Georgian Mathematical Journal Volume 10 (2003), Number 3, 467–480

INTEGRAL REPRESENTATIONS FOR THE SOLUTION OF

DYNAMIC BENDING OF A PLATE WITH DISPLACEMENT-TRACTION BOUNDARY DATA

IGOR CHUDINOVICH AND CHRISTIAN CONSTANDA

Abstract. The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

2000 Mathematics Subject Classification: 74H20, 74K20, 35L55, 45F15, 31A10. Key words and phrases: Boundary integral equations, dynamic problems, elastic plates.

1. Introduction

Mathematical models play an essential role in the study of mechanical phenomena and processes. Owing to their complexity, solutions of initial-boundary value problems for such models are usually approximated by means of various numerical procedures. But before methods of this type can be developed and applied, the practitioner must have a guarantee that the problem is well posed; in other words, that it has a unique solution depending continuously on the data.

In this paper we address the questions of existence, uniqueness, and stability of the solution to the time-dependent bending of a plate with transverse shear deformation where displacements are prescribed on one part of the boundary and the bending and twisting moments and shear force are given on the remaining part. The analysis is performed in Sobolev spaces, whose norms are particularly well suited to the construction of global error estimates.

The corresponding equilibrium problem has been fully investigated in [1]. A discussion of other mathematical models in elasticity can be found in [2]–[4].

2. Formulation of the Problem

Consider a homogeneous and isotropic elastic plate of thickness h_{0} = const > 0, which occupies a domain S̄ × [−h_{0}/2, h_{0}/2] in ℝ^{3}, where S is a domain in ℝ^{2} with a simple closed boundary ∂S. In the transverse shear deformation model proposed in [5] it is assumed that the displacement vector at (x, x_{3}), x = (x_{1}, x_{2}) ∈ ℝ^{2}, at time t ≥ 0, is of the form

(x_{3}u_{1}(x, t), x_{3}u_{2}(x, t), u_{3}(x, t))^{T},

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

where the superscript T denotes matrix transposition. Then u = (u_{1}, u_{2}, u_{3})^{T} satisfies the equation of motion

B(∂_{t}^{2}u)(x, t) + (Au)(x, t) = q(x, t), (x, t) ∈ G = S × (0, ∞);

here B = diag{ρh^{2}, ρh^{2}, ρ}, h^{2} = h_{0}^{2}/12, ρ is the density of the material, ∂_{t} = ∂/∂t,

A = (\begin{matrix}−h^{2}μΔ − h^{2}(λ + μ)∂_{1}^{2} + μ & −h^{2}(λ + μ)∂_{1}∂_{2} & μ∂_{1} \\ −h^{2}(λ + μ)∂_{1}∂_{2} & −h^{2}μΔ − h^{2}(λ + μ)∂_{2}^{2} + μ & μ∂_{2} \\ −μ∂_{1} & −μ∂_{2} & −μΔ \end{matrix}) ,

∂_{α} = ∂/∂x_{α}, α = 1, 2, λ and μ are the Lamé constants satisfying λ + μ > 0 and μ > 0, and q is a combination of forces and moments acting on the plate and its faces x_{3} = ±h_{0}/2.

In what follows we work with three-component distributions; however, for simplicity, we use the same symbols for their spaces and norms as in the scalar case.

We denote by H_{m,p}(ℝ^{2}), m ∈ ℝ, p ∈ ℂ, the space that coincides with H_{m}(ℝ^{2}) as a set but is endowed with the norm

∥u∥_{m,p} = { ∫_{ℝ^{2}} (1 + |p|^{2} + |ξ|^{2})^{m}|ũ(ξ)|^{2} dξ}^{1/2},

where ũ is the distributional Fourier transform of u ∈ S′(ℝ^{2}). Next, H̊_{m,p}(S) is the subspace of H_{m,p}(ℝ^{2}) consisting of all u ∈ H_{m,p}(ℝ^{2}) with supp u ⊂ S̄, and H_{m,p}(S) is the space of the restrictions to S of all v ∈ H_{m,p}(ℝ^{2}). The norm of u ∈ H_{m,p}(S) is defined by

∥u∥_{m,p;S} = inf_{v∈H_{m,p}(ℝ^{2}): v|_{S}=u}∥v∥_{m,p}.

Also, H_{−m,p}(ℝ^{2}) is the dual of H_{m,p}(ℝ^{2}) with respect to the duality generated by the inner product (· , ·)_{0} in L^{2}(ℝ^{2}); the dual of H̊_{m,p}(S) is H_{−m,p}(S). Let γ be the trace operator that maps H_{1,p}(S) continuously to the space H_{1/2,p}(∂S), which coincides as a set with H_{1/2}(∂S) but is endowed with the norm

∥f∥_{1/2,p;∂S} = inf_{u∈H_{1,p}(S): γu=f} ∥u∥_{1,p;S}.

The continuity of γ from H_{1,p}(S) to H_{1/2,p}(∂S) is uniform with respect to p ∈ ℂ. Finally, H_{−1/2,p}(∂S) is the dual of H_{1/2,p}(∂S) with respect to the duality generated by the inner product (· , ·)_{0;∂S} in L^{2}(∂S).

We fix κ > 0 and introduce the complex half-plane ℂ_{κ} = {p = σ + iτ ∈ ℂ : σ > κ}. Consider the space H_{m,k,κ}^{L}(S), m, k ∈ ℝ, of all û(x, p), x ∈ S, p ∈ ℂ_{κ}, such that U(p) = û(· , p) is a holomorphic mapping from ℂ_{κ} to H_{m}(S) (which implies that U(p) also belongs to H_{m,p}(S) for every p ∈ ℂ_{κ}) and for which

∥û∥_{m,k,κ;S}^{2} = sup_{σ>κ} ∫_{−∞}^{∞}(1 + |p|^{2})^{k}∥U(p)∥_{m,p;S}^{2} dτ < ∞.

The norm on H_{m,k,κ}^{L}(S) is defined by this equality. In what follows, we use the symbol û(x, p) when we want to emphasize that this is a distribution in H_{m,p}(S), and the symbol U(p) when we need to regard it as a mapping from ℂ_{κ} to H_{m}(S). The spaces H_{±1/2,k,κ}^{L}(∂S) and its norm ∥ · ∥_{±1/2,k,κ;∂S} are introduced similarly.

Let H_{m,k,κ}^{L^{−1}}(G) and H_{±1/2,k,κ}^{L^{−1}}(Γ), Γ = ∂S × (0, ∞), be the spaces of the inverse Laplace transforms u and f of all û ∈ H_{m,k,κ}^{L}(S) and f̂ ∈ H_{±1/2,k,κ}^{L}(∂S), with norms

∥u∥_{m,k,κ;G} = ∥û∥_{m,k,κ;S}, ∥f∥_{±1/2,k,κ;∂G} = ∥f̂∥_{±1/2,k,κ;∂S}.

We assume that ∂S is a C^{2}-curve consisting of two arcs ∂S_{ν}, ν = 1, 2, such that ∂S = \overline{∂S}_{1} ∪ \overline{∂S}_{2}, ∂S_{1} ∩ ∂S_{2} = ∅, and mes ∂S_{ν} > 0, ν = 1, 2. Let S^{+} and S^{−} be the interior and exterior domains into which ∂S divides ℝ^{2}, and let G^{±} = S^{±} × (0, ∞) and Γ_{ν} = ∂S_{ν} × (0, ∞), ν = 1, 2. We denote by γ^{±} the trace operators corresponding to S^{±}. For simplicity, we use the same symbols for the trace operators in the spaces of originals and in those of their Laplace transforms. Thus, γ^{±} also denote the trace operators mapping H_{1,k,κ}^{L^{−1}}(G^{±}) continuously to H_{1/2,k,κ}^{L^{−1}}(Γ) for any k ∈ ℝ. Also, we denote by π_{ν}, ν = 1, 2, the operators of restriction from Γ to Γ_{ν} (and from ∂S to ∂S_{ν}), and write γ_{ν}^{±} = π_{ν}γ^{±}, ν = 1, 2. Finally, let π^{±} be the operators of restriction from ℝ^{2} × (0, ∞) to G^{±} (or from ℝ^{2} to S^{±}).

We introduce the subspace H̊_{1/2,p}(∂S_{ν}) of H_{1/2,p}(∂S) consisting of all φ ∈ H_{1/2,p}(∂S) such that supp φ ∈ \overline{∂S}_{ν}, ν = 1, 2, and denote by H_{1/2,p}(∂S_{ν}) the space of restrictions from ∂S to ∂S_{ν} of the elements of H_{1/2,p}(∂S). The norm of φ ∈ H_{1/2,p}(∂S_{ν}) is defined by

∥φ∥_{1/2,p;∂S_{ν}} = inf_{f∈H_{1/2,p}(∂S):π_{ν}f=φ} ∥f∥_{1/2,p;∂S}, ν = 1, 2.

Let l_{ν}, ν = 1, 2, be extension operators from ∂S_{ν} to ∂S which map H_{1/2,p}(∂S_{ν}) to H_{1/2,p}(∂S) continuously and uniformly with respect to p and satisfy

∥l_{ν}f∥_{1/2,p;∂S} ≤ c∥f∥_{1/2,p;∂S_{ν}} ∀f ∈ H_{1/2,p}(∂S_{ν}).

Also, let l^{±} be operators of extension from ∂S to S^{±} which map H_{1/2,p}(∂S) to H_{1,p}(S^{±}) continuously and uniformly with respect to p.

By H̊_{−1/2,p}(∂S_{ν}) and H_{−1/2,p}(∂S_{ν}), ν =1, 2, we denote the duals of H_{1/2,p}(∂S_{ν}) and H̊_{1/2,p}(∂S_{ν}), respectively, with respect to the duality generated by the inner product in [L^{2}(∂S_{ν})]^{3}; their norms are ∥ · ∥_{−1/2,p;∂S} and ∥ · ∥_{−1/2,p;∂S_{ν}}. The corresponding spaces H_{±1/2,k,κ}^{L^{−1}}(Γ_{ν}) and H̊_{±1/2,k,κ}^{L^{−1}}(Γ_{ν}), ν = 1, 2, and their norms ∥ · ∥_{±1/2,k,κ,Γ_{ν}} and ∥ · ∥_{±1/2,k,κ,Γ}, k ∈ ℝ, κ > 0, are introduced in the usual way.

In what follows we denote by

a_{±}(u, v) = 2 ∫_{S^{±}} E(u, v) dx

the sesquilinear form of the internal energy density, where

2E(u, v) = h^{2}E_{0}(u, v) + h^{2}μ(∂_{2}u_{1} + ∂_{1}u_{2})(∂_{2}v̄_{1} + ∂_{1}v̄_{2})

+ μ[(u_{1} + ∂_{1}u_{3})(v̄_{1} + ∂_{1}v̄_{3}) + (u_{2} + ∂_{2}u_{3})(v̄_{2} + ∂_{2}v̄_{3})],

E_{0}(u, v) = (λ + 2μ)[(∂_{1}u_{1})(∂_{1}v̄_{1}) + (∂_{2}u_{2})(∂_{2}v̄_{2})]

+ λ[(∂_{1}u_{1})(∂_{2}v̄_{2}) + (∂_{2}u_{2})(∂_{1}v̄_{1})].

The classical formulation of the dynamic mixed problems (DM^{±}) consists in finding u ∈ C^{2}(G^{±}) ∩ C^{1}(Ḡ^{±}) that satisfies

B(∂_{t}^{2}u)(x, t) + (Au)(x, t) = 0, (x, t) ∈ G^{+} or G^{−},
u(x, 0+) = (∂_{t}u)(x, 0+) = 0, x ∈ S^{+} or S^{−},
u^{±}(x, t) = f_{1}(x, t), (x, t) ∈ Γ_{1}, (T u)^{±}(x, t) = g_{2}(x, t), (x, t) ∈ Γ_{2},

where T is the moment-force boundary operator defined by

(\begin{matrix}h^{2}(λ + 2μ)n_{1}∂_{1} + h^{2}μn_{2}∂_{2} & h^{2}μn_{2}∂_{1} + h^{2}λn_{1}∂_{2} & 0 \\ h^{2}λn_{2}∂_{1} + h^{2}μn_{1}∂_{2} & h^{2}μn_{1}∂_{1} + h^{2}(λ + 2μ)n_{2}∂_{2} & 0 \\ μn_{1} & μn_{2} & μ(n_{1}∂_{1} + n_{2}∂_{2}) \end{matrix}) ,

n = (n_{1}, n_{2}) is the outward unit normal to ∂S, the superscripts ± denote the limiting values of the corresponding functions as (x, t) → Γ from inside G^{±} (or x → ∂S from inside S^{±}), and f_{1} and g_{2} are given functions.

We call u ∈ H_{1,0,κ}^{L^{−1}}(G^{±}) a weak solution of the corresponding problem (DM^{±}) if it satisfies

\begin{matrix}∫_{0}^{∞}\{a_{±}(u, v) − (B^{1/2}∂_{t}u, B^{1/2}∂_{t}v)_{0;S^{±}}\} dt = ± ∫_{0}^{∞}(g_{2}, v)_{0;∂S_{2}} dt, \\ γ_{1}^{±}u = f_{1} \end{matrix}  (1)

for all v ∈ C_{0}^{∞}(Ḡ^{±}) such that γ_{1}^{±}v = 0.

3. Solvability of the Problems

In what follows we use the same symbol c for all positive constants that occur in various estimates and are independent of the functions in those estimates and of p ∈ ℂ_{κ} (but may depend on κ).

Theorem 1. For every κ > 0, f_{1} ∈ H_{1/2,1,κ}^{L^{−1}}(Γ_{1}), and g_{2} ∈ H_{−1/2,1,κ}^{L^{−1}}(Γ_{2}), problems (1) have a unique solution u ∈ H_{1,0,κ}^{L^{−1}}(G^{±}). Furthermore, if f_{1} ∈ H_{1/2,k,κ}^{L}(Γ_{1}) and g_{2} ∈ H_{−1/2,k,κ}^{L}(Γ_{2}), then u ∈ H_{1,k−1,κ}^{L}(G^{±}) and for every k ∈ ℝ

∥u∥_{1,k−1,κ;G^{±}} ≤ c(∥f_{1}∥_{1/2,k,κ;Γ_{1}} + ∥g_{2}∥_{−1/2,k,κ;Γ_{2}}).

Proof. We prove the assertion for (DM^{+}); the case of (DM^{−}) is treated similarly. We begin by rewriting (1) in terms of Laplace transforms.

Let H̊_{1,p}(S^{+}, ∂S_{ν}) be the subspace of H_{1,p}(S^{+}) of all elements u such that π_{3−ν}γ^{+}u = 0, ν = 1, 2; that is, γ^{+}u ∈ H̊_{1/2,p}(∂S_{ν}). Going over to Laplace transforms, (1) becomes the problem (DM_{p}^{+}) of seeking u ∈ H_{1,p}(S^{+}) such that for every p ∈ ℂ_{κ}

\begin{matrix}p^{2}(B^{1/2}u, B^{1/2}v)_{0;S^{+}} + a_{+}(u, v) = (g_{2}, v)_{0;∂S_{2}} ∀v ∈ H̊_{1,p}(S^{+}, ∂S_{2}), \\ π_{1}γ^{+}u = f_{1}. \end{matrix} (2)

Since for any v ∈ H̊_{1,p}(S^{+}, ∂S_{2})

|(g_{2}, v)_{0;∂S_{2}} | ≤ ∥g_{2}∥_{−1/2,p;∂S_{2}}∥v∥_{1/2,p;∂S} ≤ c∥g_{2}∥_{−1/2,p;∂S_{2}}∥v∥_{1,p;S^{+}},

(g_{2}, v)_{0;∂S_{2}} defines an antilinear functional on H̊_{1,p}(S^{+}, ∂S_{2}); hence, it can be written as

(g_{2}, v)_{0;∂S_{2}} = (q_{2}, v)_{0;S^{+}} ∀v ∈ H_{1,p}(S^{+}, ∂S_{2}),  (3)

where q_{2} ∈ [H̊_{1,p}(S^{+}, ∂S_{2})]^{∗} and

∥q_{2}∥_{[H̊_{1,p}(S^{+},∂S_{2})]^{∗}} ≤ c∥g_{2}∥_{−1/2,p;∂S_{2}}.  (4)

First, we assume that f_{1} = 0. Taking (3) into account, we write (2) in the form

p^{2}(B^{1/2}u_{0}, B^{1/2}v)_{0;S^{+}} + a_{+}(u_{0}, v) = (q_{2}, v)_{0;S^{+}} ∀v ∈ H̊_{1,p}(S^{+}, ∂S_{2}), (5)

where u_{0} ∈ H̊_{1,p}(S^{+}, ∂S_{2}) is an unknown vector function. The unique solvability of (5) and the estimate

∥u_{0}∥_{1,p;S^{+}} ≤ c|p| ∥q_{2}∥_{[H̊_{1,p}(S^{+},∂S_{2})]^{∗}}  (6)

are proved in the usual way (see, for example, [6]).

If f_{1} ≠ 0, then we set w = l^{+}l_{1}f_{1} ∈ H_{1,p}(S^{+}) and remark that

∥w∥_{1,p;S^{+}} ≤ c∥f_{1}∥_{1/2,p;∂S_{1}}.  (7)

We seek the solution u of (2) as u = u_{0} +w. Clearly, u_{0} ∈ H̊_{1,p}(S^{+}, ∂S_{2}) satisfies

p^{2}(B^{1/2}u_{0}, B^{1/2}v)_{0;S^{+}} + a_{+}(u_{0}, v)
= (q_{2}, v)_{0;S^{+}} − p^{2}(B^{1/2}w, B^{1/2}v)_{0;S^{+}} − a_{+}(w, v)
∀v ∈ H̊_{1,p}(S^{+}, ∂S_{2}).  (8)

By (7),

|p^{2}(B^{1/2}w, B^{1/2}v)_{0,S^{+}} + a_{+}(w, v)| ≤ c∥w∥_{1,p;S^{+}}∥v∥_{1,p;S^{+}}
≤ c∥f_{1}∥_{1/2,p;∂S_{1}}∥v∥_{1,p;S^{+}};

hence,

p^{2}(B^{1/2}w, B^{1/2}v)_{0;S^{+}} + a_{+}(w, v) = (q_{1}, v)_{0;S^{+}} ∀v ∈ H̊_{1,p}(S^{+}, ∂S_{2}),

where q_{1} ∈ [H̊_{1,p}(S^{+}, ∂S_{2})]^{∗} and

∥q_{1}∥_{[H̊_{1,p}(S^{+},∂S_{2})]^{∗}} ≤ c∥f_{1}∥_{1/2,p;∂S_{1}}.  (9)

Equation (8) now takes the form

p^{2}(B^{1/2}u_{0}, B^{1/2}v)_{0;S^{+}} + a_{+}(u_{0}, v) = (q_{2} − q_{1}, v)_{0;S^{+}} ∀v ∈ H̊_{1,p}(S^{+}, ∂S_{2}). (10)

We have already stated that (10) has a unique solution u_{0} ∈ H̊_{1,p}(S^{+}, ∂S_{2}). By (6), (4), and (9),

∥u_{0}∥_{1,p;S^{+}} ≤ c|p|∥q_{1} − q_{2}∥_{[H̊_{1,p}(S^{+},∂S_{2})]^{∗}}
≤ c|p|(∥f_{1}∥_{1/2,p;∂S_{1}} + ∥g_{2}∥_{−1/2,p;∂S_{2}}).

Therefore, for u = u_{0} + w we have

∥u∥_{1,p;S^{+}} ≤ c|p|(∥f_{1}∥_{1/2,p;∂S_{1}} + ∥g_{2}∥_{−1/2,p;∂S_{2}}).  (11)

Returning to the spaces of originals and repeating the scheme used in [6] and [7], we complete the proof of the assertion. □

4. Dynamic Plate Potentials and their Properties

Consider a matrix D(x, t) of fundamental solutions for the equation of motion, that is, a (3 × 3)-matrix such that

B(∂_{t}^{2}D)(x, t) + AD(x, t) = δ(x, t)I, (x, t) ∈ ℝ^{2} × ℝ,
D(x, t) = 0, (x, t) ∈ ℝ^{2} × (−∞, 0),

where δ is the Dirac delta distribution and I is the identity (3 × 3)-matrix. Clearly, its Laplace transform D̂(x, p) satisfies

Bp^{2}D̂(x, p) + AD̂(x, p) = δ(x)I, x ∈ ℝ^{2}.

The explicit form of D̂(x, p) can be found in [7].

Let α, β ∈ C^{2}(∂S × ℝ) be functions with compact support in Γ̄, and let α̂ and β̂ be their Laplace transforms. We define the single-layer and double-layer potentials

(V_{p}α̂)(x, p) = ∫_{∂S} D̂(x − y, p)α̂(y, p) ds_{y}, x ∈ ℝ^{2}, p ∈ ℂ_{0},

(W_{p}β̂)(x, p) = ∫_{∂S} (T_{y}D̂(y − x, p))^{T}β̂(y, p) ds_{y}, x ∈ S^{+} ∪ S^{−}, p ∈ ℂ_{0},

where T_{y} is the boundary operator T acting with respect to y.

Since D̂(x, p) has a polynomial growth with respect to p ∈ ℂ_{κ}, κ > 0, we may now define the dynamic (retarded) single-layer and double-layer potentials

(V α)(x, t) = (L^{−1}V_{p}α̂)(x, t)

= ∫_{0}^{∞}∫_{∂S} D(x − y, t − τ)α(y, τ) ds_{y} dτ, (x, t) ∈ ℝ^{2} × (0, ∞),

(W β)(x, t) = (L^{−1}W_{p}β̂)(x, t)

= ∫_{0}^{∞}∫_{∂S} (T_{y}D(y − x, t − τ))^{T}β(y, τ) ds_{y} dτ, (x, t) ∈ G^{+} ∪ G^{−}.

We consider the boundary operators V_{0}, W^{±}, and N defined by

V_{0}α = γ^{±}π^{±}V α, W^{±}β = γ^{±}π^{±}W β, Nβ = T^{±}π^{±}W β.

Their corresponding transformed versions are defined by

V_{p,0}α̂ = γ^{±}π^{±}V_{p}α̂, W_{p}^{±}β̂ = γ^{±}π^{±}W_{p}β̂, N_{p}β̂ = T^{±}π^{±}W_{p}β̂.

These operators can be extended by continuity to much wider classes of densities [7]. For convenience, we gather here some of the main results.

First we introduce the Poincaré–Steklov operators T_{p}^{±} acting in Sobolev spaces with a parameter. Let f ∈ H_{1/2,p}(∂S), p ∈ ℂ_{0}, and let u ∈ H_{1,p}(S^{±}) be the weak solution [8] of the problem

\begin{matrix}p^{2}(B^{1/2}u, B^{1/2}v)_{0,;S^{±}} + a_{±}(u, v) = 0 ∀v ∈ H̊_{1,p}(S^{±}), \\ γ^{±}u = f. \end{matrix}  (12)

Also, let φ ∈ H_{1/2,p}(∂S) be arbitrary and let w ∈ H_{1,p}(S^{±}) be such that γ^{±}w = φ. We define the Poincaré–Steklov operators T_{p}^{±} depending on the parameter p ∈ ℂ_{0} by

(T_{p}^{±}f, φ)_{0;∂S} = ±{p^{2}(B^{1/2}u, B^{1/2}w)_{0;S^{±}} + a_{±}(u, w)}.  (13)

It is obvious that (13) defines T_{p}^{±} correctly. For if w_{1}, w_{2} ∈ H_{1,p}(S^{±}) are such that γ^{±}w_{1} = γ^{±}w_{2} = φ, then v = w_{1} − w_{2} ∈ H̊_{1,p}(S^{±}) and, by (12),

p^{2}(B^{1/2}u, B^{1/2}w_{1})_{0,S^{±}} + a_{±}(u, w_{1}) =p^{2}(B^{1/2}u, B^{1/2}(v + w_{2}))_{0;S^{±}} +a_{±}(u, v + w_{2})
=p^{2}(B^{1/2}u, B^{1/2}w_{2})_{0;S^{±}} + a_{±}(u, w_{2}).

Lemma 1. For every p ∈ ℂ_{0}, the operators T_{p}^{±} are homeomorphisms from H_{1/2,p}(∂S) to H_{−1/2,p}(∂S), and for every κ > 0 and p ∈ ℂ̄_{κ}

∥T_{p}^{±}f∥_{−1/2,p;∂S} ≤ c|p|∥f∥_{1/2,p;∂S},  (14)
∥f∥_{1/2,p;∂S} ≤ c|p|∥T_{p}^{±}f∥_{−1/2,p;∂S}.  (15)

Returning to the spaces of originals, we introduce the Poincaré–Steklov operators T^{±}. From (14) and (15) we easily deduce the following assertion.

Theorem 2. For any κ > 0 and k ∈ ℝ, the operators T^{±} define injective maps

T^{±} : H_{1/2,k,κ}^{L^{−1}}(Γ) → H_{−1/2,k−1,κ}^{L^{−1}}(Γ).

Their inverses, extended by continuity to the corresponding spaces, define injections

(T^{±})^{−1} : H_{−1/2,k,κ}^{L^{−1}}(Γ) → H_{1/2,k−1,κ}^{L^{−1}}(Γ).

We now use the above results to investigate the properties of the singlelayer and double-layer potentials. Obviously, both these potentials satisfy the equation

Bp^{2}û(x, p) + Aû(x, p) = 0, x ∈ S^{+} ∪ S^{−} .

Lemma 2. For any p ∈ ℂ_{0}, the operator V_{p,0}, extended by continuity from C^{2}(∂S) to the space H_{−1/2,p}(∂S), is a homeomorphism from H_{−1/2,p}(∂S) to H_{1/2,p}(∂S), and for every p ∈ ℂ̄_{κ}

∥V_{p,0}α̂∥_{1/2,p;∂S} ≤ c|p|∥α̂∥_{−1/2,p;∂S},
∥α̂∥_{−1/2,p;∂S} ≤ c|p|∥V_{p,0}α̂∥_{1/2,p;∂S}.

For any density α̂ ∈ H_{−1/2,p}(∂S), the single-layer potential is continuous in the sense that γ^{+}V_{p}α̂ = γ^{−}V_{p}α̂ = V_{p,0}α̂. Also, the jump formulas

T_{p}^{+}V_{p,0}α̂ = \frac{1}{2} α̂ + (T_{p}^{+}V_{p,0}α̂)_{0},
T_{p}^{−}V_{p,0}α̂ = −\frac{1}{2} α̂ + (T_{p}^{+}V_{p,0}α̂)_{0}

hold, where (T_{p}^{+}V_{p,0}α̂)_{0} is the integral operator generated by the direct value on ∂S of the corresponding singular integral.

Lemma 3. For any p ∈ ℂ_{0}, the operators W_{p}^{±}, extended by continuity from C^{2}(∂S) to H_{1/2,p}(∂S), are homeomorphisms from H_{1/2,p}(∂S) to H_{1/2,p}(∂S), and for every p ∈ ℂ̄_{κ}

∥W_{p}^{±}β̂∥_{1/2,p;∂S} ≤ c|p|^{2}∥β̂∥_{1/2,p;∂S},
∥β̂∥_{1/2,p;∂S} ≤ c|p|^{2}∥W_{p}^{±}β̂∥_{1/2,p;∂S}.

The jump formulas for the double-layer potential with β̂ ∈ H_{1/2,p}(∂S) have the form

W_{p}^{+}β̂ = −\frac{1}{2} β̂ + (W_{p}β̂)_{0},
W_{p}^{−}β̂ = \frac{1}{2} β̂ + (W_{p}β̂)_{0},

where (W_{p}β̂)_{0} is the integral operator generated by the direct value on ∂S of the corresponding singular integral.

Finally, we consider the operator N_{p} = T_{p}^{+}W_{p}^{+} = T_{p}^{−}W_{p}^{−}, which is an integral (pseudodifferential) operator with a hypersingular kernel.

Lemma 4. For any p ∈ ℂ_{0}, the operator N_{p}, extended by continuity from C^{2}(∂S) to H_{1/2,p}(∂S), is a homeomorphism from H_{1/2,p}(∂S) to H_{−1/2,p}(∂S), and for every p ∈ ℂ̄_{κ}

∥N_{p}β̂∥_{−1/2,p;∂S} ≤ c|p|^{3}∥β̂∥_{1/2,p;∂S},
∥β̂∥_{1/2,p;∂S} ≤ c|p|∥N_{p}β̂∥_{−1/2,p;∂S}.

We remark that if α̂ ∈ H_{−1/2,p}(∂S) and β̂ ∈ H_{1/2,p}(∂S), then the single-layer and double-layer potentials belong to H_{1,p}(S^{±}).

Returning to the spaces of originals, we establish the properties of the boundary operators, which are summarized in the next assertion.

Theorem 3. (i) The operator V_{0}, extended by continuity from C_{0}^{2}(Γ) to H_{−1/2,k,κ}^{L^{−1}}(Γ), is continuous and injective from H_{−1/2,k,κ}^{L^{−1}}(Γ) to H_{1/2,k−1,κ}^{L^{−1}}(Γ) for every κ > 0 and k ∈ ℝ, and its range is dense in H_{1/2,k−1,κ}^{L^{−1}}(Γ). The inverse

DYNAMIC BENDING OF A PLATE WITH MIXED BOUNDARY DATA 475

V_{0}^{−1}, extended by continuity from the range of V_{0} to H_{1/2,k,κ}^{L^{−1}}(Γ), is continuous and injective from H_{1/2,k,κ}^{L^{−1}}(Γ) to H_{−1/2,k−1,κ}^{L^{−1}}(Γ) for any k ∈ ℝ, and its range is dense in H_{−1/2,k−1,κ}^{L^{−1}}(Γ). Furthermore,

∥π^{+}V α∥_{1,k−1,κ;G^{+}} + ∥π^{−}V α∥_{1,k−1,κ;G^{−}} ≤ c∥α∥_{−1/2,k,κ;Γ}.

(ii) The operators W^{±}, extended by continuity from C_{0}^{2}(Γ) to H_{1/2,k,κ}^{L^{−1}}(Γ), are continuous and injective from H_{1/2,k,κ}^{L^{−1}}(Γ) to H_{1/2,k−2,κ}^{L^{−1}}(Γ) for any κ > 0 and k ∈ ℝ, and their ranges are dense in H_{1/2,k−2,κ}^{L^{−1}}(Γ). The inverses (W^{±})^{−1}, extended by continuity from the ranges of W^{±} to H_{1/2,k,κ}^{L^{−1}}(Γ), are continuous and injective from H_{1/2,k,κ}^{L^{−1}}(Γ) to H_{1/2,k−2,κ}^{L^{−1}}(Γ) for all k ∈ ℝ, and their ranges are dense in H_{1/2,k−2,κ}^{L^{−1}}(Γ). Furthermore,

∥π^{+}W β∥_{1,k−2,κ;G^{+}} + ∥π^{−}W β∥_{1,k−2,κ;G^{−}} ≤ c∥β∥_{1/2,k,κ;Γ}.

(iii) The operator N, extended by continuity from C_{0}^{2}(Γ) to H_{1/2,k,κ}^{L^{−1}}(Γ), is continuous and injective from H_{1/2,k,κ}^{L^{−1}}(Γ) to H_{−1/2,k−3,κ}^{L^{−1}}(Γ) for any κ > 0 and k ∈ ℝ, and its range is dense in H_{−1/2,k−3,κ}^{L^{−1}}(Γ). The inverse N^{−1}, extended by continuity from the range of N to H_{−1/2,k,κ}^{L^{−1}}(Γ), is continuous and injective from H_{−1/2,k,κ}^{L^{−1}}(Γ) to H_{1/2,k−1,κ}^{L^{−1}}(Γ) for any k ∈ ℝ and its range is dense in H_{1/2,k−1,κ}^{L^{−1}}(Γ).

(iv) For any α ∈ H_{−1/2,k,κ}^{L^{−1}}(Γ), κ > 0, and k ≥ 1, the potential V α is a weak solution of the homogeneous equation (1) in G^{+} ∪ G^{−} with homogeneous initial conditions. For any β ∈ H_{1/2,k,κ}^{L^{−1}}(Γ), κ > 0, and k ≥ 2, the potential W β is a weak solution of the homogeneous equation (1) in G^{+} ∪ G^{−} with homogeneous initial conditions.

The jump formulas for the single-layer and double-layer dynamic potentials are obvious and we omit them.

5. Integral Representations of the Solutions

For every p ∈ ℂ_{0} we introduce three pairs of special boundary operators. Let α, β = 1, 2, α ≠ β. For f ∈ H_{1/2,p}(∂S) we define the operators π_{αβ}^{±} by

π_{αβ}^{±}f = {π_{α}f, π_{β}T_{p}^{±}f}.

For g ∈ H_{−1/2,p}(∂S), we define the operators θ_{αβ}^{±} by

θ_{αβ}^{±}g = π_{αβ}^{±}(T_{p}^{±})^{−1}g.

Finally, for {f_{α}, g_{β}} ∈ H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}) we define the operators ρ_{βα}^{±} by

ρ_{βα}^{±}{f_{α}, g_{β}} = π_{βα}^{±}(π_{αβ}^{±})^{−1}{f_{α}, g_{β}}

We denote by ∥{f_{α}, g_{β}}∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}} the norm of {f_{α}, g_{β}} in the space H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}); that is,

∥{f_{α}, g_{β}}∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}} = ∥f_{α}∥_{1/2,p;∂S_{α}} + ∥g_{β}∥_{−1/2,p;∂S_{β}}.

Lemma 5. (i) The operators π_{αβ}^{±} are homeomorphisms from H_{1/2,p}(∂S) to H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}), and for every κ > 0 and p ∈ ℂ̄_{κ}

∥π_{αβ}^{±}f∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}} ≤ c|p|∥f∥_{1/2,p;∂S},  (16)
∥f∥_{1/2,p;∂S} ≤ c|p|∥π_{αβ}^{±}f∥_{1/2,p,∂S_{α};−1/2,p;∂S_{β}}.  (17)

(ii) The operators θ_{αβ}^{±} are homeomorphisms from H_{−1/2,p}(∂S) to the space H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}), and for every κ > 0 and p ∈ ℂ̄_{κ}

∥θ_{αβ}^{±}g∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}} ≤ c|p|∥g∥_{−1/2,p;∂S},  (18)
∥g∥_{−1/2,p;∂S} ≤ c|p|∥θ_{αβ}^{±}g∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}}.  (19)

(iii) The operators ρ_{αβ}^{±} are homeomorphisms from the space H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}) to H_{1/2,p}(∂S_{β}) × H_{−1/2,p}(∂S_{α}), and for every κ > 0, p ∈ ℂ̄_{κ}

∥ρ_{αβ}^{±}{f_{α}, g_{β}}∥_{1/2,p;∂S_{β};−1/2,p,∂S_{α}} ≤ c|p|∥{f_{α}, g_{β}}∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}},  (20)
∥{f_{α}, g_{β}}∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}} ≤ c|p|∥ρ_{αβ}^{±}{f_{α}, g_{β}}∥_{1/2,p;∂S_{β};−1/2,p;∂S_{α}}.  (21)

Proof. (i) The continuity of π_{αβ}^{±} and (16) follow from Lemma 1. Let u ∈ H_{1,p}(S^{±}) be a unique solution of the problem

\begin{matrix}p^{2}(B^{1/2}u, B^{1/2}v)_{0;S^{±}} + a_{±}(u, v) = ±(g_{β}, v)_{0;∂S_{β}} ∀v ∈ H̊_{1,p}(S^{±}, ∂S_{β}), \\ π_{α}γ^{±}u = f_{α}, \end{matrix} (22)

where {f_{α}, g_{β}} ∈ H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}). By (11),

∥u∥_{1,p;S^{±}} ≤ c|p|∥{f_{α}, g_{β}}∥_{1/2,p,∂S_{α};−1/2,p;∂S_{β}}.  (23)

If f = γ^{±}u ∈ H_{1/2,p}(∂S), then π_{αβ}^{±}f = {f_{α}, g_{β}}; hence, π_{αβ}^{±} are surjective. The trace theorem and (23) imply that (17) holds.

(ii) From Lemma 1 and (i) it follows that θ_{αβ}^{±} are homeomorphisms from H_{−1/2,p}(∂S) to H_{1/2,p}(∂S_{α}) × H_{−1/2,p}(∂S_{β}). Let u ∈ H_{1,p}(S^{±}) be the unique solution of the problem

p^{2}(B^{1/2}u, B^{1/2}v)_{0;S^{±}} + a_{±}(u, v) = ±(g, v)_{0;∂S} ∀v ∈ H_{1,p}(S^{±}), where g ∈ H_{−1/2,p}(∂S). By Theorem 3 in [8],

∥u∥_{1,p;S^{±}} ≤ c|p|∥g∥_{−1/2,p;∂S}.

If f = (T_{p}^{±})^{−1}g, then ∥f∥_{1/2,p;∂S} ≤ c|p| ∥g∥_{−1/2,p;∂S}. We now have ∥π_{α}f∥_{1/2,p;∂S_{α}} + ∥π_{β}g∥_{−1/2,p;∂S_{β}} ≤ ∥f∥_{1/2,p;∂S} + ∥g∥_{−1/2,p;∂S} ≤ c|p|∥g∥_{−1/2,p;∂S}, which proves (18).

Next, let u be the solution of (2), γ^{±}u = f and T_{p}^{±}f = g. From the estimate [7]

∥g∥_{−1/2,p;∂S} ≤ c∥u∥_{1,p;S^{±}}

and (11) it follows that (19) is holds.

(iii) The definition of ρ_{αβ}^{±} and assertion (i) indicate that these operators are homeomorphisms from H_{1/2,p}(∂S_{α})×H_{−1/2,p}(∂S_{β}) to H_{1/2,p}(∂S_{β})×H_{−1/2,p}(∂S_{α}). Let u be the solution of (2), f = γ^{±}u, and g = T_{p}^{±}f. We have

∥π_{α}g∥_{−1/2,p;∂S_{α}} + ∥π_{β}f∥_{1/2,p;∂S_{β}} ≤ ∥g∥_{−1/2,p;∂S} + ∥f∥_{1/2,p;∂S} ≤ c∥u∥_{1,p;S^{±}}
≤ c|p| ∥{f_{α}, g_{β}}∥_{1/2,p;∂S_{α};−1/2,p;∂S_{β}},

which proves (20). Estimate (21) is proved similarly. □

We now consider four representations for the solutions of problems (DM^{±}) in terms of dynamic potentials and prove the unique solvability of the corresponding systems of boundary integral equations. We begin with the representation

u(x, t) = (V α)(x, t), (x, t) ∈ G^{+} or G^{−},  (24)

which yields the system of boundary equations

\begin{aligned}(π_{1}V_{0}α)(x, t) & = f_{1}(x, t), (x, t) ∈ Γ_{1}, \\ (π_{2}T^{±}V_{0}α)(x, t) & = g_{2}(x, t), (x, t) ∈ Γ_{2}. \end{aligned}  (25)

Theorem 4. For every κ > 0, k ∈ ℝ, f_{1} ∈ H_{1/2,k,κ}^{L^{−1}}(Γ_{1}), and g_{2} ∈ H_{−1/2,k,κ}^{L^{−1}}(Γ_{2}) system (25) has a unique solution α ∈ H_{−1/2,k−2,κ}^{L^{−1}}(Γ), in which case u defined by (24) belongs to H_{1,k−1,κ}^{L^{−1}}(G^{±}). If k ≥ 1, then u is the solution of (DM^{±}).

Proof. We give the proof of this assertion and of the next three ones only for (DM^{+}). The exterior problem (DM^{−}) is treated similarly.

In terms of Laplace transforms, (25) takes the form

π_{1}V_{p,0}α̂ = f_{1}, π_{2}T_{p}^{+}V_{p,0}α̂ = g_{2},

or

π_{12}^{+} V_{p,0}α̂ = {f_{1}, g_{2}}.  (26)

By Lemmas 5 and 2, (26) has a unique solution α̂ ∈ H_{−1/2,p}(∂S) and

∥α̂∥_{−1/2,p;∂S} ≤ c|p|^{2}∥{f_{1}, g_{2}}∥_{1/2,p;∂S_{1};−1/2,p,∂S_{2}}  (27)

for all p ∈ ℂ_{κ}. Taking (27) and Theorem 1 into account, we complete the proof by the standard scheme used in [8]. □

The second representation is

u(x, t) = (W β)(x, t), (x, t) ∈ G^{+} or G^{−},  (28)

and it leads to the system

\begin{aligned}(π_{1}W^{±}β)(x, t) & = f_{1}(x, t), (x, t) ∈ Γ_{1}, \\ (π_{2}Nβ)(x, t) & = g_{2}(x, t), (x, t) ∈ Γ_{2}. \end{aligned}  (29)

Theorem 5. For all κ > 0, k ∈ ℝ, f_{1} ∈ H_{1/2,k,κ}^{L^{−1}}(Γ_{1}), and g_{2} ∈ H_{−1/2,k,κ}^{L^{−1}}(Γ_{2}) system (29) has a unique solution β ∈ H_{1/2,k−2,κ}^{L^{−1}}(Γ), in which case u defined by (28) belongs to H_{1,k−1,κ}^{L^{−1}}(G^{±}). If k ≥ 1, then u is the solution of (DM^{±}).

Proof. In the case of (DM^{+}), (29) written in terms of Laplace transforms takes the form

π_{1}W_{p}^{+}β̂ = f_{1}, π_{2}N_{p}β̂ = g_{2},

or

π_{12}^{+}W_{p}^{+}β̂ = {f_{1}, g_{2}}.  (30)

If g = N_{p}β̂, then W_{p}^{+}β̂ = (T_{p}^{+})^{−1}g and (30) can be rewritten as

θ_{12}^{+}g = {f_{1}, g_{2}}.  (31)

By Lemmas 5 and 1, (31)—hence, also (30)—have a unique solution β̂ ∈ H_{1/2,p}(∂S) and

∥β̂∥_{1/2,p;∂S} ≤ c|p|^{2}∥{f_{1}, g_{2}}∥_{1/2,p;∂S_{1};−1/2,p;∂S_{2}}  (32)

for all p ∈ ℂ_{κ}. Theorem 1 and (32) now enable us to complete the proof in the usual way. □

The third representation is

u(x, t) = (V α_{1})(x, t) + (W β_{2})(x, t), (x, t) ∈ G^{+} or G^{−},  (33)

where α_{1} ∈ H̊_{−1/2,k,κ}^{L^{−1}}(Γ_{1}) and β_{2} ∈ H̊_{1/2,k,κ}^{L^{−1}}(Γ_{2}). This representation yields the system of boundary equations

\begin{aligned}(π_{1}V_{0}α_{1})(x, t) + (π_{1}W^{±}β_{2})(x, t) & = f_{1}(x, t), (x, t) ∈ Γ_{1}, \\ (π_{2}T^{±}V_{0}α_{1})(x, t) + (π_{2}Nβ_{2})(x, t) & = g_{2}(x, t), x, t ∈ Γ_{2}. \end{aligned}  (34)

Theorem 6. For all κ > 0, k ∈ ℝ, f_{1} ∈ H_{1/2,k,κ}^{L^{−1}}(Γ_{1}), and g_{2} ∈ H_{−1/2,k,κ}^{L^{−1}}(Γ_{2}) system (34) has a unique solution {α_{1}, β_{2}} ∈ H̊_{−1/2,k−1,κ}^{L^{−1}}(Γ_{1}) × H̊_{1/2,k−1,κ}^{L^{−1}}(Γ_{2}), in which case u defined by (33) belongs to H_{1,k−1,κ}^{L^{−1}}(G^{±}). If k ≥ 1, then u is the solution of (DM^{±}).

Proof. Going over to Laplace transforms, we see that in the case of (DM^{+}) system (34) becomes

π_{12}^{+}(V_{p,0}α̂_{1} + W_{p}^{+}β̂_{2}) = {f_{1}, g_{2}}.  (35)

We claim that {α̂_{1}, β̂_{2}}, where

α̂_{1} = [(θ_{12}^{+})^{−1} − (θ_{12}^{−})^{−1}]{f_{1}, g_{2}},
β̂_{2} = [(π_{12}^{−})^{−1} − (π_{12}^{+})^{−1}]{f_{1}, g_{2}},

is a solution of (35). Indeed, from the definition of π_{12}^{±} and θ_{12}^{±} it follows that π_{1}[(π_{12}^{+})^{−1} − (π_{12}^{−})^{−1}] = 0; hence, β̂_{2} ∈ H̊_{1/2,p}(∂S_{2}). Analogously, α̂_{1} ∈ H̊_{−1/2,p}(∂S_{1}). Then, since (θ_{12}^{±})^{−1} = T_{p}^{±}(π_{12}^{±})^{−1} and V_{p,0}(T_{p}^{+} − T_{p}^{−}) = I, we have

V_{p,0}α̂_{1} + W_{p}^{+}β̂_{2} = V_{p,0}{(θ_{12}^{+})^{−1} − (θ_{12}^{−})^{−1} + T_{p}^{−}[(π_{12}^{−})^{−1} − (π_{12}^{+})^{−1}]}{f_{1}, g_{2}}
= V_{p,0}{(θ_{12}^{+})^{−1} − T_{p}^{−}(π_{12}^{+})^{−1}}{f_{1}, g_{2}} = (π_{12}^{+})^{−1}{f_{1}, g_{2}},

which proves the assertion.

The difference {α̃_{1}, β̃_{2}} ∈ H̊_{−1/2,p}(∂S_{1}) × H̊_{1/2,p}(∂S_{2}) of two solutions of (35) will satisfy

π_{12}^{+}(V_{p,0}α̃_{1} + W_{p}^{+}β̃_{2}) = {0, 0},

so

V_{p,0}α̃_{1} + W_{p}^{+}β̃_{2} = 0.

We take u(x, p) = (V_{p}α̃_{1})(x, p)+(W_{p}β̃_{2})(x, p). This function is a solution of both (DM_{p}^{+}) and (DM_{p}^{−}) with f_{1} = 0 and g_{2} = 0, respectively; therefore, u(x, p) = 0 in S^{+} and in S^{−}. Then β̃_{2} = γ^{−}u − γ^{+}u = 0 and α̃_{1} = T_{p}^{+}γ^{+}u − T_{p}^{−}γ^{−}u = 0, which proves the uniqueness of the solution.

The estimates

∥α̂_{1}∥_{−1/2,p;∂S} ≤ c|p| ∥{f_{1}, g_{2}}∥_{1/2,p;∂S_{1};−1/2,p;∂S_{2}},
∥β̂_{2}∥_{1/2,p;∂S} ≤ c|p| ∥{f_{1}, g_{2}}∥_{1/2,p;∂S_{1};−1/2,p;∂S_{2}}

follow from (17) and (19). The proof is completed by following the standard procedure. □

The fourth representation is

u(x, t) = (W β_{1})(x, t) + (V α_{2})(x, t), (x, t) ∈ G^{+} or G^{−},  (36)

where β_{1} ∈ H̊_{1/2,k,κ}^{L^{−1}}(Γ_{1}) and α_{2} ∈ H̊_{−1/2,k,κ}^{L^{−1}}(Γ_{2}). This yields the system

\begin{aligned}(π_{1}W^{±}β_{1})(x, t) + (π_{1}V_{0}α_{2})(x, t) & = f_{1}(x, t), (x, t) ∈ Γ_{1}, \\ (π_{2}Nβ_{1})(x, t) + (π_{2}T^{±}V_{0}α_{2})(x, t) & = g_{2}(x, t), (x, t) ∈ Γ_{2}. \end{aligned}  (37)

Theorem 7. For all κ > 0, k ∈ ℝ, f_{1} ∈ H_{1/2,k,κ}^{L^{−1}}(Γ_{1}), and g_{2} ∈ H_{−1/2,k,κ}^{L^{−1}}(Γ_{2}) system (37) has a unique solution {β_{1}, α_{2}} ∈ H̊_{1/2,k−2,κ}^{L^{−1}}(Γ_{1}) × H̊_{−1/2,k−2,κ}^{L^{−1}}(Γ_{2}), in which case u defined by (36) belongs to H_{1,k−1,κ}^{L^{−1}}(G^{±}). If k ≥ 1, then u is the solution of (DM^{±}).

Proof. In terms of Laplace transforms, for (DM^{+}) (37) takes the form

π_{12}^{+}(W_{p}^{+}β̂_{1} + V_{p,0}α̂_{2}) = {f_{1}, g_{2}}.  (38)

We claim that

β̂_{1} = [(π_{12}^{−})^{−1}ρ_{21}^{+} − (π_{12}^{+})^{−1}]{f_{1}, g_{2}},
α̂_{2} = [(θ_{12}^{+})^{−1} − (θ_{12}^{−})^{−1}ρ_{21}^{+}]{f_{1}, g_{2}}

is the solution of (38). Indeed, we have

W_{p}^{+}β̂_{1} + V_{p,0}α̂_{2}
= V_{p,0}{T_{p}^{−}[(π_{12}^{−})^{−1}ρ_{21}^{+} − (π_{12}^{+})^{−1}] + (θ_{12}^{+})^{−1} − (θ_{12}^{−})^{−1}ρ_{21}^{+}}{f_{1}, g_{2}}
= V_{p,0}{T_{p}^{+}(π_{12}^{+})^{−1} − T_{p}^{−}(π_{12}^{+})^{−1}}{f_{1}, g_{2}} = (π_{12}^{+})^{−1}{f_{1}, g_{2}};

hence, (38) holds.

Since π_{2}[(π_{12}^{−})^{−1}ρ_{21}^{+}−(π_{12}^{+})^{−1}]{f_{1}, g_{2}} = 0, we have β̂_{1} ∈ H̊_{1/2,p}(∂S_{1}). Similarly, α̂ ∈ H̊_{−1/2,p}(∂S_{2}). To prove the unique solvability of (38), we repeat (with obvious changes) the proof of Theorem 6. From (17), (19), and (20) it follows that for all p ∈ ℂ_{κ}

∥β̂_{1}∥_{1/2,p;∂S} ≤ c|p|^{2}∥{f_{1}, g_{2}}∥_{1/2,p;∂S_{1};−1/2,p;∂S_{2}},
∥α̂_{2}∥_{−1/2,p;∂S} ≤ c|p|^{2}∥{f_{1}, g_{2}}∥_{1/2,p;∂S_{1};−1/2,p;∂S_{2}}.

The proof is now completed in the usual way. □

Acknowledgement

One of the authors (I.C.) wishes to thank the authorities of the University of Guanajuato and his colleagues for arranging his visit to Salamanca and making his stay there a pleasurable and fruitful experience.

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(Received 6.03.2003)

Authors’ addresses: Igor Chudinovich Department of Mechanical Engineering University of Guanajuato Salamanca, Mexico E-mail: chudinovych@salamanca.ugto.mx Christian Constanda Department of Mathematical and Computer Sciences The University of Tulsa Tulsa, Oklahoma, USA E-mail: christian-constanda@utulsa.edu