GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 53-68

ON THE SOLVABILITY OF A DARBOUX TYPE NON-CHARACTERISTIC SPATIAL PROBLEM FOR THE

WAVE EQUATION

S. KHARIBEGASHVILI

Abstract. The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given.

In the space of variables x_{1}, x_{2}, t let us consider the wave equation

□u ≡ \frac{∂^{2}u}{∂t^{2}} − \frac{∂^{2}u}{∂x_{1}^{2}} − \frac{∂^{2}u}{∂x_{2}^{2}} = F,  (1)

where F and u are known and desired real functions, respectively.

We denote by D : k_{1}t < x_{2} < k_{2}t, 0 < t < t_{0}, −1 < k_{i} = const < 1, i = 1, 2, k_{1} < k_{2}, a domain lying in the half-space t > 0 and bounded by the plane surfaces S_{i} : k_{i}t − x_{2} = 0, 0 ≤ t ≤ t_{0}, i = 1, 2, of the temporal type and by the plane t = t_{0}.

We shall consider a Darboux type problem formulated as follows: In the domain D find a solution u(x_{1}, x_{2}, t) of equation (1) by the boundary conditions

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

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

It should be noted that in [1–5] Darboux type problems are studied for the cases where at least one of the surfaces S_{1} and S_{2} is the characteristic surface of equation (1) passing through the Ox_{1}-axis. Other multi-dimensional analogues of the Darboux problem are treated in [6–8].

53

1991 Mathematics Subject Classification. 35L20.

Key words and phrases. Non-characteristic spatial problem of Darboux type, wave equation, a priori estimate.

1072-947X/96/0100-0053$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

As distinct from the cases considered in [1–5], the fact that none of the surfaces S_{1} and S_{2} is characteristic results in the nonavailability of an integral representation for regular solutions of problem (1), (2). The latter circumstance somehow complicates the investigation of this problem. Below we shall prove the existence and uniqueness theorems for regular as well as for strong solutions of problem (1), (2) belonging to the class W_{2}^{1}.

Let C_{∗}^{∞}(\overline{D}) denote a 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 similarly.

The well-known Sobolev spaces will be denoted by W_{2}^{1}(D), W_{2}^{2}(D), W_{2}^{1}(S_{i}), i = 1, 2. Note that the space C_{∗}^{∞}(\overline{D}) is an everywhere dense subspace of the spaces W_{2}^{1}(D) and W_{2}^{2}, while C_{∗}^{∞}(S_{i}) is an everywhere dense 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 function u ∈ W_{2}^{1}(D) is called a strong solution of problem (1), (2) belonging to 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), □u_{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, ∥□u_{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 −1 < k_{1} < 0, 0 < k_{2} < 1, then for any u ∈ W_{2}^{1}(D) there holds the a priori estimate

∥u∥_{W_{2}^{1}(D)} ≤ C( ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{i})} + ∥F ∥_{L_{2}(D)}),  (3)

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

Proof. Since the space C_{∗}^{∞}(D) (C_{∗}^{∞}(S_{i})) is an everywhere dense subspace of the spaces W_{2}^{1}(D) and W_{2}^{2}(D) (W_{2}^{1}(S_{i})), by virtue of the well-known theorems on 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 for us to prove that the a priori estimate (3) 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_{1} = cos({ \atop n, x_{1}}), α_{2} = cos({ \atop n, x_{2}}), α_{3} = cos(n̂, t).

Here n = (α_{1}, α_{2}, α_{3}) denotes the external normal unit vector to ∂D_{τ}. One can easily verify that

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

n|_{D_{0τ}} = (0, 0, 1).

Therefore for −1 < k_{1} < 0, 0 < k_{2} < 1 we have

α_{3}|_{S_{iτ}} < 0, i = 1, 2, α_{3}^{−1}(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{i}} > 0, i = 1, 2.  (4)

On multiplying both parts of equation (1) by 2u_{t}, where u ∈ C_{∗}^{∞}(\overline{D}), F = □u, and integrating the obtained expression with respect to D_{τ}, we obtain due to (4)

2 ∫_{D_{τ}} F u_{t} dx dt = ∫_{D_{τ}} (\frac{∂u_{t}^{2}}{∂t} + 2u_{x_{1}}u_{tx_{1}} + 2u_{x_{2}}u_{tx_{2}})dx dt −

−2 ∫_{S_{τ}} (u_{x_{1}}u_{t}α_{1} + u_{x_{2}}u_{t}α_{2})ds = ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx +

+ ∫_{S_{τ}} [(u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})α_{3} − 2(u_{x_{1}}u_{t}α_{1} + u_{x_{2}}u_{t}α_{2})]ds =

= ∫_{D_{0τ}}(u_{t}^{2}+u_{x_{1}}^{2} +u_{x_{2}}^{2})dx+∫_{S_{τ}} α_{3}^{−1}[(α_{3}u_{x_{1}} −α_{1}u_{t})^{2}+(α_{3}u_{x_{2}} −α_{2}u_{t})^{2}+

+(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})u_{t}^{2}]ds ≥ ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx +

+ ∫_{S_{τ}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds.  (5)

Assuming

w(τ) = ∫_{D_{0τ}} (u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx, ũ_{i} = α_{3}u_{x_{i}} − α_{i}u_{t}, i = 1, 2,

C_{1} = max (\frac{\sqrt{1 + k_{1}^{2}}}{|k_{1}|} , \frac{\sqrt{1 + k_{2}^{2}}}{|k_{2}|} ) ,

we find by (5) that

w(τ) ≤ C_{1} ∫_{S_{τ}} (u_{1}^{2} + u_{2}^{2})ds + ∫_{D_{τ}} (F^{2} + u_{t}^{2})dx dt ≤

≤ C_{1} ∫_{S_{τ}} (u_{1}^{2} + u_{2}^{2})ds + ∫_{0}^{τ} dξ ∫_{D_{0ξ}} u_{t}^{2}dx + ∫_{D_{τ}} F^{2}dx dt ≤

≤ C_{1} ∫_{S_{τ}} (u_{1}^{2} + u_{2}^{2})ds + ∫_{0}^{τ} w(ξ)dξ + ∫_{D_{τ}} F^{2}dx dt.  (6)

Let (x, τ_{x}) be the point at which the surface S_{1} ∪ S_{2} intersects with the straight line parallel to the t-axis and 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^{2}(x, τ)dx ≤ 2 ∫_{D_{0τ}} u^{2}(x, τ_{x})dx + 2|τ − τ_{x}| ∫_{D_{0τ}} dx ∫_{τ_{x}}^{τ} u_{t}^{2}(x, t)dt =

= 2 ∫_{S_{τ}} α_{3}^{−1}u^{2}ds + 2|τ − τ_{x}| ∫_{D_{τ}} u_{t}^{2}dx dt ≤

≤ C_{2}( ∫_{S_{τ}} u^{2}ds + ∫_{D_{τ}} u_{t}^{2}dx dt),  (7)

where C_{2} = 2 max(C_{1}, t_{0}).

On introducing the notation

w_{0}(τ) = ∫_{D_{0τ}} (u^{2} + u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx

and combining inequalities (6) and (7), we obtain

w_{0}(τ) ≤ C_{2}[ ∫_{S_{τ}} (u^{2} + u_{1}^{2} + u_{2}^{2})ds + ∫_{0}^{τ} w_{0}(ξ)dξ + ∫_{D_{τ}} F^{2}dx dt],

which by virtue of Gronwall’s lemma implies

w_{0}(τ) ≤ C_{3}[ ∫_{S_{τ}} (u^{2} + u_{1}^{2} + u_{2}^{2})ds + ∫_{D_{τ}} F^{2}dx dt],  (8)

where C_{3} = const > 0.

One can easily verify that the operator α_{3} \frac{∂}{∂x_{i}} −α_{i} \frac{∂}{∂t} is the internal differential operator on the surface S_{τ}. Therefore the following inequality holds by virtue of (2):

∫_{S_{τ}} (u^{2} + u_{1}^{2} + u_{2}^{2})ds ≤ C_{4} ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2}, C_{4} = const > 0.  (9)

From (8) and (9) we obtain

w_{0}(τ) ≤ C_{5}( ∑_{i=1}^{2} ∥f_{i}∥_{W_{2}^{1}(S_{iτ})}^{2} + ∥F ∥_{L_{2}(D_{τ})}^{2}), C_{5} = const > 0. (10)

The integration of both parts of inequality (10) with respect to τ gives us estimate (3).

In the sequel it will be assumed that −1 < k_{1} < 0, 0 < k_{2} < 1, i.e., that inequalities (4) are fulfilled.

Lemma 2. The dependence domain for the point P_{0}(x_{1}^{0}, x_{2}^{0}, t^{0}) ∈ D of the solution u(x_{1}, x_{2}, t) of problem (1), (2) belonging to the class C^{2}(\overline{D}) or W_{2}^{2}(D) is contained within the characteristic cone of the past ∂K_{P_{0}} : t = t^{0} − \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}} with the 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,

where K_{P_{0}} : t < t^{0} − \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}} is the interiority of the characteristic cone ∂K_{P_{0}}.

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}}} ≡ □u|_{Ω_{P_{0}}} = 0,  (11)

then u|_{Ω_{P_{0}}} = 0.

Let us first consider the case with u ∈ C^{2}(\overline{D}). We denote by S_{3P_{0}} the remainder part of the boundary of the domain Ω_{P_{0}}, i.e., S_{3P_{0}} = ∂Ω_{P_{0}}\(S_{1P_{0}} ∪ S_{2P_{0}}). Since by our construction the surface S_{3P_{0}} is part of the characteristic cone ∂K_{P_{0}} of equation (1), we have

α_{3}|_{S_{3P_{0}}} = const > 0, (α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{3P_{0}}} = 0,  (12)

where n = (α_{1}, α_{2}, α_{3}) is the external normal unit vector to ∂Ω_{P_{0}}.

On multiplying both parts of equation (1) by 2u_{t} and integrating the obtained expression with respect to the domain Ω_{P_{0}}, we obtain due to (4), (11), (12) and the arguments used in deriving (5) the following inequality:

0=2 ∫_{Ω_{P_{0}}} F u_{t}dx dt= ∫_{∂Ω_{P_{0}}} [(u_{t}^{2}+u_{x_{1}}^{2} +u_{x_{2}}^{2})α_{3}−2(u_{x_{1}}u_{t}α_{1}+u_{x_{2}}u_{t}α_{2})]ds=

= ∫_{∂Ω_{P_{0}}} α_{3}^{−1}[(α_{3}u_{x_{1}} −α_{1}u_{t})^{2}+(α_{3}u_{x_{2}} −α_{2}u_{t})^{2}+(α_{3}^{2}−α_{1}^{2}−α_{2}^{2})u_{t}^{2}]ds≥

≥ ∫_{S_{3P_{0}}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds.  (13)

To obtain (13) we used the fact that the operator α_{3} \frac{∂}{∂x_{i}} −α_{i} \frac{∂}{∂t} is the internal differential operator on the surface ∂Ω_{P_{0}} and, in particular, on S_{1P_{0}} ∪ S_{2P_{0}}. By virtue of (11) we have the equalities

(α_{3} \frac{∂u}{∂x_{i}} − α_{i} \frac{∂u}{∂t})|_{S_{1P_{0}} ∪S_{2P_{0}}} = 0, i = 1, 2.

Since α_{3} > 0 on S_{3P_{0}}, (13) implies

(α_{3}u_{x_{i}} − α_{i}u_{t})|_{S_{3P_{0}}} = 0, i = 1, 2.  (14)

Taking into account that u ∈ C^{2}(\overline{D}) and the internal differential operators α_{3} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t}, i = 1, 2, are linearly independent on the two-dimensional connected surface S_{3P_{0}}, we immediately find by (14) that

u|_{S_{3P_{0}}} ≡ const.  (15)

But by (11)

u|_{S_{3P_{0}} ∩(S_{1P_{0}} ∪S_{2P_{0}}} ) = 0,

which on account of (15) gives us

u|_{S_{3P_{0}}} ≡ 0.  (16)

From (16) it follows in particular that u(P_{0}) = 0.

If now we take an arbitrary point Q ∈ Ω_{P_{0}}, then (11) implies that the above equalities will hold if the point P_{0} is replaced by the point Q. Therefore, on repeating our previous reasoning for the domain Ω_{Q}, we find that u(Q) = 0. Thus for the case with u ∈ C^{2}(\overline{D}) we obtain u|_{Ω_{P_{0}}} = 0.

Let now u ∈ W_{2}^{2}(D) and equalities (11) be fulfilled. One can easily verify that inequality (13), where the point P_{0} is replaced by the point Q, also holds for any point Q ∈ Ω_{P_{0}}, i.e.,

∫_{S_{3Q}} α_{3}^{−1}[(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds ≤ 0.

Hence by virtue of the fact that α_{3}|_{S_{3Q}} = const > 0 we obtain

∫_{S_{3Q}} [(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds = 0.  (17)

Let Γ_{Q} denote a piecewise-smooth curve which is the boundary of the twodimensional connected surface S_{3Q}. It is obvious that

Γ_{Q} = S_{3Q} ∩ (S_{1Q} ∪ S_{2Q}).  (18)

Recalling the fact that on the surface S_{3Q} the internal differential operators α_{3} \frac{∂}{∂x_{i}} − α_{i} \frac{∂}{∂t}, i = 1, 2, are independent for any v ∈ W_{2}^{1}(S_{3Q}) it is not difficult to obtain the estimate

∫_{S_{3Q}} v^{2}ds ≤ C( ∫_{Γ_{Q}} v^{2}ds +

+ ∫_{S_{3Q}} [(α_{3}v_{x_{1}} − α_{1}v_{t})^{2} + (α_{3}v_{x_{2}} − α_{2}v_{t})^{2}]ds),  (19)

where C = const > 0 does not depend on v and the trace v|_{Γ_{Q}} ∈ L_{2}(Γ_{Q}) is correctly defined by virtue of the respective embedding theorem.

Since u ∈ W_{2}^{2}(D), the traces u|_{S_{3Q}} ∈ W_{2}^{1}(S_{3Q}), u|_{Γ_{Q}} ∈ L_{2}(Γ_{Q}) are correctly defined by virtue of the embedding theorems. Therefore by (11) and (18) we have

u|_{Γ_{Q}} = 0.  (20)

From (17), (19) and (20) we obtain

∫_{S_{3Q}} u^{2}ds ≤ C( ∫_{Γ_{Q}} u^{2}ds +

+ ∫_{S_{3Q}} [(α_{3}u_{x_{1}} − α_{1}u_{t})^{2} + (α_{3}u_{x_{2}} − α_{2}u_{t})^{2}]ds) = 0

which immediately implies

∫_{S_{3Q}} u^{2}ds = 0, u|_{S_{3Q}} = 0, ∀Q ∈ Ω_{P_{0}}.  (21)

Since u ∈ W_{2}^{2}(D), we conclude due to (21) and Fubini’s theorem that

u|_{Ω_{P_{0}}} = 0.

Remark 1. Lemma 2 implies that the wave process described by problem (1), (2) propagates at finite velocity. Therefore if u ∈ C^{∞}(\overline{D}) is a solution of problem (1), (2) for f_{i} ∈ C_{∗}^{∞}(S_{i}), i = 1, 2, F ∈ C_{∗}^{∞}(\overline{D}), then we have u ∈ C_{∗}^{∞}(\overline{D}).

For our further discussion we shall need

Lemma 3. Let G be a bounded subdomain a of D having a piecewisesmooth boundary and bounded from above by the plane t = t_{0} and from the sides by the planes S_{1}, S_{2} and the piecewise-smooth surfaces S_{3}, S_{4} of the temporal type on which the following inequalities are fulfilled:

α_{3}|_{S_{3}} < 0, α_{3}|_{S_{4}} < 0,  (22)

where n = (α_{1}, α_{2}, α_{3}) is the unit normal vector to ∂G and S_{3} ∩ S_{4} = ∅. Let K_{P_{0}}^{+} : t > t^{0} + \sqrt{(x_{1} − x_{1}^{0})^{2} + (x_{2} − x_{2}^{0})^{2}} be the domain bounded by the characteristic cone of the future with the vertex at the point P_{0}(x_{1}^{0}, x_{2}^{0}, t^{0}). Let u_{0} ∈ C^{∞}(\overline{G}) and g_{i} = u_{0}|_{∂G∩S_{i}}, i = 1, 2, F_{0} = □u_{0}, X = supp g_{1} ∪

supp g_{2} ∪ supp F_{0}, Y = ∪_{P_{0}∈X}K_{P_{0}}^{+}.

We denote by S_{3}^{ε}, S_{4}^{ε} ε-neighborhoods of the surfaces S_{3}, S_{4}, where ε is a fixed sufficiently small positive number. Then if

u_{0}|_{S_{3}∪S_{4}} = 0,  (23)
Y ∩ (S_{3}^{3} ∪ S_{4}^{3}) = ∅,  (24)

then the function

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

is a solution of problem (1), (2) of the class C_{∗}^{∞}(\overline{D}) for

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

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

Proof. To prove this lemma it is sufficient to show that the function u_{0} ∈ C^{∞}(\overline{G}) vanishes on the set G ∩ (S_{3}^{ε} ∪ S_{4}^{ε}).

Let P_{0} ∈ G ∩ (S_{3}^{ε} ∪ S_{4}^{ε}) be an arbitrary point of this set. We shall show that u_{0}(P_{0}) = 0.

Let us use the notation of Lemma 2:

Ω_{P_{0}} = G ∩ K_{P_{0}}, S_{iP_{0}} = S_{i} ∩ ∂Ω_{P_{0}}, i = 1, 2, 3, 4, S_{5P_{0}} = ∂K_{P_{0}} ∩ ∂Ω_{P_{0}}.

It is obvious that ∂Ω_{P_{0}} = ∪^{5}_{i=1}S_{iP_{0}}.

By the assumptions of Lemma 3 we have

α_{3}|_{S_{iP_{0}}} <0, i=1, 2, 3, 4; α_{3}^{−1}(α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{iP_{0}}} >0, i=1, 2, 3, 4; (25)
α_{3}|_{S_{5P_{0}}} > 0, (α_{3}^{2} − α_{1}^{2} − α_{2}^{2})|_{S_{5P_{0}}} = 0,  (26)

where n = (α_{1}, α_{2}, α_{3}) is the unit normal vector to ∂Ω_{P_{0}}.

Due to (23), (24) and P_{0} ∈ G ∩ (S_{3}^{ε} ∪ S_{4}^{ε}) we have

u_{0}|_{S_{iP_{0}}} = 0, i = 1, 2, 2, 4; □u_{0}|_{Ω_{P_{0}}} = F_{0}|_{Ω_{P_{0}}} = 0.  (27)

On multiplying both sides of the equation □u_{0} = F_{0} by 2u_{0t} and on integrating the obtained expression by Ω_{P_{0}}, we find by virtue of (25)–(27) and the arguments used in deriving inequalities (5) and (13) that

0 = 2 ∫_{Ω_{P_{0}}} F_{0} \frac{∂u_{0}}{∂t} dx dt = ∫_{∂Ω_{P_{0}}} α3^{1}[(α_{3} \frac{∂u_{0}}{∂x_{1}} − α_{1} \frac{∂u_{0}}{∂t} )^{2} +

+(α_{3} \frac{∂u_{0}}{∂x_{2}} − α_{2} \frac{∂u_{0}}{∂t} )^{2} + (α_{3}^{2} − α_{1}^{2} − α_{2}^{2})(\frac{∂u_{0}}{∂t} )^{2}]ds ≥

≥ ∫_{S_{5P_{0}}} α_{3}^{−1}[(α_{3} \frac{∂u_{0}}{∂x_{1}} − α_{1} \frac{∂u_{0}}{∂t} )^{2} + (α_{3} \frac{∂u_{0}}{∂x_{2}} − α_{2} \frac{∂u_{0}}{∂t} )^{2}]ds.

Since α_{3}|_{S_{5P_{0}}} > 0, the latter formula gives us

(α_{3} \frac{∂u_{0}}{∂x_{i}} − α_{i} \frac{∂u_{0}}{∂t} )|_{S_{5P_{0}}} = 0, i = 1, 2.

The rest of the reasoning repeats the proof of Lemma 2. Therefore u(P_{0}) = 0.

Remark 2. One can easily verify that Lemma 3 remains valid if conditions (22) are not fulfilled on some set ω ⊂ S_{3} ∪ S_{4} of the zero two-dimensional measure, i.e., α_{3}|_{ω} = 0. In particular, if ω = ∪^{m}_{i=1}γ_{i} is the union of the finite

number of smooth curves γ_{i} ⊂ S_{3} ∪ S_{4} and α_{3}|_{ω} = 0, α_{3}|_{(S_{3}∪S_{4})\ω} < 0, then Lemma 3 remains valid. We shall use this fact in the sequel when proving Theorem 1.

Remark 3. Also note that Lemmas 2 and 3 actually provide us with the technique for constructing the solution of problem (1), (2) to be described below when proving Theorem 1. This technique consists in reducing the initial problem (1), (2) to the mixed problem for a second-order hyperbolic equation in the finite cylinder.

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

f_{i}|_{Γ} = 0, i = 1, 2.  (28)

Functions of the class W_{2}^{1}(S_{i}) which satisfy equality (28) will be denoted

by W^{∘} _{2}^{1}(S_{i}, Γ), i.e.,

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

We have

Theorem 1. 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), (2) of the class W_{2}^{1} for which estimate (3) holds.

Proof. We denote by S_{i}^{0} : k_{i}t − x_{2} = 0, 0 ≤ t < +∞, i = 1, 2, the halfplane containing the support S_{i} from the boundary conditions (2), and by D_{0} the dihedral angle between the half-planes S_{1}^{0} and S_{2}^{0}. It is known that

the function f_{i} ∈ W^{∘} _{2}^{1}(S_{i}, Γ) can be extended into 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 assume that

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

Obviously, 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 are everywhere dense 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_{0})} = lim_{n→∞} ∥f̃_{i} − f_{in}∥_{W_{2}^{1}(S_{i}^{0})} = 0, i = 1, 2. (29)

In the plane of the variables x_{2}, t we introduce the polar coordinates r, φ. The t-axis is assumed to be the polar axis, while the polar angle φ is counted from the t-axis and assumed to be positive in the clockwise direction. We denote by φ_{i} the value of the dihedral angle between the half-planes S_{i}^{0} and x_{2} = 0, 0 ≤ t < +∞, i = 1, 2. Since the half-planes S_{i}^{0} are of the temporal type (−1 < k_{1} < 0, 0 < k_{2} < 1), we have 0 < φ_{i} < \frac{π}{4} , i = 1, 2.

In passing from the Cartesian cordinates x_{1}, x_{2}, t to the system of coordinates x_{1}, τ = log r, φ, the dihedral angle D_{0} transforms to an infinite layer

H = { − ∞ < x_{1} < ∞, −∞ < τ < ∞, −φ_{1} < φ < φ_{2}}  (30)

while in terms of the previous notations for the functions u and F equation (1) takes the form

e^{−2τ} L(τ, φ, ∂)u = F,  (31)

where ∂ = (\frac{∂}{∂x_{1}} , \frac{∂}{∂τ} , \frac{∂}{∂φ}), and L(τ, φ, ∂) is a second-order hyperbolic type differential operator with respect to τ with infinitely differentiable coefficients which depend on τ and φ.

In the half-plane x_{1}, φ let us consider a convex domain Ω of the class C^{∞} bounded by the straight line segments ℓ_{1} : φ = −φ_{1}, ℓ_{2} : φ = φ_{2} and the curves γ_{1} : x_{1} = g(φ), −φ_{1} ≤ φ ≤ φ_{2}, γ_{2} : x_{2} = −g(φ), −φ_{1} ≤ φ ≤ φ_{2}. Here g(φ) ∈ C^{∞}(−φ_{1}, φ_{2}) ∩ C[−φ_{1}, φ_{2}], g(φ) > 0 for −φ_{1} ≤ φ ≤ φ_{2}, g′(φ) > 0 for −φ_{1} < φ < 0, g′(0) = 0, g′(φ) < 0 for 0 < φ < φ_{2}, g′′(φ) < 0 for −φ_{1} < φ < φ_{2}, and

min (g(−φ_{1}), g(φ_{2})) > 1 + t_{0} + d,  (32)

where

d = max(d_{1}, d_{2}, d_{3}), d_{i} = sup_{(x_{1},x_{2},t)∈supp f_{i}} |x_{1}|, i = 1, 2,

d_{3} = sup_{(x_{1},x_{2},t)∈supp F} |x_{1}|.

We denote by H_{0} ⊂ H a cylindrical domain Ω×(−∞, ∞) of the class C^{∞} where (−∞, ∞) is the τ-axis, and by ∂H_{0} its lateral surface ∂Ω × (−∞, ∞). When the inverse transformation (x_{1}, τ, φ) → (x_{1}, x_{2}, t) takes place, the cylindrical domain H_{0} will transform into the infinite domain G_{0} ⊂ D_{0} bounded by the surfaces S̃_{i} = S_{i}^{0} ∩ ∂G_{0}, i = 1, 2, and also by the surfaces S̃_{3} and S̃_{4}.

We shall show below that the surfaces S̃_{3} and S̃_{4} are of the temporal type and the following conditions are fulfilled on them:

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

where ω is the union of two smooth curves ω_{1} and ω_{2} lying on S̃_{3} ∪ S̃_{4}.

Indeed, one can easily verify that the surfaces S̃_{1} and S̃_{2} are the images of the cylindrical surfaces S′_{1} = ℓ_{1}×(−∞, ∞) ⊂ ∂H_{0} and S′_{2} = ℓ_{2}×(−∞, ∞) ⊂ ∂H_{0}, while the surfaces S̃_{3} and S̃_{4} are the images of the surfaces S_{3}^{0} = γ_{1} × (−∞, ∞) ⊂ ∂H_{0} and S_{4}^{0} = γ_{2} × (−∞, ∞) ⊂ ∂H_{0} when the inverse transformation (x_{1}, τ, φ) → (x_{1}, x_{2}, t) takes place. We divide the surface S_{3}^{0} into two parts S_{3}^{0} = S_{3+}^{0} ∪ S_{3−}^{0}, where

S_{3+}^{0} = γ_{1+} × (−∞, ∞), S_{3−}^{0} = γ_{1−} × (−∞, ∞),
γ_{1+} : x_{1} = g(φ), 0 < φ < φ_{2}, γ_{1−} : x_{1} = g(φ), −φ_{1} < φ < 0.

It is easy to see that when the inverse transformation (x_{1}, τ, φ) → (x_{1}, x_{2}, t) occurs, the image S̃_{3+} ⊂ S̃_{3} of the surface S_{3+}^{0} admits the parametric representation

S̃_{3+} : x_{1} = g(φ), x_{2} = σ sin φ, t = σ cos φ; 0 < φ < φ_{2}, 0 < σ < +∞.

Hence for the unit normal vector n = (α_{1}, α_{2}, α_{3}) to ∂G_{0} on the segment S̃_{3+} we obtain the expression

n|_{S_{3+}} = (\frac{σ}{\sqrt{σ^{2} + g^{′2}(φ)}}, \frac{−g′(φ) cos φ}{\sqrt{σ^{2} + g^{′2}(φ)}}, \frac{g′(φ) sin φ}{\sqrt{σ^{2} + g^{′2}(φ)}}) . (34)

Taking into account the structure of the domain Ω, by (34) we find that S̃_{3+} is a surface of the temporal type on which α_{3}|_{S_{3+}} < 0. Similar statements

are proved also for the other segments S̃_{3−}, S̃_{4+} and S̃_{4−} of the surfaces S̃_{3} and S̃_{4}. To prove finally that condition (33) is fulfilled, it remains for us only to note that component α_{3} of the unit normal vector n vanishes on the curves

ω_{1} = ∂S̃_{3+} ∩ ∂S̃_{3−}, ω_{2} = ∂S̃_{4+} ∩ ∂S̃_{4−}

which are the images of the straight lines ω̃_{1} : x_{1} = g(0), φ = 0, −∞ < τ < ∞, and ω̃_{2} : x_{1} = −g(0), φ = π, −∞ < τ < ∞, and the third component α_{3} of the unit normal vector n is equal to zero.

On the boundary ∂G_{0} of the domain G_{0} we define a function ν_{n}(x_{1}, x_{2}, t) of the class C^{∞} as follows:

ν_{n}|_{S_{i}} = f_{in}, i = 1, 2, ν_{n}|_{S_{3}} = ν_{n}|_{S_{4}} = 0, n = 1, 2, ….

The fact that the function ν_{n} ∈ C_{0}^{∞}(∂G_{0}) is implied by the structure of G_{0}, by inequality (32), and also by the smoothness and positioning of the carriers of the functions f_{in} ∈ C_{0}^{∞}(S_{i}^{0}), i = 1, 2.

In passing to the variables x_{1}, τ, φ, the functions ν_{n} and F_{n} will transform into some functions for which we shall use the previous notation. It is obvious that

ν_{n} ∈ C_{0}^{∞}(∂H_{0}), F_{n} ∈ C_{0}^{∞}(H_{0}).  (35)

By virtue of (35) there are numbers h_{in} = const, h_{1n} < h_{2n} such that

supp ν_{n} ⊂ ∂H_{0} ∩ {h_{1n} < τ < h_{2n}},

supp F_{n} ⊂ H_{0} ∩ {h_{1n} < τ < h_{2n}} = H_{n}. (36)

Note that when the inverse transformation J^{−1} : (x_{1}, τ, φ) → (x_{1}, x_{2}, t) takes place, the upper base ∂H_{n} ∩ {τ = h_{2n}} of the finite cylinder H_{n} will transform into the surface lying higher than the plane t = t_{0}, i.e., inf t > t_{0} for (x_{1}, x_{2}, t) ∈ ∂(J^{−1}(H_{n})) ∩ {log r = h_{2n}}.

Assume that

h_{n1}^{0} = h_{1n} − 1, h_{2n}^{0} = h_{2n} + 1, H_{n}^{0} = H_{0} ∩ {h_{1n}^{0} < τ < h_{2n}^{0}}.

Denote by S_{0n} the lateral surface of the finite cylinder H_{n}^{0}, and by Ω_{0} the lower base of H_{n}^{0}.

For the hyperbolic equation (31) with F = F_{n} let us consider, in the cylinder H_{n}^{0}, the following mixed problem:

e^{−2τ} L(τ, φ, ∂)v = F_{n},  (37)
v|_{Ω_{0}} = 0, \frac{∂v}{∂τ} |_{Ω_{0}} = 0,  (38)
v|_{S_{0n}} = ν_{n}.  (39)

By virtue of (35) and the results from [9], [10] the mixed problem (37)– (39) has a unique solution v = v_{n} of the class C^{∞}(\overline{H}_{n}^{0}). Note that if

H_{n1} = H_{0} ∩ {h_{1n} − 1 < τ < h_{1n}} ⊂ H_{n}^{0},

then by virtue of (36) this solution is identically zero in the cylinder H_{n1}, i.e., v_{n}|_{H_{n1}} = 0. We assume H_{n}^{−} = H_{0} ∩ {−∞ < τ < h_{2n}^{0}} and

u_{n}(θ) = \begin{cases}v_{n}(θ), & θ ∈ H_{n}^{0}, \\ 0, & θ ∈ H_{n}^{−}\H_{n}^{0}. \end{cases}

Since v_{n}|_{H_{n1}} = 0, the function u_{n} belongs to the class C^{∞}(H_{n}^{−}), vanishes for τ < h_{1n} and is a solution of the following mixed problem in the semi-infinite cylinder H_{n}^{−} with the Cauchy zero data for τ = −∞:

e^{−2τ} L(τ, φ, ∂)u_{n} = F_{n},
u_{n}|_{∂H_{n}^{−}∩∂H_{0}} = ν_{n}.

Returning to the initial variables x_{1}, x_{2}, t when the inverse transformation J^{−1} : (x_{1}, τ, φ) → (x_{1}, x_{2}, t) takes place and retaining the previous notation for the functions u_{n} and F_{n}, we find that:

(1) the function u_{n} in the domain G_{n} = J^{−1}(H_{n}^{−}) ∩ {0 < t < t_{0}} belongs to the class C^{∞}(\overline{G}_{n}) and satisfies the equation

□u_{n} = F_{n};

by the construction of the domains Ω, H_{0}, and G_{0}, the domain G_{n} does not depend on the number n and therefore will be denoted below by G;

(2) the function u_{n} on the lateral part ∪^{4}_{i=1}S̃_{i}^{0} of the boundary ∂G satisfies the conditions

u_{n}|_{_{S_{3}^{0}}∪_{S_{4}^{0}}} = 0, u_{n}|_{S_{i}^{0}} = f_{in}, 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 is a part of the surface S̃_{i}, figuring in conditions (33), for i = 3, 4.

Thus by virtue of (32), (33), Lemma 3, and Remark 2, the function

u_{n}^{0}(P) = \begin{cases}u_{n}(P), & P ∈ G, \\ 0, & P ∈ D\G \end{cases}

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

Due to inequality (3) we have

∥u_{n}^{0} − u_{m}^{0}∥_{W_{2}^{1}(D)} ≤ C( ∑_{i=1}^{2} ∥f_{in} − f_{im}∥_{W_{2}^{1}(S_{i})} + ∥F_{n} − F_{m}∥_{L_{2}(D)}). (40)

From (29) and (40) we conclude that the sequence of functions u_{n}^{0} is fundamental in the space W_{2}^{1}(D). Therefore by virtue of the fact that the space W_{2}^{1}(D) is complete there is a function u ∈ W_{2}^{1}(D) such that u_{n}^{0} → u in W_{2}^{1}(D), □u_{n}^{0} → F in the space L_{2}(D), and u_{n}^{0}|_{S_{i}} → f_{i} in W_{2}^{1}(S_{i}), i = 1, 2, for n → ∞. Therefore the function u is the strong solution of problem (1),

(2) of the class W_{2}^{1}. The uniqueness of the strong solution of problem (1),

(2) belonging to the class W_{2}^{1} follows from inequality (3).

Let us now consider the case where equation (1) contains the lowest terms

L_{0}u ≡ □u + au_{x_{1}} + bu_{x_{2}} + cu_{t} + du = F,  (41)

where the coefficients a, b, c, and d are the known bounded measurable functions in the domain D.

In the space W_{2}^{1}(D) we introduce the equivalent norm which depends on the parameter γ

∥u∥_{D,1,γ}^{2} = ∫_{D} e^{−γt}(u^{2} + u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2})dx dt, γ > 0.

Arguments similar to those used in [4] allow us to prove

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

∥u∥_{D,1,γ} ≤ \frac{C}{\sqrt{γ}} ( ∑_{i=1}^{2} ∥f_{i}∥_{S_{i},1,γ} + ∥F ∥_{D,0,γ}),  (42)

where f_{i} = u|_{S_{i}}, F = □u, and the positive constant C does not depend on u and the parameter γ.

By virtue of estimate (42) the lowest terms in equation (41) for the above-introduced equivalent norms of the spaces L_{2}(D), W_{2}^{1}(D), W_{2}^{1}(S_{i}), i = 1, 2, give arbitrarily small perturbations for a sufficiently large value of the parameter γ, which fact enables one to prove by Theorem 1 and the results from [4] that problem (41), (2) has a unique solution in the class W_{2}^{1}.

The following theorem is valid:

Theorem 2. 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 (41), (2) in the class W_{2}^{1}, for which estimate (3) holds.

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(Received 10.02.1994)

Author’s address: I. Vekua Institute of Applied Mathematics Tbilisi State University 2,University St., Tbilisi 380043 Republic of Georgia