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The first order system equations of a principal type on the plane.

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УДК 517.9
THE FIRST ORDER SYSTEM EQUATIONS
OF A PRINCIPAL TYPE ON THE PLANE
N.A. Zhura
Lebedev Physical Institute of the Russian Academy of Sciences,
Leninskij pr., 53, LPI, Moscow, 119991, Russian Federation, e-mail: nikzhura@gmail.com
Abstract. Boundary value problems for the system equations of a principal type with constant
coefficients on the plane are studied. The half-infinite domains with noncharacteristic boundary and
finite domains with such property are considered. The representation solutions of this systems through
solutions of canonical elliptic and hyperbolic systems is obtained. Also the index formula for associated
problems in Holder weighted classes is founded.
Keywords: principal type equations, noncharacteristic boundary, index formula, function theoretical
approach, canonical systems of first order.
1
Integral representation
On the (x1 , x2 ) - plane R2 we consider a system of linear partial differential equations
∂u
∂u
−a
= 0,
∂x2
∂x1
(1)
where u(x) is an unknown l - vector-valued function and a ∈ Rl×l is a constant matrix. The
system (1) is said to be of a composite type (principal one [1, 2]) if it’s characteristic equation
det(a − ν) = 0
(2)
has s1 ≥ 1 complex roots with the positive imaginary part and s2 ≥ 1 real roots, 2s1 + s2 = l.
Let b1 ∈ C s1 ×l , b2 ∈ Rs2 ×l be constant matrices such that nonsingular matrix b = (b1 |b1 |b2 )
reduces a to the Jordan normal form
b−1 ab = diag (J1 , J1 , J2 ),
(3)
where the block matrixes Jk ∈ Csk ×sk , k = 1, 2, are composed from Jordan cells. Here J1 has
complex eigenvalues with positive imaginary part and J2 ∈ Rs2 ×s2 has only real eigenvalues.
Let k2 ≤ s2 denote the maximum of orders of Jordan cells composing J2 .
It is valid the following representation theorem [3].
Theorem 1. Any regular solution u of the system (1) can be represented in the form
u = 2Re b1 Φ + b2 Ψ,
(4)
where Φ is a regular solution of the canonical elliptic system
∂Φ
∂Φ
= J1
,
∂x2
∂x1
(5)
The work was supported in part by the National Natural Science Foundation of China (NSFC) in the
framework of the bilateral project "Complex Analysis and its applications"(project No. 08-01-92208-GFEN)
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НАУЧНЫЕ ВЕДОМОСТИ
№13(68). Выпуск 17/1 2009
and Ψ is a regular solution of the canonical hyperbolic system
∂Ψ
∂Ψ
= J2
.
∂x2
∂x1
(6)
Solutions of the system (5) are said to be a J1 – analytical functions. It is known [4] that a
general solution of this systems can be represented in the form
∂
Φ(x) = exp(x2 J1,0 )
φ(x1 + νx2 ),
∂x1
where J1 = ν + J10 is decomposition of the matrix J1 into diagonal ν and nilpotent parts. Here
the s1 − vector φ(x1 + νx2 ) consists from components φj (x1 + νj x2 ), 1 ≤ j ≤ s1 , where the
functions φj (z) are analytic in the corresponding domain of the complex plane. The similar
representation
∂
ψ(x),
(7)
Ψ(x) = exp(x2 J2,0 )
∂x1
there exists for a s2 -vector-valued function ψ = (ψj , 1 ≤ j ≤ s2 ), where J2 = ν + J2,0 and
ψj (x) = ψ̃k (x1 + νj x2 ). Note that ψj satisfies the hyperbolic equation
∂ψj
∂ψj
= νj
,
∂x2
∂x1
2
νj ∈ R.
Fredholm solvability in the half-infinite domain
Let C µ (D) be the space of functions satisfying the Holder condition on the the closed domain
D with exponent 0 < µ ≤ 1 (and bounded if D is infinite). The space C µ,n (D) consists of the
functions with partial derivatives in C µ,n−1 , n ≥ 1, (C µ,0 = C µ ). These spaces are Banach
with respect to the corresponding norm. It is convenient to write C µ+0,n for the class ∪ε>0 C µ+ε,n .
If the domain D is infinite we also use the space C µ,n (D̂) for the set D̂ = D∪{∞} considered
as the compact on the Riemann sphere Ĉ. These definitions also applies to the classes C µ,n on
curves Γ ⊆ C.
Let D be a half-infinite domain on the complex plane i.e. it is a simple connected domain with
smooth boundary Γ on the Riemann sphere. So the curve Γ permits a smooth parametrization
z = γ(t), t ∈ R, where
γ ′ (t) ∈ C µ,k2 (R̂).
(8)
We consider a boundary value problem
Cu = f on Γ,
(9)
for the system (1) where C is a (s1 + s2 ) × l matrix-valued function, and f is a (s1 + s2 ) vectorvalued function on Γ = ∂D. This problem is considered in the class C µ,1 (D) of solutions (1)
such that the functions Φ and Ψ belong to this class in the representation (4). More exactly we
say that the vector-valued function Ψ defined by (7) belongs to the class C µ,1 if the components
of ψ̃ belong to the class C s2 +1−j,µ , j = 1, . . . , s2 , as functions of one variable. For brevity it is
assumed here that J2 consists from one Jordan cell, in the general case this definition is regarded
N.A. Zhura. The first order system equations of ...
128
with respect to each Jordan’s block of J2 . In what follows it is assumed that the characteristics
x1 + νj x2 = const of the system (6) don’t tangent of the curve Γ, i.e.
Reγ ′ (t) + νj Imγ ′ (t) 6= 0,
t ∈ R̂, 1 ≤ j ≤ s2 .
Moreover it is assumed that Γ coincide with a straight line in a neighborhood of ∞.
It is assumed also, that
C, f ∈ C µ,k2 (Γ̂)
(10)
(11)
and
|(b−1 u)1| ≤ const(|x|)−1
as |x| → ∞, where by (b−1 u)1 we denote the first s1 elements of the vector b−1 u.
Let us put
C 1 s1
f1 s1
C=
,
f=
.
C 2 s2
f2 s2
(12)
(13)
Without loss of generality we can assume that
det C2 b2 6= 0 on Γ.
(14)
A = C1 (1 − b2 (C2 b2 )−1 C2 )b1 .
(15)
det A 6= 0 on Γ .
(16)
Let us consider
We say that (1),(9) is a normal type problem if
Theorem 2. Suppose that the conditions (8), (10) for the countour Γ and condition (11)
for C, f are fullfilled. Then the problem (1), (8) is fredholmian in C 1,µ (D) if and only if the
normality condition (16) is satisfied, and its index is
1
æ = − arg det A + s1 .
(17)
π
Γ
3
The case of the basic domain
Let the hyperbolic system (7) be such that the nilpotent part J20 of the matrix J2 is equal to
0 and the diagonal matrix ν = (νi δij ) is composed from two real numbers. Suppose that the
boundary ∂D of the finite domain D ⊆ C consists of two noncharacteristic smooth curves Γ1
and Γ2 that connect two corner points τ1 and τ2 . We consider the following boundary value
problem
Cj u = fj on Γj , j = 1, 2,
(18)
for the system (1), where Cj is a (s1 + s2 ) × l matrix and fj is a (s1 + s2 ) vector.
Let us introduce the weighted Holder space Cλµ (D) = Cλµ (D; τ1 , τ2 ), λ = (λ1 , λ2 ) ∈ R2 , of
all functions ϕ(z) such that
ϕ(z) = |z − τ1 |λ1 −µ |z − τ2 |λ2 −µ ϕ∗ (z),
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НАУЧНЫЕ ВЕДОМОСТИ
№13(68). Выпуск 17/1 2009
where ϕ∗ (z) ∈ C µ (D) and ϕ∗ (τ1 ) = ϕ∗ (τ2 ) = 0. The classes Cλ1,µ of differentiable functions are
introduced by induction under the condition
ϕ ∈ Cλµ ,
µ
∂ϕ/∂xi ∈ Cλ−1
.
We consider the problem (1),(18) in the class Cλ1,µ (D) of solutions (1) such that the functions
Φ and Ψ belong to this class in the representation (4).
Let γj (t) ∈ C 1,µ [0, 1] be the smooth parametrization [0, 1] → Γj , j = 1, 2 and the complex
numbers q2i−1 = γi′ (0), q2i = −γi′ (1) are the tangent vectors at the points τ1 , τ2 . By θj denote
the angle of the sector corresponding to the corner τj . Evidently, θj = arg qk − arg qr , 0 <
arg q < 2π, 0 < θj < 2π, Pj = k, r, j = 1, 2, (more presicely, P1 = 1, 3, P2 = 4, 2), where the
rotation from vector qk to qr about τj within domain is clock-wise.
Let us put the functions of matrices
mj (ζ) = (Re qk + (Im qk )J1 )ζ (Re qr + (Im qr )J1 )−ζ , k, r = Pj ,
and let be
−1 A1 (τ1 )w1 (ζ) + A1 (τ1 )v1 (ζ)w1(ζ) ,
−1 −1
A2 (τ2 )v2 (ζ)w2(ζ) + A2 (τ2 )w2 (ζ) ,
x1 (ζ) = e2πiζ − 1
x2 (ζ) = e2πiζ
where
vj =
0
m−1
j (ζ)
mj (ζ)
0
!
,
(19)
wj (ζ) = e2πiζ vj (ζ) − 1, j = 1, 2,
Aj = cj,1(1 − b2 (cj,2b2 )−1 cj,2 )b1 , j = 1, 2.
Theorem 3. Suppose that the conditiones (8), (10) for the curves Γ1 and Γ2 including the
corner τ1 , τ2 are fullfilled. Let C, f belong to Cλµ and the normality condition
det Aj (t) 6= 0,
t ∈ Γj , j = 1, 2
(20)
be satisfied.
Then the problem (1), (18) is Fredholm in Cλ1,µ (D) if and only if
det xk (ζ) 6= 0,
Reζ = λk , k = 1, 2,
(21)
and its index is
1
λk +i∞
X
1
1
−1
æ = − arg det(A1 (t)A2 (t)) −
arg det xk (ζ)
− s1 .
π
2π k=1,2
0
ζ=λk −i∞
4
(22)
Some generalazations
We now consider the problem
Cj u = fj on Γj , j = 1, 2,
(23)
in finite domains D, whose boundary ∂D consists of two curves Γ1 and Γ2 with the corners τ1
and τ2 . We assume that the matrix C1 (C2 ) of order (s1 + s2 ) × l (s1 × l), and the vector f1 (f2 )
N.A. Zhura. The first order system equations of ...
130
k
of order s1 + s2 (s1 ) are prescribed on Γ1 (Γ2 ) and fj , Cj are belonged Hµ,λ
, k = s2 , j = 1, 2.
Here the curves Γj satisfy conditions (9), (10).
Theorem 4. The assertion of theorem 3 remains in force also for the problem (1), (23),
provided only that Aj mean matrices A1 = C1,1 (1 − b2 (C1,2 b2 )−1 C1,2 )b1 , A2 = C2 b1 , and the last
term −s1 in the formula (22) must be replaced by the s1 .
We also studied the questions of asymptotics of the solutions near the corner points and the
smoothness of the solutions up to the boundary. We generalized this approach for the systems
of higher order and for a class of the admissible finite domain with piecewise smooth boundary.
If the order of C1 in the last problem is not equal to s1 + s2 then we investigated this problem
only for the case k2 = 1.
Our study is carried out in the framework of the function-theoretic method [5]. The scheme
of this method is as follows. First of all we express a general regular solution in terms of regular
solutions Φ and Ψ and use an anologue of a theorem of Vekua on integral representations of Φ
and some notions about Ψ which arises from (7). By substituting that into the corresponding
boundary conditions we reduce the problem to system of integral equations on the boundary
of the domain. Another approuch see in [6].
Bibliography
1. J. Hadamard. Proprietes d’une equation line’aire aux derivees partielles du quatriemes
ordre, Tohoky Math. Journ. – 1933.– v. 37. – P. 133 – 150.
2. L. Hormander. Linear partial differenrial operators. N.-Y.,1963.
3. N.A. Zhura. General boundary value problems for Douglis -Nirenberg elliptic systems in
the domains with piese- wise smoothh boundary, Dokl. Akad. Nauk Russian 330 (1993),
151-154 (in Russian).
4. A.P. Soldatov. A function theory method in boundary value probles 1n the plane. I: The
smooth case. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 1070-1100;(in Russian).
5. A.P. Soldatov. Singular integral operators and boundary value problems of functions
theory. Vysh.Shkola, Moscow, 1991.
6. G.I. Eskin. Boundary value problems for equations with constant coefficients on the plane,
Matem.sbornik Russian 59 (1962), 67-124 (in Russian).
131
НАУЧНЫЕ ВЕДОМОСТИ
№13(68). Выпуск 17/1 2009
СИСТЕМЫ УРАВНЕНИЙ ПЕРВОГО ПОРЯДКА
ГЛАВНОГО ТИПА НА ПЛОСКОСТИ
Н.А. Жура
Физический институт имени П.Н.Лебедева Российской академии наук,
Ленинский проспект, 53, ФИАН, Москва, 119991, Россия, e-mail: nikzhura@gmail.com
Аннотация. В работе изучаются краевые задачи для систем уравнений первого порядка
главного типа с постоянными коэффициентами на плоскости. При этом рассматриваются как полубесконечные области с нехарактеристической границей, так и конечные области типа луночки.
Дано представление решений этих систем через решения более простых, так называемых, канонических систем первого порядка эллиптического и гиперболического типов. Получены также
формулы для индекса соответствующих задач в весовых классах Гёльдера.
Ключевые слова: уравнения главного типа, нехарактеристическая граница, канонические
системы первого порядка.
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