct inspection, that

(4.34) (Q∇u) · ∇u = −2x₁x₂∂_x1u + (x2₂− x₁2)∂_x2u2(x2₁+ x2₂)−2.

Finally, Condition 2 ensures that

(4.35) (Z· n) ≥ 0 on S.

Now we need to investigate the behavior of A±_δ + B±_δ as δ → 0. Due to (4.27), we have

(4.36) u(x) = a+x−1/2₁ cos(ω ln x₁+ b+) + O(x1/2₁ ), x∈ Ω+_τ, x₁→ +0,

and

u(x) = a−(−x₁)−1/2cos(ω ln(−x₁) + b−) + O(x1/2₁ ), x∈ Ω−_τ, x₁→ −0,

where a±and b± are some real constants. Consider, for definiteness, A+_δ + B_δ+. Then we employ Theorem 3.4 and obtain

∂_x1u(x) = a+x−3/2₁ −2−1cos(ω ln x₁+ b)− ω sin(ω ln x₁+ b) + O(x−1/2₁ ), x∈ Ω+_τ, x₁→ +0,

∂_x2u(x) = O(x−1/2₁ ), x∈ Ω+_τ, x₁→ +0.

Moreover, we have from (4.33)

Z₁(x) = x₁+ O(x2₁), x∈ Ω+_τ, x₁→ +0, and Z₂(x) = O(x2₁), x∈ Ω+_τ, x₁→ +0, and consequently  ∂_x1u(x)2+∂_x2(x)u2 

Z₁(x)− 2(Z(x) · ∇u(x) + Hu(x))∂_x1u(x)

=−∂_x1u(x)x₁∂_x1u(x) + u(x)+ O(x−1₁ )

=−(a+)2x−2₁ −2−1cos(ω ln x₁+ b)− ω sin(ω ln x₁+ b)

×2−1cos(ω ln x₁+ b)− ω sin(ω ln x₁+ b)+ O(x−1₁ )

= (a+)2x−2₁ 4−1cos2(ω ln x₁+ b+)− ω2sin2(ω ln x₁+ b+)+ O(x−1₁ )

= (a+)2x−2₁ 4−1+ ω2cos2(ω ln x₁+ b+)− ω2+ O(x−1₁ ), x∈ Ω+_τ, x₁→ +0.

Next, using

φ(x₁) = κx2₁/2 + O(x3₁), we get

(4.37) A+_δ = 2−1(a+)2κ4−1+ ω2cos2(ω ln δ + b+)− ω2+ O(δ) as δ→ +0.

ON THE LINEAR WATER WAVE PROBLEM 4241 For B_δ+, we have, using (4.36),

B_δ+=−ν(a+)2cos2(ω ln δ + b+) + O(δ) as δ→ +0. Finally using (4.13), we obtain

(4.38) A+_δ + B+_δ =−2−1(a+)2κω2+ O(δ) as δ→ +0. In the same way we derive,

(4.39) A−_δ + B−_δ =−2−1(a−)2κω2+ O(δ) as δ→ +0. Now we can pass to the limit in (4.31) as δ→ 0. We get

(4.40) 0 =  Ω (Q∇u) · ∇udx −  S|∇u| 2_{(Z· n)ds − 2}−1_κω2_(a−₎2_{+ (a}+₎2_.

As result we conclude via (4.34) and (4.35) that u≡ 0.

Applying the same arguments to imaginary part of v, we obtain the same re-sult.

Corollary 4.5. It follows from the proof of Theorem 4.4 that Condition 2

implies Condition 1.

Now Theorem 4.3 easily follows.

Proof of Theorem 4.3. It follows from the properties of scattering matrix S (see

Theorem 4.2) that it is enough to prove only one of the claims of Theorem 4.3. Let us prove that det S_(2,1) is not zero. If it is not so, then there is a nonzero row

a = (a₁, a₂) such that aS_(2,1)= (0, 0), and consequently the function u = a₁η₃+ a₂η₄

is not identically zero and satisfies the conditions of Theorem 4.4. This delivers a contradiction.

Theorem 4.3 allows us to formulate other existence and uniqueness results. Theorem 4.6. Suppose Condition 2 is satisfied and ν > κ/8. Assume further

that{f, g} ∈ L₂(Ω)× H1/2_{(∂Ω) and have compact supports separated from the origin.}

Then there is the unique solution of (1.4)–(1.6), u, in the space H2

loc(Ω)∩ L∞(Ω). Moreover, (4.41) u = 2  j=1 c+_ju+_j + 2  j=1 c−_ju−_j +u, where u ∈ H2

γ(Ω) for any γ and c±j, j = 1, 2, are constants.

Proof. Theorem 3.7 delivers us the unique solution of (1.4)–(1.6), v∈ H2

γ,0(Ω) for any γ > 0, i.e., (4.42) v = 4  j=1 d_ju−_j +v, wherev ∈ H2

γ(Ω) ∀γ > 0. Consider the function

(4.43) u := v− a₁η₁− a₂η₂,

where a row a = (a₁, a₂) solves

aS_(1,2)= (d₃, d₄).

The solution exists due to Theorem 4.3. It is easy to see that u defined by (4.43) is a solution of (1.4)–(1.6) and has the structure (4.41) withu ∈ H2

γ(Ω) for any γ > 0.

The inclusion u ∈ H2

γ(Ω), for γ ≤ 0, follows from Theorems 3.3 and 3.4. Clearly

u∈ H2

loc(Ω)∩ L∞(Ω).

Let us discuss uniqueness. Consider some u∈ H2

loc(Ω)∩L∞(Ω) which is a solution

of (1.4)–(1.6). If we additionally know that representation (4.41) is valid for u, then uniqueness follows immediately from Theorem 4.4.

Let us prove that representation (4.41) is valid. It follows from Theorems 3.3 and 3.4 that u∈ H2

γ(Ωτ) for any γ, since f and g have support separated from the

origin and u∈ H2

loc(Ω). Now we need to show that the representation (4.41) is valid

at infinity. But this is the same as proving that any solution v∈ H2

loc(R2+)∩ L∞(R2+)

of (2.1), (2.2) with compactly supported f ∈ L₂(R2

+) can be represented as (4.44) v = 2  j=1 c+_ju+_j + 2  j=1 c−_ju−_j +v, where v ∈ ˙H2_(R2

+). Consider the solution v1 of (2.1), (2.2) in the space with

ra-diation conditions. Clearly representation (4.44) is valid for v₁ (in fact coefficients next to outgoing waves are zero) and v₁ ∈ H2

loc(R2+)∩ L∞(R2+). Then w := v−

v₁ ∈ H2

loc(R2+)∩ L∞(R2+) is a solution of (2.1), (2.2) with zero right-hand side, and

consequently w is a linear combination of functions e−iνx1−νx2 _{and e}iνx1−νx2_{. We}

see that representation (4.44) is valid for w, v₁and consequently is valid for v. This completes the proof.

In a similar way we prove the next result.

Theorem 4.7. Suppose Condition 2 is satisfied and ν > κ/8. Assume further

that{f, g} ∈ L₂(Ω)× H1/2_{(∂Ω) and have compact support separated from the origin.}

Then there is a unique solution of (1.4)–(1.6), w, in the space

H_loc2 (Ω\ O) ∩ Lp(Ω), p∈ (2, 6).

Moreover, w∈ H2

γ, for any γ > 1/2 and can be represented as

(4.45) w = 4  j=3 c+_ju+_j + 4  j=3 c−_ju−_j +w, wherew∈ H2

γ(Ω) for any γ > 0 and c±j, j = 1, 2, are some constants.

Proof. Theorem 3.7 delivers us the unique solution of (1.4)–(1.6), v∈ H2

γ,0(Ω) for any γ > 0, i.e., (4.46) v = 4  j=1 d_ju−_j +v, wherev ∈ H2

γ(Ω) for any γ > 0. Consider the function

(4.47) w := v− a₁η₃− a₂η₄,

where the row a = (a₁, a₂) solves

aS_(2,1)= (d₁, d₂).