GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 4, 1996, 363-378

ON A DARBOUX TYPE MULTIDIMENSIONAL PROBLEM FOR SECOND-ORDER HYPERBOLIC SYSTEMS

S. KHARIBEGASHVILI

Abstract. The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.

In the space of variables x_{1}, …, x_{n}, t we consider the system of linear differential equations of second-order

Lu ≡ u_{tt} − ∑_{i,j=1}^{n} A_{ij}u_{x_{i}x_{j}} + ∑_{i=1}^{n} B_{i}u_{x_{i}} + Cu = F, n > 2,  (1)

where A_{ij}, B_{i}, C are given real m × m matrices, F the given and u the unknown m-dimensional vector, m > 1.

It will be assumed below that the matrices A_{ij} are symmetrical and constant, and the inequality

∑_{i,j=1}^{n} A_{ij}η_{i}η_{j} ≥ c_{0} ∑_{i=1}^{n} |η_{i}|^{2}, c_{0} = const > 0,  (2)

holds for any m-dimensional real vectors η_{i}, i = 1, …, n.

Condition (2) readily implies that system (1) is hyperbolic.

Let G be a dihedral angle in the space R^{n+1} of variables x_{1}, …, x_{n}, t with a temporal type noncharacteristic boundary ∂G, i.e.,

(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})ηη ≤ 0, η ∈ R^{m},  (3)

363 where the vector (α_{1}, …, α_{n}, α_{n+1}) is the unit normal vector to ∂G at points differing from points of the edges of G, and E is the m × m unit matrix.

1991 Mathematics Subject Classification. 35L20.

Key words and phrases. Darboux type problem, hyperbolic systems, Sobolev space, a priori estimate.

1072-947X/96/0700-0363$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Let Γ be a hyperplane parallel to the edges γ of the angle G and severing from this angle a subdomain D ⊂ G whose sections by two-dimensional planes perpendicular to the edge γ are triangular.

It is further assumed that in system (1) elements of the matrices B_{i}, C are bounded measurable functions in the domain D, and the right-hand part of this system F ∈ L_{2}(D).

Consider a Darboux type problem formulated as follows: In the domain D find a solution u(x_{1}, …, x_{n}, t) of system (1) by the boundary condition

u|_{∂D∩∂G} = f,  (4)

where f is the known real vector.

Note that for the case where at least one face of the angle G is characteristic, problems of this type for a second-order hyberbolic equation are studied in [1–4]. In [5] a Darboux type problem for a wave equation is studied on the assumption that both faces of G are of temporal type. Some multidimensional analogues of the Darboux problems are treated in [6–8]. Also note [9] where a Goursat type multidimensional problem for hyperbolic systems (1) is considered in the conical domain. In this paper, problem (1),

(4) is investigated in the Sobolev space W_{2}^{1}(D).

For simplicity let D : k_{1}t < x_{n} < k_{2}t, 0 < t < t_{0}; k_{i} = const, i = 1, 2, k_{1} < k_{2}, be the domain lying in the half-space t > 0 and bounded by the plane hypersurfaces S_{i} : k_{i}t − x_{n} = 0, 0 ≤ t ≤ t_{0}, i = 1, 2, of temporal type and by the hyperplane t = t_{0}. It is obvious that for such a domain D condition (3), along with the property of the surfaces S_{1} and S_{2} being noncharacteristic, is equivalent to the inequalities

k_{i}^{2} < min(μ_{1}, …, μ_{m}), i = 1, 2,  (5)

where μ_{j}, j = 1, …, m, are the eigenvalues of the symmetrical matrix A_{nn} which by virtue of (2) is positively definite. Then the boundary condition (4) takes the form

u|_{S_{i}} = f_{i}, i = 1, 2,  (6)

where f_{i}, i = 1, 2, are the known real vector-functions on S_{i} and (f_{1} − f_{2})|_{S_{1}∩S_{2}} = 0.

Denote by C_{∗}^{∞}(\overline{D}) the space of functions belonging to the class C^{∞}(\overline{D}) and having bounded supports, i.e.,

C_{∗}^{∞}(\overline{D}) = \{u ∈ C^{∞}(\overline{D}) : diam supp u < ∞\}.

The spaces C_{∗}^{∞}(S_{i}), i = 1, 2, are defined in a similar manner.

Denote by W_{2}^{1}(D), W_{2}^{2}(D), W_{2}^{1}(S_{i}), i = 1, 2, the known Sobolev spaces. Note that the space C_{∗}^{∞}(\overline{D}) is a dense everywhere subspace of the spaces W_{2}^{1}(D) and W_{2}^{2}(D), while C_{∗}^{∞}(S_{i}) is a dense everywhere subspace of the space W_{2}^{1}(S_{i}), i = 1, 2.

Definition. Let f_{i} ∈ W_{2}^{1}(S_{i}), i = 1, 2, F ∈ L_{2}(D). A vector-functon u ∈ W_{2}^{1}(D) will be called a strong solution of problem (1), (6) from the class W_{2}^{1} if there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}) such that u_{n} → u in the space W_{2}^{1}(D), Lu_{n} → F in the space L_{2}(D), and u_{n}|_{S_{i}} → f_{i} in W_{2}^{1}(S_{i}), i = 1, 2, i.e., for n → ∞

∥u_{n} − u∥_{W_{2}^{1}(D)} → 0, ∥Lu_{n} − F ∥_{L_{2}(D)} → 0,
∥u_{n}|_{S_{i}} − f_{i}∥_{W_{2}^{1}(S_{i})} → 0, i = 1, 2.

We have

Lemma 1. If inequalities (5) are fulfilled, k_{1} < 0 and k_{2} > 0, then for any u ∈ W_{2}^{2}(D) the a priori estimate

∥u∥_{W_{2}^{1}(D)} ≤ c( ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{i})} + ∥F ∥_{L_{2}(D)})  (7)

holds, where f_{i} = u|_{S_{i}}, i = 1, 2, F = Lu, and the positive constant c does not depend on u.

Proof. Since the space C_{∗}^{∞}(\overline{D}) (C_{∗}^{∞}(S_{i})) is a dense everywhere subspace of the spaces W_{2}^{1}(D) and W_{2}^{2}(D) (W_{2}^{1}(S_{i})), by virtue of the familiar theorems of embedding the space W_{2}^{2}(D) into the space W_{2}^{1}(D) and the space W_{2}^{2}(D) into W_{2}^{1}(S_{i}) it is sufficient to show that the a priori estimate (7) holds for functions u of the class C_{∗}^{∞}(\overline{D}).

We introduce the notation

D_{τ} = {(x, t) ∈ D : t < τ}, D_{0τ} = ∂D_{τ} ∩ {t = τ}, 0 < τ ≤ t_{0},

S_{iτ} = ∂D_{τ} ∩ S_{i}, i = 1, 2, S_{τ} = S_{1τ} ∪ S_{2τ} ,
α_{i} = cos(ν̂, x_{i}), i = 1, …, n, α_{n+1} = cos(ν̂, t).

Here ν = (α_{1}, …, α_{n}, α_{n+1}) is the unit normal vector to ∂D_{τ} and, as one can readily see,

ν|_{S_{1τ}} = (0, …, 0, \frac{−1}{\sqrt{1 + k_{1}^{2}}} , \frac{k_{1}}{\sqrt{1 + k_{1}^{2}}} ),

ν|_{S_{2τ}} = (0, …, 0, \frac{1}{\sqrt{1 + k_{2}^{2}}} , \frac{−k_{2}}{\sqrt{1 + k_{2}^{2}}} ), ν|_{D_{0τ}} = (0, …, 0, 1).

Therefore for k_{1} < 0, k_{2} > 0 we find by virtue of inequality (3)

α_{n+1}|_{S_{τ}} < 0, α_{n+1}^{−1}(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})ηη|_{S_{τ}} ≥ 0, η ∈ R^{m}. (8)

After performing a scalar multiplication of both sides of system (1) by the vector 2u_{t}, where u ∈ C_{∗}^{∞}(\overline{D}), F = Lu, and integrating the resulting expression with respect to D_{τ}, we obtain

2 ∫_{D_{τ}} (F − ∑_{i=1}^{n} B_{i}u_{x_{i}} − Cu)u_{t}dxdt =

= ∫_{D_{τ}} (\frac{∂(u_{t}u_{t})}{∂t} + 2 ∑_{i,j=1}^{n} A_{ij}u_{x_{j}}u_{tx_{i}})dxdt − 2 ∫_{S_{τ}} ∑_{i,j=1}^{n} A_{ij}u_{t}u_{x_{j}}α_{i}ds =

= ∫_{D_{0τ}} (u_{t}u_{t} + ∑_{i,j=1}^{n} A_{ij}u_{x_{i}}u_{x_{j}})dx +

+ ∫_{S_{τ}} α_{n+1}^{−1}[ ∑_{i,j=1}^{n} A_{ij}(α_{n+1}u_{x_{i}} − α_{i}u_{t})(α_{n+1}u_{x_{j}} − α_{j}u_{t}) +

+(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})u_{t}u_{t}]ds.  (9)

Assuming that

w(τ) = ∫_{D_{0τ}} (u_{t}u_{t} + ∑_{i,j=1}^{n}A_{ij}u_{x_{i}}u_{x_{j}})dx, ũ_{i} = α_{n+1}u_{x_{i}} − α_{i}u_{t}, i = 1, …, n,

and using (8), from (9) we have

w(τ) ≤ c_{1}[ ∫_{S_{τ}} ∑_{i,j=1}^{n} A_{ij}ũ_{i}ũ_{j}ds + ∫_{0}^{τ} w(t)dt +

+ ∫_{D_{τ}} uudxdt + ∫_{D_{τ}} F F dxdt], c_{1} = const > 0.  (10)

Let (x, τ_{x}) be the point at which the surface S_{1}∪S_{2} intersects the straight line parallel to the t-axis passing through the point (x, 0). We have

u(x, τ) = u(x, τ_{x}) + ∫_{τ_{x}}^{τ} u_{t}(x, t)dt,

which implies

∫_{D_{0τ}} u(x, τ)u(x, τ)dx ≤

≤ 2 ∫_{D_{0τ}} u(x, τ_{x})u(x, τ_{x})dx + 2|τ − τ_{x}| ∫_{D_{0τ}} dx ∫_{τ_{x}}^{τ} u_{t}(x, t)u_{t}(x, t)dt =

= 2 ∫_{S_{τ}} |α_{3}^{−1}|uuds + 2|τ − τ_{x}| ∫_{D_{τ}} u_{t}u_{t}dxdt ≤
≤ c_{2}( ∫_{S_{τ}} uuds + ∫_{D_{τ}} u_{t}u_{t}dxdt), c_{2} = const > 0.  (11)

By introducing the notation

w_{0}(τ) = ∫_{D_{0τ}} (uu + u_{t}u_{t} + ∑_{i,j=1}^{n} A_{ij}u_{x_{i}}u_{x_{j}})dx

and combining inequalities (10) and (11) we obtain

w_{0}(τ) ≤ c_{3}[ ∫_{S_{τ}} (uu + ∑_{i,j=1}^{n} A_{ij}ũ_{i}ũ_{j})ds + ∫_{0}^{τ} w_{0}(t)dt + ∫_{D_{τ}} F F dxdt],

which by the Gronwall lemma leads us to

w_{0}(τ) ≤ c_{4}[ ∫_{S_{τ}} (uu + ∑_{i,j=1}^{n} A_{ij}ũ_{i}ũ_{j})ds + ∫_{D_{τ}} F F dxdt],  (12)

where c_{i} = const > 0, i = 3, 4.

One can readily see that the operator α_{n+1} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t} is the internal differential operator on the surface S_{τ}. Therefore the inequality

∫_{S_{τ}} (uu + ∑_{i,j=1}^{n} A_{ij}ũ_{i}ũ_{j})ds ≤ c_{5} ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2}, c_{5} = const > 0, (13)

is fulfilled by virtue of (6).

Using condition (2) we have

c_{6}(uu + u_{t}u_{t} + ∑_{i=1}^{n} u_{x_{i}}u_{x_{i}}) ≤ uu + u_{t}u_{t} + ∑_{i,j=1}^{n} A_{ij}u_{x_{i}}u_{x_{j}},  (14)
c_{6} = const > 0.

By virtue of (13) and (14) inequality (12) implies

∫_{D_{0τ}} (uu + u_{t}u_{t} + ∑_{i=1}^{n} u_{x_{i}}u_{x_{i}})dx ≤

≤ c_{7}( ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2} + ∥F ∥_{L_{2}(D_{τ})}^{2}), c_{7} = const > 0.  (15)

The integration of both sides of (15) with respect to τ gives estimate (7).

For simplicity, when considering below the dependence domain and the existence theorem for a solution of problem (1), (6), we shall limit our consideration to the case where system (1) does not contain the lowest terms, namely,

L_{0}u ≡ u_{tt} − ∑_{i,j=1}^{n} A_{ij}u_{x_{i}x_{j}} = F.  (16)

Let K : −∞ < t ≤ |x|g(\frac{x}{|x|}) be the conical domain lying in the halfspace t < 0 and bounded by the surface ∂K : t = |x|g(\frac{x}{|x|}) with vertex at the origin O(0, …, 0), where g is a completely defined negative smooth function given on the unit sphere in R^{n}. If P_{0}(x_{1}^{0}, …, x_{n}^{0}, t^{0}) ∈ R^{n+1}, then we denote by K_{P_{0}} the conical domain K drawn from the point P_{0} towards the decreasing values of time, i.e., K_{P_{0}} : −∞ < t ≤ t^{0} + |x − x^{0}|g(\frac{x−x^{0}}{|x−x^{0}|}).

It will be assumed below that the characteristic matrix of system (1) is nonnegative on the cone ∂K, i.e.,

(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})ηη ≥ 0, η ∈ R^{m},  (17)

where (α_{1}, …, α_{n}, α_{n+1}) is the external unit normal vector to the cone ∂K at points different from the vertex.

We shall clarify the geometrical meaning of condition (17).

By condition (2) the symmetrical matrix ∑_{i,j=1}^{n} A_{ij}ξ_{i}ξ_{j}, (ξ_{1}, …, ξ_{n}) ∈ R^{n} is positively definite. Therefore there is an orthogonal matrix T = T (ξ_{1}, …, ξ_{n}) such that the matrix T^{−1}(∑_{i,j=1}^{n} A_{ij}ξ_{i}ξ_{j})T is diagonal and its elements σ_{1}, …, σ_{m} on the diagonal are positive, i.e., σ_{i} = λ_{i}^{2}(ξ_{1}, …, ξ_{n}) > 0, λ_{i} > 0, i = 1, …, m. Therefore the real numbers ξ_{n+1} = ±λ_{i}(ξ_{1}, …, ξ_{n}), i = 1, …, m, are the roots of the characteristic polynomial p_{0}(ξ)=det(Eξ_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}ξ_{i}ξ_{j}) of system (1). As one can easily verify, λ_{0}(ξ_{1}, …, ξ_{n}) = max_{1≤i≤m} λ_{i}(ξ_{1}, …, ξ_{n}) is a continuous positive homogeneous function of first order with respect to the variables ξ_{1}, …, ξ_{n}. For system (1), the internal cavity of the cone of normals [10] lying in the half-space ξ_{n+1} ≥ 0 is the convex cone [11]

Γ = {(ξ_{1}, …, ξ_{n+1}) ∈ R^{n+1} : ξ_{n+1} = λ_{0}(ξ_{1}, …, ξ_{n})}.

Since

(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})ηη =

= (Eα_{n+1}^{2} − T^{−1}( ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})T )T^{−1}ηT^{−1}η,

where (Eα_{n+1}^{2} −T^{−1}(∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})T) is a diagonal matrix with elements (α_{n+1}^{2} − λ_{i}^{2}) on the diagonal, condition (17) is equivalent to the condition

|α_{n+1}| ≥ λ_{0}(α_{1}, …, α_{n}).  (18)

By virtue of the assumptions for the cone K we have

α_{n+1}|_{K} > 0  (19)

and therefore (18) can be rewritten as

α_{n+1} ≥ λ_{0}(α_{1}, …, α_{n}).  (20)

Denote by Γ^{∗} = ∩_{η∈Γ}{ξ ∈ R^{n+1} : ξη ≤ 0} a conical domain dual to Γ. As

is known, ∂Γ^{∗} is a convex shell of the cone of rays of system (1) lying in the direction of decreasing values of the time t = ξ_{n+1} [10, 11].

Using inequality (20), one can prove, as in [9], that for a convex smooth cone K condition (17) is equivalent to the condition K ⊃ Γ^{∗}.

It will always be assumed below that k_{1} < 0, k_{2} > 0 and (3) is fulfilled. Hence, as we have seen, it follows that inequalities (8) are valid.

Lemma 2. For a point P_{0}(x^{0}, t^{0}) ∈ D of the solution u(x, t) of problem (16), (6) of the class C^{2}(\overline{D}) or W_{2}^{2}(D) the domain of dependence is contained in the conical domain K_{P_{0}} with vertex at the point P_{0}.

Proof. We set

Ω_{P_{0}} = D ∩ K_{P_{0}}, S_{iP_{0}} = S_{i} ∩ ∂Ω_{P_{0}}, i = 1, 2.

To prove the lemma it is sufficient to show that if

f_{i}|_{S_{iP_{0}}} ≡ u|_{S_{iP_{0}}} = 0, i = 1, 2, F |_{Ω_{P_{0}}} ≡ L_{0}u|_{Ω_{P_{0}}} = 0,  (21)

then u|_{Ω_{P_{0}}} = 0.

First we shall consider the case u ∈ C^{2}(\overline{D}). Let S_{3P_{0}} denote the remaining boundary of the domain Ω_{P_{0}}, i.e., S_{3P_{0}} = ∂Ω_{P_{0}}\(S_{1P_{0}} ∪S_{2P_{0}}). By virtue of (17) and (19) we have

α_{n+1}|_{S_{3P_{0}}} > 0, α_{n+1}^{−1}(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})ηη|_{S_{3P_{0}}} ≥ 0. (22)

After multiplying scalarly both sides of equations (16) by the vector 2u_{t} and integrating the obtained expression with respect to the domain Ω_{P_{0}}, by virtue of (2), (8), (21), (22) and the reasoning used in proving (9) we obtain the inequality

0 = 2 ∫_{Ω_{P_{0}}} F u_{t}dxdt =

= ∫_{∂Ω_{P_{0}}} [(u_{t}u_{t} + ∑_{i,j=1}^{n} A_{ij}u_{x_{i}}u_{x_{j}})α_{n+1} − 2 ∑_{i,j=1}^{n} A_{ij}u_{t}u_{x_{j}}α_{i}]ds =

= ∫_{∂Ω_{P_{0}}} α_{n+1}^{−1}[ ∑_{i,j=1}^{n} A_{ij}(α_{n+1}u_{x_{i}} − α_{i}u_{t})(α_{n+1}u_{x_{j}} − α_{j}u_{t})]ds +

+ ∫_{∂Ω_{P_{0}}} α_{n+1}^{−1}(Eα_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}α_{i}α_{j})u_{t}u_{t}ds ≥

≥ c_{0} ∫_{S_{3P_{0}}} α_{n+1}^{−1} ∑_{i=1}^{n} ∥α_{n+1}u_{x_{i}} − α_{i}u_{t}∥^{2}ds.  (23)

In deriving (23) we used the fact that the operator α_{n+1} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t} is the internal differential operator on the boundary ∂Ω_{P_{0}} and, in particular, the equalities

(α_{n+1} \frac{∂u}{∂x_{i}} − α_{i} \frac{∂u}{∂t})|_{S_{1P_{0}} ∪S_{2P_{0}}} = 0, i = 1, …, n,

hold due to (21).

Since inf_{|α|=1} λ_{0}(α_{1}, …, α_{n}) > 0, by (20) we have inf_{S_{3P_{0}}} α_{n+1} > 0, which by (23) implies

(α_{n+1}u_{x_{i}} − α_{i}u_{t})|_{S_{3P_{0}}} = 0, i = 1, …, n.  (24)

Taking into account that u ∈ C^{2}(\overline{D}) and the internal differential operators α_{n+1} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t}, i = 1, …, n, are linearly independent on the ndimensional connected surface S_{3P_{0}}, we immediately find from (24) that

u|_{S_{3P_{0}}} ≡ const .  (25)

But on account of (21) u|_{S_{3P_{0}} ∩(S_{1P_{0}} ∪S_{2P_{0}} )} = 0, and hence by (25) we obtain

u|_{S_{3P_{0}}} ≡ 0.  (26)

From (26), in particular, it follows that u(P_{0}) = 0.

Taking now an arbitrary point Q ∈ Ω_{P_{0}}, by (21) we conclude that the above equalities hold provided that the point P_{0} is replaced by the point Q. Thus by repeating the above reasoning for the domain Ω_{Q} we obtain u(Q) = 0. Therefore for the case u ∈ C^{2}(\overline{D}) we have u|_{Ω_{P_{0}}} = 0.

By a similar slightly modified reasoning and embedding theorem for the Sobolev spaces one can prove this lemma for the case u ∈ W_{2}^{2}(D) [5].

Let K^{+} : |x|g_{+}(\frac{x}{|x|}) ≤ t < +∞ denote a conical domain lying in the halfspace t > 0 and bounded by the surface ∂K_{+} : t = |x|g_{+}(\frac{x}{|x|}) with vertex at the origin O(0, …, 0), where g_{+} is a completely defined smooth function given on the unit sphere R^{n}. For P_{0}(x^{0}, t^{0}) ∈ R^{n+1} we denote by K_{P_{0}}^{+} a conical domain K^{+} drawn from the point P_{0} in the direction of increasing values of time, i.e.,

K_{P_{0}}^{+} : t^{0} + |x − x^{0}|g_{+}(\frac{x − x^{0}}{|x − x^{0}|}) ≤ t < ∞.

As in the case of the conical domain K, it will be assumed that the characteristic matrix of system (1) is non-negative on ∂K^{+}, i.e., that (17) holds. It is obvious that conical domains K^{+} are obtained from conical domains K by the central symmetry with respect to the origin. If Γ_{+}^{∗} is a conical domain centrally symmetrical to Γ^{∗} with respect to the origin, then, as in the case of the domain K, one can prove that for the convex smooth cone K^{+} condition (17) is equivalent to the condition K^{+} ⊃ Γ_{+}^{∗}.

Lemma 3. Let D_{0} be a bounded subdomain of D with a piecewise smooth boundary bounded from above by a hyperplane t = t_{0}, and from the sides by hyperplanes S_{1}, S_{2} and by piecewise smooth hypersurfaces S_{3}, S_{4} of temporal type on which the inequalities

α_{n+1}|_{S_{3}} < 0, α_{n+1}|_{S_{4}} < 0  (27)

hold, where ν = (α_{1}, …, α_{n+1}) is the external unit vector to ∂D_{0} and S_{3} ∩ S_{4} = ∅. Let u_{0} ∈ C^{∞}(\overline{D}_{0}) and g_{i} = u_{0}|_{∂D_{0}∩S_{i}}, i = 1, 2, F_{0} = L_{0}u_{0}, X = supp g_{1} ∪ supp g_{2} ∪ supp F_{0}, Y = ∪_{P_{0}∈X}K_{P_{0}}^{+}. Denote by S_{3}^{ε}, S_{4}^{ε} the ε-neighborhoods of hypersurfaces S_{3}, S_{4}, where ε is a fixed sufficiently small number. Then if u_{0}|_{S_{3}∪S_{4}} = 0, Y ∩ (S_{3}^{ε} ∪ S_{4}^{ε}) = ∅, the function

u(P) = \begin{cases}u_{0}(P), & P ∈ D_{0}, \\ 0, & P ∈ D\D_{0} \end{cases}

is a solution of problem (16), (6) of the class C_{∗}^{∞}(\overline{D}) for

f_{i}(P) = \begin{cases}g_{i}(P), & P ∈ ∂D_{0} ∩ S_{i}, \\ 0, & P ∈ S_{i}\(∂D_{0} ∩ S_{i}), \end{cases} , i = 1, 2,

F (P) = \begin{cases}F_{0}(P), & P ∈ D_{0}, \\ 0, & P ∈ D\D_{0}. \end{cases}

The proof of Lemma 3 repeats in the main the proof of the corresponding lemma from [5].

Remark 1. Note that Lemma 3 remains valid in the case where conditions (27) are not fulfilled on some set ω ⊂ S_{3} ∪S_{4} of zero n-dimensional measure, i.e., α_{n+1}|_{ω} = 0. In particular, Lemma 3 remains true if ω = ∪_{i=1}^{p}γ_{i} is the union of a finite number of smooth (n − 1)-dimensional submanifolds γ_{i} ⊂ S_{3} ∪ S_{4} and α_{n+1}|_{ω} = 0, α_{n+1}|_{(S_{3}∪S_{4})\ω} < 0. This fact will be used below in proving the existence theorem for problem (16), (6).

Remark 2. Also note that Lemmas 2 and 3 actually suggest a method for constructing a solution of problem (16), (6) which will be described below in proving Theorem 1. The method consists in reducing the initial problem (16), (6) to a mixed-type problem for a second-order hyperbolic system in a cylinder.

Let ξ_{n+1} = ±λ_{i}(ξ_{1}, …, ξ_{n}), λ_{i} > 0, i = 1, …, m, be the roots of the characteristic polynomial det(Eξ_{n+1}^{2} − ∑_{i,j=1}^{n} A_{ij}ξ_{i}ξ_{j}) of system (1) with respect to ξ_{n+1}. The functions λ_{0}(ξ_{1}, …, ξ_{n}) = max_{1≤i≤m} λ_{i}(ξ_{1}, …, ξ_{n}), λ_{0}^{−}(ξ_{1}, …, ξ_{n}) = min_{1≤i≤m} λ_{i}(ξ_{1}, …, ξ_{n}) are obviously continuous positive homogeneous functions of first order with respect to the variables ξ_{1}, …, ξ_{n}. Consider the cones

Γ = {(ξ_{1}, …, ξ_{n+1}) ∈ R^{n+1} : ξ_{n+1} = λ_{0}(ξ_{1}, …, ξ_{n})},
Γ_{−} = {(ξ_{1}, …, ξ_{n+1}) ∈ R^{n+1} : ξ_{n+1} = −λ_{0}^{−}(ξ_{1}, …, ξ_{n})}.

As is known, the interior of the convex cone Γ is a set of all spatial type normals lying in the half-plane ξ_{n+1} ≥ 0, while ∂Γ_{−}^{∗} is the interior of the cone of rays of system (1) lying in the direction of increasing values of the

time t = ξ_{n+1}, where Γ_{−}^{∗} = ∩_{η∈Γ_{−}}{ξ ∈ R^{n+1} : ξη ≤ 0} [10]. Let Γ^{∘} denote

the domain bounded by the cone Γ, i.e., the interior of Γ. Below it will be assumed without loss of generality that

Γ_{−}^{∗} ⊂ Γ^{∘},  (28)

since otherwise this can be achieved by a subsequent change of the independent variables x′_{i} = εx_{i}, i = 1, …, n, t′ = t in system (1) for a sufficiently small ε = const.

For k_{1} ≤ k ≤ k_{2} we denote by l_{k} a ray coming out of the origin with the direction vector (0, …, 0, k, 1), i.e., l_{k} : τ(0, …, 0, k, 1), 0 < τ < +∞. By virtue of the assumptions made for the supports of the data of problem (1),

(6) we have

l_{k} ⊂ Γ^{∘}_{−}^{∗}.  (29)

Denote by H_{k} an arbitrary noncharacteristic hyperplane of temporal type containing the ray l_{k}. Take an arbitrary point P ∈ l_{k} and choose a Cartesian system x_{1}^{0}, …, x_{n}^{0}, t^{0} which is connected with this point and has the vertex at the point P so that the t^{0}-axis could be directed along the ray l_{k}, and the x_{n}^{0}-axis along the normal to H_{k} at this point towards increasing values of time. Denote by H_{k}^{+} that part of the half-space R^{n+1} with the boundary H_{k} which contains the positive x_{n}^{0}-semiaxis. Denote by Q^{0}(ξ) and p̃_{0}(ξ) = det Q^{0}(ξ) respectively the characteristic matrix and the polynomial of system (1) written in terms of the coordinate system x_{1}^{0}, …, x_{n}^{0}, t^{0} connected with an arbitrarily chosen point P ∈ l_{k}.

By (29) there is a convex cone Γ_{1} which is tangential to the hyperplane H_{k} along the ray l_{k}, lies in the half-space H_{k}^{+}, and is contained

in the set Γ^{∘}_{−}^{∗} ∪ O, where O = O(0, …, 0) is the vertex of Γ_{−}. Hence by (28) and the arguments from [12] it follows that when system (1) is strictly hyperbolic, exactly m characteristic hyperplanes of system (1) pass inwards the angle t^{0} > 0, x_{n}^{0} > 0 through the (n − 1)-dimensional plane t^{0} = x_{n}^{0} = 0 connected with an arbitrary point P ∈ l_{k}. Hence in turn it follows that for Re s > 0 the number of roots λ_{j}(ξ_{1}, …, ξ_{n−1}, s) is equal to m, i = \sqrt{−1}, provided that we take into account the multiplicity of the polynomial p̃_{0}(iξ_{1}, …, iξ_{n−1}, λ, s) with Re λ_{j} < 0 [12]. The polynomial p̃_{0}(iξ_{1}, …, iξ_{n−1}, λ, s) can be written in the form of product Δ_{−}(s)Δ_{+}(s), 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−1}) ∈ R^{n−1}, ξ_{1}^{2} +⋯+ξ_{n−1}^{2} +|s|^{2} = 1 [13]. Denote by Q̃^{0}(iξ_{1}, …, iξ_{n−1}, λ, s) an m × m matrix whose element Q̃_{pq}^{0}(iξ_{1}, …, iξ_{n−1}, λ, s) is equal to an algebraic complement to the element Q_{pq}^{0}(iξ_{1}, …, iξ_{n−1}, λ, s) in the characteristic matrix Q^{0}(iξ_{1}, …, iξ_{n−1}, λ, s) of system (1).

Let us consider

Condition 1. System (1) is strictly hyperbolic. For an arbitrary noncharacteristic hyperplane of temporal type H_{k}, passing through the ray l_{k} : τ(0, …, 0, k, 1), 0 < τ < +∞ when k_{1} ≤ k ≤ k_{2}, and for an arbitrary point P ∈ l_{k} and any s, Re s ≥ 0, and (ξ_{1}, …, ξ_{n−1}) ∈ R^{n−1} such that ξ_{1}^{2} + ⋯ + ξ_{n−1}^{2} + |s|^{2} = 1, the rows of the matrix Q̃^{0}(iξ_{1}, …, iξ_{n−1}, λ, s) are linearly independent as polynomials of λ modulo the polynomial Δ_{−}(λ).

It will be assumed below that the functions f_{1} and f_{2} in the boundary conditions (6) vanish on γ_{0} = S_{1} ∩ S_{2}, i.e.,

f_{i}|_{γ_{0}} = 0, i = 1, 2.  (30)

Denote by W^{∘} _{2}^{1}(S_{i}, γ_{0}) functions of the class W_{2}^{1}(S_{i}) which satisfy equality

(30), i.e., W^{∘} _{2}^{1}(S_{i}, γ_{0}) = {f ∈ W_{2}^{1}(S_{i}) : f|_{γ_{0}} = 0}, i = 1, 2.

We have

Theorem 1. Let condition 1 be fulfilled. Then for any f_{i} ∈ W^{∘} _{2}^{1}(S_{i}, γ_{0}), i = 1, 2, F ∈ L_{2}(D) there exists a unique strong solution u of problem (16),

(6) of the class W_{2}^{1}, for which estimate (7) is valid.

Proof. Here only a short scheme of the proof is given. The detailed proof of Theorem 1 for one equation of hyperbolic type is given in [5].

Denote by S_{i}^{0} : k_{i}t − x_{n} = 0, 0 ≤ t < +∞, i = 1, 2, a half-plane containing the support S_{i} of the boundary conditions (6), and by D_{0} the dihedral angle between the half-planes S_{1}^{0} and S_{2}^{0}. As is known, the function f_{i} ∈ W_{2}^{1}(S_{i}, Γ) can be extended in the half-plane S_{i}^{0} as a function f̃_{i} of the

class W^{∘} _{2}^{1}(S_{i}^{0}), i.e., (f_{i} − f̃_{i})|_{S_{i}} = 0, f̃_{i} ∈ W^{∘} _{2}^{1}(S_{i}^{0}), i = 1, 2. We set

F̃(P) = \begin{cases}F (P), & P ∈ D, \\ 0, & P ∈ D_{0}\D. \end{cases}

It is obvious that F̃ ∈ L_{2}(D_{0}). If C_{0}^{∞}(D_{0}), C_{0}^{∞}(S_{i}^{0}), i = 1, 2, are the spaces of finite infinitely differentiable functions, then, as we know, these spaces will

be dense everywhere in the spaces L_{2}(D_{0}), W^{∘} _{2}^{1}(S_{i}^{0}), i = 1, 2, respectively. Therefore there are sequences F_{n} ∈ C_{0}^{∞}(D_{0}) and f_{in} ∈ C_{0}^{∞}(S_{i}^{0}), i = 1, 2, such that

lim_{n→∞} ∥F̃ − F_{n}∥_{L_{2}(D)} = lim_{n→∞} ∥f̃_{i} − f̃_{in}∥_{W_{2}^{1}(S_{i}^{0})} = 0, i = 1, 2. (31)

In the domain of the variables x_{n}, t we introduce the polar coordinates r, φ, taking the t-axis as a polar axis, and counting the polar angle φ from the polar axis and assuming it positive in the clockwise direction. Let φ_{i} be the dihedral angle formed by the half-planes S_{i}^{0} and x_{n} = 0, 0 ≤ t < +∞, i = 1, 2.

In passing from the Cartesian system x_{1}, x_{2}, …, x_{n}, t to the system x_{1}, ..., x_{n−1}, τ = log r, φ the dihedral angle D_{0} becomes an infinite layer

H = { − ∞ < x_{i} < ∞, i = 1, …, n − 1, −∞ < τ < ∞, −φ_{1} < φ < φ_{2}},

and equation (16) in the previous terms for the functions u and F takes the form

e^{−2τ} L_{1}(τ, φ, ∂)u = F,  (32)

where ∂ = (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n−1}} , \frac{∂}{∂τ} , \frac{∂}{∂φ}), L_{1}(τ, φ, ∂) is a second-order matrix differential operator of hyperbolic type with respect to τ with infinitely differentiable coefficients depending on τ and φ.

Denote by H_{β_{k}} ⊂ H, β_{k} = const > 0, a cylindrical domain Ω_{k} ×(−∞, ∞) of the class C^{∞}, where (−∞, ∞) is the τ-axis, and by ∂H_{β_{k}} its lateral surface ∂Ω_{k} × (−∞, ∞). It is assumed that the cylinder H_{β_{k}}^{∗} = {|x_{i}| < β_{k}, i = 1, …, n − 1, −∞ < τ < ∞, −φ_{1} < φ < φ_{2}} is contained in H_{β_{k}}. Under the inverse transformation (x_{1}, …, x_{n−1}, τ, φ) → (x_{1}, …, x_{n}, t) the cylindrical domain H_{β_{k}} will transform to the infinite domain G_{β_{k}} ⊂ D_{0} bounded by the surfaces S̃_{i} = S_{i}^{0} ∩ ∂G_{β_{k}}, i = 1, 2, and also by the surfaces S̃_{3}, S̃_{4}. It can be shown that the surfaces S̃_{3} and S̃_{4} are of temporal type and the following conditions are fulfilled on them:

α_{n+1}|_{(_{S_{3}}∪_{S_{4}})\ω} < 0, α_{n+1}|_{ω} = 0,  (33)

where ω ⊂ S̃_{3} ∪ S̃_{4} is some set of zero n-dimensional measure and ν = (α_{1}, …, α_{n}, α_{n+1}) is the external unit normal vector to ∂G_{β_{k}}.

Choose a number β_{k} so large that

supp f_{ik} ⊂ S̃_{i}, i = 1, 2, supp F_{k} ⊂ G_{β_{k}}.

Hence the function g_{k}(x, t) defined on the boundary ∂G_{β_{k}} of the domain G_{β_{k}} as

g_{k}|_{S_{i}} = f_{ik}, i = 1, 2, g_{k}|_{S_{3}} = g_{k}|_{S_{4}} = 0  (34)

will belong to the class C_{0}^{∞}(∂G_{β_{k}}).

In passing to the variables x_{1}, …, x_{n−1}, τ, φ, the functions g_{k} and F_{k} will transform to some functions which will be denoted as previously. It is obvious that

g_{k} ∈ C_{0}^{∞}(∂H_{β_{k}}), F_{k} ∈ C_{0}^{∞}(H_{β_{k}}).  (35)

For the hyperbolic equation (32) with F = F_{k} we consider, in the cylinder H_{β_{k}}, the following mixed-type problem with the ”Cauchy zero conditions” for τ = −∞:

e^{−2τ} L_{1}(τ, φ, ∂)v = F_{k},  (36)
v|_{∂H_{β_{k}}} = g_{k}.  (37)

Using condition 1 and the results from [13], [14], the mixed-type problem (36), (37) has, by virtue of (35), a unique solution v = v_{k} of the class C^{∞}(\overline{H}_{β_{k}}) which identically vanishes for τ < −M, where M = const is a sufficiently large positive number.

Returning to the initial variables x_{1}, …, x_{n}, t and using the previous notations for the functions v_{k} and F_{k}, we obtain the following facts:

(1) the function u_{k}^{0} = v_{k}|_{G_{β_{k}}^{∗}}, where G_{β_{k}}^{∗} = G_{β_{k}} ∩ D, belongs to the class

C^{∞}(\overline{G}_{β_{k}}^{∗}) and satisfies the equation L_{0}u_{k}^{0} = F_{k};

(2) on the lateral part ∪_{i=1}^{4}S̃_{i}^{0} of the boundary G_{β_{k}}^{∗} the function u_{k}^{0} satisfies the conditions u_{k}^{0}|_{_{S_{3}^{0}}∪_{S_{4}^{0}}} = 0, u_{k}^{0}|_{S_{i}^{0}} = f_{ik}, i = 1, 2, where, as one

can easily verify, the surface S̃_{i}^{0} is a part of the surface S_{i} for i = 1, 2, and a part of the surface S̃_{i} for i = 3, 4 figuring in conditions (33).

Therefore, choosing a value of β_{k} sufficiently large and applying (33), (34), Lemma 3, and Remark 1 we conclude that the function

u_{k}(P) = \begin{cases}u_{k}^{0}(P), & P ∈ G_{β_{k}}^{∗}, \\ 0, & P ∈ D\G_{β_{k}}^{∗}, \end{cases}

belongs to the class C_{∗}^{∞}(\overline{D}) and is a solution of problem (16), (6) for f_{i} = f_{ik}, i = 1, 2, and F = F_{k}.

By virtue of inequality (7) we obtain

∥u_{k} − u_{p}∥_{W_{2}^{1}(D)} ≤
≤ c( ∑_{i=1}^{2} ∥f_{ik} − f_{ip}∥_{W_{2}^{1}(S_{i})} + ∥F_{k} − F_{p}∥_{L_{2}(D)}).  (38)

It follows from (31) and (38) that the sequence of functions u_{k} is fundamental in the space W_{2}^{1}(D). The completeness of the space W_{2}^{1}(D) implies that there exists a function u ∈ W_{2}^{1}(D) such that u_{k} → u in the space W_{2}^{1}(D), L_{0}u_{k} → F in the space L_{2}(D), and u_{k}|_{S_{i}} → f_{i} in W_{2}^{1}(S_{i}), i = 1, 2, for k → ∞. Therefore the function u is a strong solution of problem (16),

(6) of the class W_{2}^{1}. The uniqueness of this strong solution follows from inequality (7).

Let us now turn to problem (1), (6). In the space W_{2}^{1}(D) we introduce a norm depending on the parameter λ

∥u∥_{D,1,λ}^{2} = ∫_{D} e^{−λt}(uu + u_{t}u_{t} + ∑_{i=1}^{n} u_{x_{i}}u_{x_{i}})dxdt, λ > 0.

In a similar manner we introduce norms ∥F ∥_{D,0,λ} and ∥f_{i}∥_{S_{i},1,λ} in the spaces L_{2}(D) and W_{2}^{1}(S_{i}), i = 1, 2.

Arguments similar to those in [4] enable us to prove

Lemma 4. For any u ∈ W_{2}^{2}(D) the a priori estimate

∥u∥_{D,1,λ} ≤ \frac{c_{1}}{\sqrt{λ}}( ∑_{i=1}^{2} ∥f_{i}∥_{S_{i},1,λ} + ∥F ∥_{D,0,λ})  (39)

holds, where f_{i} = u|_{S_{i}}, F = L_{0}u, and the positive constant c_{1} does not depend on u and the parameter λ.

By virtue of estimate (39), for a sufficiently large value of λ the lowest terms in equation (1) give arbitrarily small perturbations in the sense of the above-introduced equivalent norms of the spaces L_{2}(D), W_{2}^{1}(D), W_{2}^{1}(S_{i}), i = 1, 2, which due to Theorem 1 and the resuls of [4] enables us to prove the unique solvability of problem (1), (6) in the class W_{2}^{1}.

Theorem 2. Let condition 1 be fulfilled. Then for any f_{i} ∈ W^{∘} _{2}^{1}(S_{i}, Γ), i = 1, 2, F ∈ L_{2}(D) there exists a unique strong solution u of problem (1),

(6) of the class W_{2}^{1}, for which estimate (7) is true.

Remark 3. In Theorems 1 and 2 it is required that system (1) or (16) be strictly hyperbolic, and also that the other part of condition 1 be fulfilled. These requirements can be left out for one class of hyperbolic systems of form (1). These are systems of form (1) for which inequality (1) holds and the symmetrical constant matrices A_{ij} are pairwise permutable, i.e., A_{ij}A_{ks} = A_{ks}A_{ij}. Then there exists an orthogonal matrix T_{0} with constant elements such that T_{0}^{−1}A_{ij}T_{0} is diagonal for any i, j = 1, …, n. Therefore for the new unknown function v = T_{0}^{−1}u we shall have, instead of system (1), a second-order hyperbolic system with the split principal part. But a problem of the Darboux type with boundary conditions of form (6) on temporal hyperplanes is uniquely solvable for one hyperbolic equation of second-order with constant coefficients at higher derivatives and estimate (39) holds for its solution [5]. Therefore the same arguments that enabled us to prove Theorem 2 on the basis of Lemma 4 and Theorem 1 make it possible to prove

Theorem 3. Let the matrices A_{ij} in system (1) be pairwise permutable.

Then for any f_{i} ∈ W^{∘} _{2}^{1}(S_{i}, Γ), i = 1, 2, F ∈ L_{2}(D) there exists a unique strong solution u of problem (1), (6) of the class W_{2}^{1}, for which estimate (7) holds.

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(Received 06.07.1994)

Author’s adress: Georgian Technical University Department of Theoretical Mechanics No. 4 77, Kostava St., Tbilisi 380075 Republic of Georgia