On some boundary value problems for a class of hyperbolic systems of second order in conic domains
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For one class of hyperbolic systems of second order, we consider multidimensional versions of the Darboux problem in conic domains. A priori estimates of solutions of these problems are obtained. The existence of a solution of the Darboux problems is proved under the supplementary conditions imposed on the coefficients of the system, when the data support of the problem is of temporary type. 1. Statement of the problem In the Euclidean space Rn+1 of the variables x = (x1 ,...,xn ) and t we consider a system of linear differential equations of the kind Lu = utt −
n
Ai j uxi x j +
i, j =1
n
Bi uxi + Cu = F,
(1.1)
i=1
where Ai j (Ai j = A ji ), Bi , and C are given real (m × m)-matrices, F is a given and u is an unknown m-dimensional real vector, n ≥ 2, m > 1. Below, the matrices Ai j will be assumed to be symmetric and constant, and for any m-dimensional real vectors ηi , i = 1,...,n, we have the inequality n i, j =1
Ai j ηi η j ≥ c0
n 2 η i ,
c0 = const > 0.
(1.2)
i=1
It can be easily verified that the system (1.1) by virtue of the condition (1.2) is hyperbolic. Let D be the conic domain {(x,t) ∈ Rn+1 : |x|g(x/ |x|) < t < +∞} lying in the half-space t > 0, and bounded by the conic manifold S = {(x,t) ∈ Rn+1 : t = |x|g(x/ |x|)}, where g is an entirely definite, positive, continuous, piecewise smooth function given on the unit sphere of the space Rn . For τ > 0, by Dτ := {(x,t) ∈ Rn+1 : |x|g(x/ |x|) < t < τ } we denote the domain lying in the half-space t > 0, bounded by the cone S and the hyperplane t = τ. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 547–567 DOI: 10.1155/JIA.2005.547
548
BVPs of hyperbolic systems in conic domains
Let S0 = ∂Dτ0 ∩ S be the conic portion of the boundary of Dτ0 for some τ0 > 0. Suppose that S1 ,...,Sk1 ,Sk1 +1 ,...,Sk1 +k2 are nonintersecting smooth conic open hypersurfaces, 2 where S1 ,...,Sk1 are the characteristic manifolds of the system (1.1), and S0 = ki=1 +k 1 Si , where Si is the closure of Si . Consider the following boundary value problem: find in the domain Dτ0 a solution u(x,t) of the system (1.1) satisfying the conditions u|S0 = f0 , Γi ut |Si = fi ,
(1.3)
i = 1,...,k1 + k2 ,
(1.4)
where fi , i = 0,1,...,k1 + k2 , are given real κi -dimensional vectors, Γi , i = 1,...,k1 + k2 , are given constant (κi × m)-matrices with κ0 = m, 0 ≤ κi ≤ m, i = 1,...,k1 + k2 . Here, the number κi , 1 ≤ i ≤ m, shows to what extent the part Si of the boundary ∂Dτ0 is occupied; in particular, κi = 0 denotes that the corresponding part Si in the boundary condition (1.4) is completely free from the boundary conditions. Below we will see that for the problem (1.1), (1.3), (1.4) to be correct, we must choose the number κi in a well-defined way, depending on the geometric properties of the hypersurface Si . It will be assumed that the elements of the matrices Bi and C in the system (1.1) are bounded, measurable functions in the domain Dτ0 , and the right-hand side of that system F ∈ L2 (Dτ0 ). Note that a part
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