Georgian Mathematical Journal 1(1994), No. 2, 141-150

ON THE CORRECT FORMULATION OF A MULTIDIMENSIONAL PROBLEM FOR STRICTLY HYPERBOLIC EQUATIONS OF HIGHER ORDER

S. KHARIBEGASHVILI

Abstract. A theorem of the unique solvability of the first boundary value problem in the Sobolev weighted spaces is proved for higherorder strictly hyperbolic systems in the conic domain with special orientation.

In the space R^{n}, n > 2, let us consider a strictly hyperbolic equation of the form

p(x, ∂)u(x) = f(x),  (1)

where ∂ = (∂_{1}, …, ∂_{n}), ∂_{j} = ∂/∂x_{j}, p(x, ξ) is a real polynomial of order 2m, m > 1, with respect to ξ = (ξ_{1}, …, ξ_{n}), f is the known function and u is the unknown function. It is assumed that in equation (1) the coefficients at higher derivatives are constant and the other coefficients are finite and infinitely differentiable in R^{n}.

Let D be a conic domain in R^{n}, i.e., D together with a point x ∈ D contains the entire beam tx, 0 < t < ∞. Denote by Γ the cone ∂D. It is assumed that D is homeomorphic onto the conic domain x_{1}^{2} + ⋯ + x_{n−1}^{2} − x_{n}^{2} < 0, x_{n} > 0 and Γ′ = Γ\O is a connected (n − 1)-dimensional manifold of the class C^{∞}, where O is the vertex of the cone Γ.

Consider the problem: Find in the domain D the solution u(x) of equation (1) by the boundary conditions

\frac{∂^{i}u}{∂ν^{i}} |_{Γ′} = g_{i}, i = 0, …, m − 1,  (2)

where ν = ν(x) is the outward normal to Γ′ at a point x ∈ Γ′, and g_{i}, i = 0, …, m − 1, are the known real functions.

Note that the problem (1), (2) is considered in [1–6] for a hyperbolictype equation of second order when Γ is a characteristic conoid. In [7] this problem is considered for a wave equation when the conic surface Γ is not characteristic at any point and has a time-type orientaton. A multidimensional analogue of the problem is treated in [8–10] for the case when one part of the cone Γ is characteristic and the other part is a time-type hyperplane. Other multidimensional analogues of the Goursat problem for hyperbolic systems of first and second order are investigated in [11–15].

1991 Mathematics Subject Classification. 35L35.

In this paper we consider the question whether the problem (1), (2) can be correctly formulated in special weighted spaces W_{α}^{k}(D) when the cone Γ is assumed not to be characteristic but having a quite definite orientation.

Denote by p_{0}(ξ) the characteristic polynomial of the equation (1), i.e., the higher homogeneous part of the polynomial p(x, ξ). The strict hyperbolicity of the equation (1) implies the existence of a vector ζ ∈ R^{n} such that the straight line ξ = λζ + η, where η ∈ R^{n} is an arbitrarily chosen vector not parallel to ζ and λ is the real parameter, intersects the cone of normals K : p_{0}(ξ) = 0 of the equation (1) at 2m different real points. In other words, the equation p_{0}(λζ + η) = 0 with respect to λ has 2m different real roots. The vector ζ is called a spatial-type normal. As is well-known, a set of all spatial-type normals form two connected centrally-symmetric convex conic domains whose boundaries K_{1} and K_{2m} give the internal cavity of the cone of normals K [3]. The surface S ⊂ R^{n} is called characteristic at a point x ∈ S if the normal to S at the point x belongs to the cone K.

Let the vector ζ be a spatial-type normal and the vector η ≠ 0 change in the plane orthogonal to ζ. Then for λ the roots of the characteristic polynomial p_{0}(λζ + η) can renumerated so that λ_{2m}(η) < λ_{2m−1}(η) < ⋯ < λ_{1}(η). It is obvious that the vectors λ_{i}(η)ζ + η cover the cavities K_{i} of K when the η changes on the plane othogonal to ζ. Since λ_{m−j}(η) = −λ_{m+j+1}(−η), 0 ≤ j ≤ m − 1, the cones K_{m−j} and K_{m+j+1} are centrally symmetric with respect to the point (0, …, 0). As is well-known, by the bicharacteristics of the equation (1) we understand straight beams whose orthogonal planes are tangential planes to one of the cavities K_{i} at the point different from the vertex.

Assume that there exists a plane π_{0} such that π_{0} ∩ K_{m} = {(0,…,0)}. This means that the cones K_{1}, …, K_{m} are located on one side of π_{0} and the cones K_{m+1}, …, K_{2m} on the other. Set K_{i}^{∗} = ∩_{η∈K_{i}}{ξ ∈ R^{n} : ξ·η < 0}, where ξ · η is the scalar product of ξ and η. Since π_{0} ∩ K_{m} = {(0, …, 0)}, K_{i}^{∗} is a conic domain and K_{m}^{∗} ⊂ K_{m−1}^{∗} ⊂ ⋯ ⊂ K_{1}^{∗}, K_{m+1}^{∗} ⊂ K_{m+2}^{∗} ⊂ ⋯ ⊂ K_{2m}^{∗}. It is easy to verify that ∂(K_{i}^{∗}) is a convex cone whose generatrices are bicharacteristics; note that in this case none of the bicharacteristics of the equation (1) comes from the point (0, …, 0) into the cone ∂(K_{m}^{∗}) or ∂(K_{m+1}^{∗}) [3].

Let us consider

Condition 1. The surface Γ′ is characteristic at none of its points and each generatrix of the cone Γ has the direction of a spatial-type normal; moreover, Γ ⊂ K_{m}^{∗} ∪ 0 or Γ ⊂ K_{m+1}^{∗} ∪ 0.

Denote by W_{α}^{k}(D), k ≥ 2m, −∞ < α < ∞, the functional space with the norm [16]

∥u∥_{W_{α}^{k}(D)}^{2} = ∑_{i=0}^{k} ∫_{D} r^{−2α−2(k−i)}∥\frac{∂^{i}u}{∂x^{i}} ∥^{2}dx,

where

r = (x_{1}^{2} + ⋯ + x_{n}^{2})^{\frac{1}{2}} , \frac{∂^{i}u}{∂x^{i}} = \frac{∂^{i}u}{∂x_{1}^{i_{1}} ⋯ ∂x_{n}^{i_{n}}} , i = i_{1} + ⋯ + i_{n}.

The space W_{α}^{k}(Γ) is defined in a similar manner.

Consider the space

V = W_{α−1}^{k+1−2m}(D) × ∏_{i=0}^{m−1} W_{α− \frac{1}{2}}^{k−i} (Γ).

Assume that to the problem (1), (2) there corresponds the unbounded operator

T : W_{α}^{k}(D) → V

with the domain of definition Ω_{T} = W_{α−1}^{k+1}(D) ⊂ W_{α}^{k}(D), acting by the formula

T u = (p(x, ∂)u, u|_{Γ′}, …, \frac{∂^{i}u}{∂ν^{i}} |_{Γ′}, …, \frac{∂^{m−1}u}{∂ν^{m−1}} |_{Γ′}), u ∈ Ω_{T} .

It is obvious that the operator T admits the closure \overline{T}.

The function u is called a strong solution of the problem (1), (2) of the class W_{α}^{k}(D) if u ∈ Ω\overline{T}, \overline{T} u = (f, g_{0}, …, g_{m−1}) ∈ V, which is equivalent to the existence of a sequence u_{i} ∈ Ω_{T} = W_{α−1}^{k+1}(D) such that u_{i} → u in W_{α}^{k}(D) and (p(x, ∂)u_{i}, u_{i}|_{Γ′}, …, \frac{∂^{m−1}u_{i}}{∂ν^{m−1}} |_{Γ′}) → (f, g_{0}, …, g_{m−1}) in V.

Below, by a solution of the problem (1), (2) of the class W_{α}^{k}(D) we will mean a strong solution of this problem in the sense as indicated above.

We will prove

Theorem. Let condition 1 be fulfilled. Then there exists a real number α_{0} = α_{0}(k) > 0 such that for α ≥ α_{0} the problem (1), (2) is uniquely solvable in the class W_{α}^{k}(D) for any f ∈ W_{α−1}^{k+1−2m}(D), g_{i} ∈ W_{α− \frac{1}{2}}^{k−i} (Γ), i = 0, …, m − 1, and to obtain the solution u we have the estimate

∥u∥_{W_{α}^{k}(D)} ≤ c( ∑_{i=1}^{m−1} ∥g_{i}∥_{W_{α− \frac{1}{2}}^{k−i} (Γ)} + ∥f∥_{W_{α−1}^{k+1−2m}(D)}),  (3)

where c is a positive constant not depending on f, g_{i}, i = 0, …, m − 1.

Proof. First we will show that the corollaries of condition 1 are the following conditions: Take any point P ∈ Γ′ and choose a Cartesian system x_{1}^{0}, …, x_{n}^{0} connected with this point and having vertex at P such that the x_{n}^{0}-axis is directed along the generatrix of Γ passing through P and the x_{n−1}^{0}-axis is directed along the inward normal to Γ at this point.

Condition 2. The surface Γ′ is characteristic at none of its points. Each generatrix of the cone Γ has the direction of a spatial-type normal, and exactly m characteristic planes of equation (1) pass through the (n − 2)- dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0 connected with an arbitrary point P ∈ Γ′ into the angle x_{n}^{0} > 0, x_{n−1}^{0} > 0.

Denote by p̃_{0}(ξ) the characteristic polynomial of the equation (1) written in terms of the coordinate system x_{1}^{0}, …, x_{n}^{0}, connected with an arbitrarily chosen point P ∈ Γ′.

Condition 3. The surface Γ′ is characteristic at none of its point. Each generatrix of the cone Γ has the direction of a spatial-type normal and for Re s > 0 the number of roots λ_{j}(ξ_{1}, …, ξ_{n−2}, s), if we take into account the multiplicity of the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) with Re λ_{j} < 0, is equal to m, i = \sqrt{−1}.

When condition 3 is fulfilled, the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) can be written as the product Δ_{−}(λ)Δ_{+}(λ), where for Re s > 0 the roots of the polynomials Δ_{−}(λ) and Δ_{+}(λ) lie, respectively, to the left and to the right of the imaginary axis, while the coefficients are continuous for s, Re s ≥ 0, (ξ_{1}, …, ξ_{n−2}) ∈ R^{n−2}, ξ_{1}^{2} + ⋯ + ξ_{n−2}^{2} + |s|^{2} = 1 [17]. On the left side of the boundary conditions (2) to the differential operator b_{j}(x, ∂), 0 ≤ j ≤ m − 1, written in terms of the coordinate system x_{1}^{0}, …, x_{n}^{0} connected with the point P ∈ Γ′, there corresponds the characteristic polynomial b_{j}(ξ) = ξ_{n−1}^{j}. Therefore, since the degree of the polynomial Δ_{−}(λ) is equal to m, the following condition will be fulfilled:

Condition 4. For any point P ∈Γ′ and any s, Re s≥ 0, and (ξ_{1}, ..., ξ_{n−2})∈ R^{n−2} such that ξ_{1}^{2}+⋯+ξ_{n−2}^{2}+|s|^{2} =1, the polynomials b_{j}(iξ_{1}, ..., iξ_{n−2}, λ, s) = λ^{j}, j = 0, …, m − 1, are linearly independent, like the polynomials of λ modulo Δ_{−}(λ).

We will now show that condition 1 implies condition 2, while the latter implies condition 3. Let us consider the case Γ ⊂ K_{m+1}^{∗} ∪ O. The second case Γ ⊂ K_{m}^{∗} ∪ O is treated similarly.

Let P ∈ Γ′ and x_{1}^{0}, …, x_{n}^{0} be the coordinate system connected with this point. Since the generatrix γ of the cone Γ passing through this point is a spatial-type normal, the plane x_{n}^{0} = 0 passing through the point P is a spatial-type plane. Denote by K_{j}^{∧} the boundary of the convex shell of the set K_{j} and by K_{j}^{⊥} the set which is the union of all bicharacteristics corresponding to the cone K_{j} and coming out of the point O along the outward normal to K_{j}, 1 ≤ j ≤ 2m. It is obvious that (K_{j}^{∧})^{∗} = K_{j}^{∗}, ∂(K_{j}^{∗}) = (K_{j}^{∧})^{⊥}. We will show that the plane π_{1}, parallel to the plane x_{n}^{0} = 0 and passing through the point (0, …, 0), is the plane of support to the cone K_{m}^{∧} at the point (0, …, 0). Indeed, it is obvious that the plane N · ξ = 0, N ∈ R^{n}\(0, …, 0), ξ ∈ R^{n} is the plane of support to K_{m}^{∧} at the point (0, …, 0) iff the normal vector N to this plane taken with the sign + or − belongs to the conic domain closure (K_{m}^{∧})^{∗} = K_{m}^{∗}. Now it remains for us to note that the conic domains K_{m}^{∗} and K_{m+1}^{∗} are centrally symmetric with respect to the point (0, …, 0), and the generatrix Γ passing through the point P is perpendicular to the plane π_{1} and, by the condition, belongs to the set K_{m+1}^{∗} ∪O. Since x_{n}^{0} = 0 is a spatial-type plane, the two-dimensional plane σ : x_{1}^{0} = ⋯ = x_{n−2}^{0} = 0 passing through the generatrix γ which is directed along the spatial-type normal intersects the cone of normals K_{p} of equation (1) with vertex at the point P by 2m different real straight lines [3]. The planes orthogonal to these straight lines and passing through the (n − 2)-dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0 give all 2m characteric planes passing through the (n − 2)-dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0. The straight lines x_{n}^{0} = 0 and x_{n−1}^{0} = 0 divide the two-dimensional plane σ into four right angles

σ_{1} : x_{n−1}^{0} > 0, x_{n}^{0} > 0; σ_{2} : x_{n−1}^{0} < 0, x_{n}^{0} > 0;
σ_{3} : x_{n−1}^{0} < 0, x_{n}^{0} < 0; σ_{4} : x_{n−1}^{0} > 0, x_{n}^{0} < 0.

One can readily see that exactly m characteristic planes of equation (1) pass through the (n−2)-dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0 into the angle x_{n}^{0} > 0, x_{n−1}^{0} iff exactly m straight lines from the intersection of σ_{4} with the twodimensional plane σ pass into the angle K_{P}. The latter fact really occurs, since: 1) the plane x_{n}^{0} = 0 is the plane of support to K_{m}^{∧} and therefore to all K_{1}, …, K_{2m}; 2) the planes x_{n}^{0} = 0, x_{n−1}^{0} = 0 are not characteristic because the generatrices of Γ have a spatial-type direction and Γ is not characteristic at the point P.

Now it will be shown that condition 2 implies condition 3. By virtue of condition 2 the plane x_{n−1}^{0} = 0 is not characteristic and therefore for λ the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) has exactly 2m roots. In this case, if Re s > 0, the number of roots λ_{j}(ξ_{1}, …, ξ_{n−2}, s), with the multiplicity of the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) taken into account, will be equal to m provided that Re λ_{j} < 0. Indeed, recalling that equation (1) is hyperbolic, the equation p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) = 0 has no purely imaginary roots with respect to λ. Since the roots λ_{j} are continuous functions of s, we can determine the number of roots λ_{j} with Re λ_{j} < 0 by passing to the limits as Re s → +∞. Since the equality

p̃_{0}(iξ_{1}, …, iξ_{n−2}λ, s) = s^{2m}p̃_{0}(i\frac{ξ_{1}}{s} , …, i\frac{ξ_{n−2}}{s} , \frac{λ}{s}, 1)

holds, it is clear that the ratios λ_{j}/s , where λ_{j} are the roots of the equation p̃_{0}(iξ_{1}, …, iξ_{n−2}, λ, s) = 0, tend to the roots μ_{j} of the equation p̃_{0}(0, … 0, μ, 1) = 0 as Re s → +∞. The latter roots are real and different because equation (1) is hyperbolic. If s is taken positive and sufficiently large, then for μ_{j} ≠ 0 we have λ_{j} = sμ_{j} + o(s). But μ_{j} ≠ 0, since the plane x_{n}^{0} = 0 is not characteristic. Therefore the number of roots λ_{j} with Re λ_{j} < 0 coincides with the number of roots μ_{j} with μ_{j} < 0. Since the characteristic planes of equation (1), passing through the (n − 2)-dimensional plane x_{n}^{0} = x_{n−1}^{0} = 0, are determined by the equalities μ_{j}x_{n−1}^{0} + x_{n}^{0} = 0, j = 1, …, 2m, condition 2 implies that for Re λ_{j} < 0 the number of roots λ_{j} is equal to m.

We give another equivalent description of the space W_{α}^{k}(D). On the unit sphere S^{n−1} : x_{1}^{2} + ⋯ + x_{n}^{2} = 1 choose a coordinate system (ω_{1}, …, ω_{n−1}) such that in the domain D the transformation

I : τ = log r, ω_{j} = ω_{j}(x_{1}, …, x_{n}), j = 1, …, n − 1,

is one-to-one, nondegenerate, and infinitely differentiable. Since the cone Γ = ∂D is strictly convex at the point O(0, …, 0), such coordinates evidently exist. As a result of the above transformation, the domain D will become the infinite cylinder G bounded by the infinitely differentiable surface ∂G = I(Γ′).

Introduce the functional space H_{γ}^{k}(G), −∞ < γ < ∞, with the norm

∥v∥_{H_{γ}^{k}(G)}^{2} = ∑_{i_{1}+j=0}^{k} ∫_{G} e^{−2γτ} ∥\frac{∂^{i_{1}+j}v}{∂τ^{i_{1}}∂ω^{j}} ∥^{2}dω dτ

where

\frac{∂^{i_{1}+j}v}{∂τ^{i_{1}}∂ω^{j}} = \frac{∂^{i_{1}+j}v}{∂τ^{i_{1}}∂ω_{1}^{j_{1}} ⋯ ∂ω_{n−1}^{j_{n−1}}} , j = j_{1} + ⋯ + j_{n−1}.

As shown in [16], a function u(x) ∈ W_{α}^{k}(D) iff ũ = u(I^{−1}(τ, ω)) ∈ H_{(α+k)− \frac{n}{2}}^{k} (G), and the estimates

c_{1}∥ũ∥_{H_{(α+k)− \frac{n}{2}}^{k} (G)} ≤ ∥u∥_{W_{α}^{k}(D)} ≤ c_{2}∥ũ∥_{H_{(α+k)− \frac{n}{2}}^{k} (G)}

hold, where I^{−1} is the inverse transformation of I and the positive constants c_{1} and c_{2} do not depend on u.

It can be easily verified that the condition v ∈ H_{γ}^{k}(G) is equivalent to the condition e^{−γτ} v ∈ W^{k}(G), where W^{k}(G) is the Sobolev space. Denote by H_{γ}^{k}(∂G) a set of ψ such that e^{−γτ} ψ ∈ W^{k}(∂G), and by W_{α− \frac{1}{2}}^{k} (Γ) a set of all φ for which φ̃ = φ(I^{−1}(τ, ω)) ∈ H_{(α+k)− \frac{n}{2}}^{k} (∂G). Assume that

∥φ∥_{W_{α− \frac{1}{2}}^{k} (Γ)} = ∥φ̃∥H_{(α+k)− \frac{n}{2}}^{k} (∂G).

Spaces W_{α}^{k}(D) possess the following simple properties:

1) if u ∈ W_{α}^{k}(D), then \frac{∂^{i}u}{∂x^{i}} ∈ W_{α}^{k−i}(D), 0 ≤ i ≤ k;

2) W_{α−1}^{k+1}(D) ⊂ W_{α}^{k}(D);

3) if u ∈ W_{α−1}^{k+1}(D), then by the well-known embedding theorems we have u|_{Γ} ∈ W_{α− \frac{1}{2}}^{k} (Γ), \frac{∂^{i}u}{∂ν^{i}} |_{Γ′} ∈ W_{α− \frac{1}{2}}^{k−i} (Γ), i = 1, …, m − 1;

4) if u ∈ W_{α−1}^{k+1}(D), then f = p(x, ∂)u ∈ W_{α−1}^{k+1−2m}(D).

In what follows we will need, in spaces W_{α}^{k}(D), W_{α− \frac{1}{2}}^{k} (Γ), other norms depending on the parameter γ = (α + k) − \frac{n}{2} and equivalent to the original norms.

Set

R_{ω,τ}^{n} = {−∞ < τ < ∞, −∞ < ω_{i} < ∞, i = 1, …, n − 1},
R_{ω,τ,+}^{n} = {(ω, τ) ∈ R_{ω,τ}^{n} : ω_{n−1} > 0}, ω′ = (ω_{1}, …, ω_{n−2}),
R_{ω′,τ}^{n−1} = {−∞ < τ < ∞, −∞ < ω_{i} < ∞, i = 1, …, n − 2}.

Denote by ṽ(ξ_{1}, …, ξ_{n−2}, ξ_{n−1}, ξ_{n}−iγ) the Fourier transform of the function e^{−γτ} v(ω, τ), i.e.,

ṽ(ξ_{1}, …, ξ_{n−1}, ξ_{n} − iγ) = (2π)^{− \frac{n}{2}} ∫ v(ω, τ)e^{−iωξ′−iτξ_{n}−γτ} dω dτ,
i = \sqrt{−1}, ξ′ = (ξ_{1}, …, ξ_{n−1}),

and by v̂(ξ, …, ξ_{n−2}, ω_{n−1}, ξ_{n} − iγ) the partial Fourier transform of the function e^{−γτ} v(ω, τ) with respect to ω′,τ.

We can introduce the following equivalent norms:

|||v|||_{R^{n},k,γ}^{2} = ∫_{R^{n}}(γ^{2} + |ξ|^{2})^{k}∥ṽ(ξ_{1}, …, ξ_{n−1}, ξ_{n} − iγ)∥^{2}dξ,

|||v|||_{R_{+}^{n},k,γ}^{2} = ∫_{0}^{∞} ∫_{R^{n−1}} ∑_{j=0}^{k}(γ^{2} + |ξ′|^{2})^{k−j} ×

×∥\frac{∂^{j}}{∂ω_{n−1}^{j}} v̂(ξ_{1}, …, ξ_{n−2}, ω_{n−1}, ξ_{n} − iγ)∥^{2}dξ′dω_{n−1},

in the above-considered spaces H_{γ}^{k}(R_{ω,τ}^{n}) and H_{γ}^{k}(R_{ω,τ,+}^{n}).

Let φ_{1}, …, φ_{N} be the partitioning of unity into G′ = G ∩ {τ = 0}, where G = I(D), i.e., ∑_{j=1}^{N} φ_{j}(ω) ≡ 1 in G′, φ_{j} ∈ C^{∞}(\overline{G}′), the supports of functions φ_{1}, …, φ_{N−1} lie in the boundary half-neighborhoods, while the support of function φ_{N} lies inside G′. Then for γ = (α+k)− \frac{n}{2} the equalities

\begin{matrix}|||u|||_{G,k,γ}^{2} = ∑_{j=1}^{N−1} |||φ_{j}u|||_{R_{+,k,γ}^{n}}^{2} + |||φ_{N}u|||_{R^{n},k,γ}^{2}, \\ |||u|||_{∂G,k,γ}^{2} = ∑_{j=1}^{N−1} |||φ_{j}u|||_{R_{ω′,τ,k,γ}^{n−1}}^{2} \end{matrix}  (4)

define equivalent norms in the spaces W_{α}^{k}(D) and W_{α− \frac{1}{2}}^{k} (Γ), where the norms on the right sides of these equalities are taken in the terms of local coordinates [17].

First we assume that equation (1) contains only higher terms, i.e., p(x, ξ) ≡ p_{0}(ξ). Equation (1) and the boundary conditions (2) written in terms of the coordinates ω, τ have the form

e^{−2mτ} A(ω, ∂)u = f,
e^{−iτ} B_{i}(ω, ∂)u|_{∂G} = g_{i}, i = 0, …, m − 1,

or

A(ω, ∂)u = f̃,  (5)
B_{i}(ω, ∂)u|_{∂G} = g̃_{i}, i = 0, …, m − 1,  (6)

where A(ω, ∂) and B_{i}(ω, ∂) are, respectively, the differential operators of orders 2m and i, with infinitely differentiable coefficients depending only on ω, while f̃ = e^{2mτ} f and g̃_{i} = e^{iτ} g_{i}, i = 0, 1, …, m − 1.

Thus, for the transformation I : D → G, the unbounded operator T of the problem (1), (2) transforms to the unbounded operator

T̃ : H_{γ}^{k}(G) → H_{γ}^{k+1−2m}(G) × ∏_{i=0}^{m−1} H_{γ}^{k−i}(∂G)

with the domain of definition H_{γ}^{k+1}(G), acting by the formula

T̃u = (A(ω, ∂)u, B_{0}(ω, ∂)u|_{∂G}, …, B_{m−1}(ω, ∂)u|_{∂G})

where γ = (α + k) − \frac{n}{2}. Note that written in terms of the coordinates ω, τ the functions f = (ω, τ) ∈ H_{γ−2m}^{k+1−2m}(G), g_{i}(ω, τ) ∈ H_{γ−i}^{k−i}(∂G), i = 0, …, m − 1, if f(x) ∈ W_{α−1}^{k+1−2m}(D), g_{i}(x) ∈ W_{α− \frac{1}{2}}^{k−i} (Γ), i = 0, …, m − 1.

Therefore the functions f̃ = e^{2mτ} f ∈ H_{γ}^{k+1−2m}(G), g̃_{i} = e^{iτ} g_{i} ∈ H_{γ}^{k−i}(∂G), i = 0, …, m − 1.

Since by condition 1 each generatrix of the cone Γ has the direction of a spatial-type normal, due to the convexity of K_{m} each beam coming from the vertex O into the conic domain D also has the direction of a spatialtype normal. Therefore equation (4) is strictly hyperbolic with respect the τ-axis. It was shown above that the fulfillment of condition 1 implies the fulfillment of condition 4. Therefore, according to the results of [17], for γ ≥ γ_{0} , where γ_{0} is a sufficiently large number, the operator \overline{T̃} has

the bounded right inverse operator \overline{T̃}^{−1}. Thus for any f̃ ∈ H_{γ}^{k+1−2m}(G), g̃_{i} ∈ H_{γ}^{k−i}(∂G), i = 0, …, m − 1, when γ ≥ γ_{0}, the problem (5), (6) is uniquely solvable in the space H_{γ}^{k}(G), and for the solution u we have the estimate

|||u|||_{G,k,γ}^{2} ≤ C( ∑_{i=0}^{m−1} |||g̃_{i}|||_{∂G,k−i,γ} + \frac{1}{γ} |||f̃|||_{G,k+1−2m,γ})  (7)

with the positive constant C not depending on γ, f and g̃_{i}, i = 0, …, m−1.

Hence it immediately follows that the theorem and the estimate (3) are valid in the case p(x, ξ) ≡ p_{0}(ξ).

Remark. The estimate (7) with the coefficient \frac{1}{γ} at |||f̃|||_{G,k+1−2m,γ}, obtained in the appropriately chosen norms (4), enables one to prove the theorem also when equation (1) contains lower terms, since the latter give arbitrarily small perturbations for sufficiently large γ.

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(Received 25.12.1992)

Author’s address:

I.Vekua Institute of Applied Mathematics

of Tbilisi State University

2 University St., 380043 Tbilisi

Republic of Georgia