CC BY
13
UDK 517.929.7
DOI: 10.20310/1810-0198-2017-22-6-1229-1234
ON POSITIVITY OF THE GREEN FUNCTION FOR POISSON PROBLEM FOR A LINEAR FUNCTIONAL DIFFERENTIAL EQUATION
© S. M. Labovskiy
Plekhanov Russian University of Economics, 36, Stremyanny lane, Moscow, Russian Federation, 117997 E-mail: labovski@gmail.com
For the Poisson problem
—Am + p(x)u — J u(s) r(x, ds) = pf, = 0
Q
equivalence of positivity of the Green function and other classical properties is showed. Here Q is an open set in 1" , and r(Q) is the boundary of the Q . For almost all x e Q , r(x, ■) is a measure satisfying certain symmetry condition. In particular this equation involves integral differential equation and the equation
m
—Au + p(x)u(x) — Pi(x)u(hi(x)) = pf,
i=i
where hi: Q ^ Q is a measurable mapping.
Keywords: Green function; Poisson problem; Vallee-Poussin theorem; Spectrum of selfadjoint operator
1. The Poisson problem 1.1. The problem
Let Q be an open set in 1n , and r(Q) be the boundary of the Q . For a function u = u(x), x e Q , let Au = UX1X1 +-----+ u'XnXn , where x = (x\,..., xn). In the Poisson problem
—Au + p^u — j u(S) r(x,ds)= (1)
n
u|r(n) =0 (2)
for almost all x e Q , the function r(x, ■) is a measure satisfying certain symmetry condition. For example, if for each x the measure is concentrated at the points hi(x), i = 1,... ,m , the equation (1) will have the form
m
—Au + p(x)u(x) — pi(x)u(hi(x)) = p(x)f (x),
i=l
where hi: Q ^ Q is a measurable mapping. The function p is a positive weight.
The functional differential equation (1) has certain mechanical interpretation if p(x) > r(x, Q). For the case n = 2 it describes the state of the loaded membrane with added special internal forces. This fact allows predict positivity of the Green function [1]. Here the condition p(x) > r(x, Q) is omitted. The boundary value problem (BVP) (1), (2) has the Fredholm property. In case of unique solvability its solution can be represented by means of the Green's function
u(x) = G(x,s)f (s) ds. Jo
1.2. Assumptions and notation
Let Q C Rn be a nonempty bounded open set, r(Q) be the boundary of the Q , and X = Q be the closure of Q . For a real function u = u(x) defined on Q and having derivative of first order,1 j , where x = (xi,..., xn). For two such functions u and v ,
uxvx : = Ux1 vx1 + ' ' ' + uxnvxn.
Let's consider the following two bilinear forms2
[u,v]:= J u'xv'x dx + J p(x)u(x)v(x) dx —J v(x)u(s) £o(dx x ds), (3)
o o OxQ
(dx := dx\ ■ ■ ■ dxn) and
{u'v):=[u^—j v(x>u(s>n(dxxds>- (4)
ox o
The domain Q is assumed to satisfy the cone condition [2].
1.2.1. The form (3) we use under following assumptions
Let M be the set of all Lebesgue measurable subsets in X = Q . Let the function r: X x M.^ R satisfy two conditions: for almost all x € X , the function r0(x, ■) is a measure on M , for any e € M , r0(^,e) is measurable on X .Let
p(x) := ro(x, Q).
The set function £0 defined by the equality
co(E) = ro(x,Ex) dx, Ex = {y: (x,y) € E} Jx
is a measure. Assume that £0 is symmetric, that is,
£o(ei x e2) = £o(e2 x ei), Vei, e2 € M. The measure n has the same properties and is defined by
n(E) = \ q(x,Ex) dx, Ex = {y: (x,y) € E},
JX
where q has properties analogous to ro . The measures £ and r(x, ■) define by
£ := £o + n, r := ro + q
1 := signifies 'is equal by definition'
2the notation £(dx x dy) is equivalent to
1.2.2. Two main spaces
We use the Sobolev spaces Wl'2(Q) and w2'2(Q) [2].
Definition 1.1. Let W be the vector subspace of all elements from W^'2(Q) satisfying [u, u] < TO .
The bilinear form [u, v] is an inner product in the Hilbert space W.
Let p(x), x e X , be a positive measurable and integrable in Q function and ¡(E) :=fE p(x) dx .
Let
(f,g) = j f (x)g(x)p(x) dx n
and L2(X,i) (or L2(Q,i)) be the Hilbert space of all i-measurable functions on X with finite
integral f f (x)2p(x) dx . n
Define the operator T: W ^ L2(Q, ¡) by the equality Tu(x) = u(x), x e Q . The operator T is continuous.
1.3. A variational form of the problem. Euler equation
The equation with relation to u in variational form
J u'xv'x dx + J p(x)u(x)v(x) dx —J v(x)u(s) £0(dx x ds), n n nxn
= f (x)v(x)p(x) dx, Vv e W, n
can be represented in the short form
[u,v] = (f,Tv), (Vv e W). (5)
The image T(W) is dense in L2(X, ¡). The operator T* has an inverse 3. For any f e L2(Q, ¡)
the equation (5) has a unique solution u = T*f e W .
The set T*(L2(X, ¡)) is the domain of the operator .
2 2
For any u e C£°(Q) or u e W0' (Q) and v e W integrating by parts obtain the identity
I u'xv'x dx = — I Au ■ v dx.
Hence, if Au = g , then
/ u'xv'x dx = — g ■ v dx, Vv e W. (6)
nn
Since C^(Q) C D(L0) the equation (6) can be used as definition of operator A on the space D(L0) in a weak sense.
Proposition 1.1. Operator has representation
L0u = 1 Au + p(x)u — J u(s) r0(x,ds. (7)
3operator can be considered as extension of the operator defined by (7)
n
n
2. Results
2.1. Eigenvalue problem and spectrum Theorem 2.1. The eigenvalue problem
—Au + pu — J u(s) r0 (x, ds) = Apu, u|p(Q) = 0
r(fi)
has in W a system of nontrivial solutions un(x) corresponding to positive eigenvalues An . That
1 2
is, Ao < A1, ■ ■ ■. This system forms an orthogonal basis in the space W0' . Note that the minimal eigenvalue A0 satisfies the relation
Ao = inf -iuuL ■
0 [u,u]=1 (Tu, Tu)
2.2. Positivity of solutions
The problem
_ A 0I _L rw I _ / 01 I o\ IY>^ I nr> no i - rt T 01 .
Ir(Q)
—Au + pu — J u(s) r0(x,ds) = pf, u|p(Q) =0 (9)
represents the equation L0u = f .
Theorem 2.2. Suppose f >^ 0 and u(x) is the solution of the problem (9). Then u(x) > 0 in Q.
Corollary 2.1. The minimal eigenvalue Ao of the problem (8) is positive and simple (Ao < Ai). It associated with a positive in Q eigenfunction u0(x).
2.3. General case
Here we consider the form (4). The equation in variational form
{u,v} = (f,Tv), Vv e W, is equivalent to the boundary value problem
Lu := L0u — Qu = f, u|r(^ = 0, (10)
where the operator Q: W ^ L2(Q, ¡j) has the representation
Qu(x) = (l/p) u(s)q(x,ds).
Jo,
We may to impose some conditions to ensure the action Q: W ^ L2(Q,j) and continuity. In this case the operator QT* will be compact. It may be showed that this operator will be compact if
q(•, Q) e L2(Q,p). (11)
Theorem 2.3. Suppose (11) is fulfilled. The eigenvalue problem
—Au + pu — J u(s) r(x,ds) = Apu, u|r(0) =0 (12)
o
has in W a system of nontrivial solutions un(x) corresponding to eigenvalues A0 < A1 < .... This system forms an orthogonal basis in the spaces W0 and in W , and in L2(Q, p).
Theorem 2.4. The following affirmations are equivalent:
1. the quadratic functional (u,u) defined by (4) is positive definite,
2. the problem (10) is uniquely resolvable, and its Green function is positive in Q x Q,
3. the inequality —Av + pv — v(s) r(x,ds) >^ 0 has positive in Q solution,
f
Jn
4. the minimal eigenvalue of the problem (12) is positive,
5. spectral radius of the operator QT* is less than unit.
REFERENCES
1. Labovskiy S., Getimane M. Poisson problem for a linear functional differential equation // Tambov University Reports. Series: Natural and Technical Sciences. Tambov, 2016. V. 21. Iss. 1. P. 76-81.
2. Adams R.A., Fournier J. Sobolev Spaces // Elsevier, 2003.
Received 3 September 2017
Labovskiy Sergei Mikhailovich, Plekhanov Russian University of Economics, Moscow, the Russian Federation, Candidate of Physics and Mathematics, Associate Professor of the Higher Mathematics Department, e-mail: labovski@gmail.com
УДК 517.929.7
DOI: 10.20310/1810-0198-2017-22-6-1229-1234
О ПОЛОЖИТЕЛЬНОСТИ ФУНКЦИИ ГРИНА ДЛЯ ЗАДАЧИ ПУАССОНА ДЛЯ ЛИНЕЙНОГО ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОГО
УРАВНЕНИЯ
© С. М. Лабовский
Российский экономический университет им. Г. В. Плеханова 117997, Российская Федерация, г. Москва, Стремянный пер., 36 E-mail: labovski@gmail.com
Для задачи Пуассона
показана эквивалентность положительности функции Грина и других классических свойств. Здесь П - открытое множество в М" , и Г(П) - граница П . Для почти всех х € П , г(х, ■) - мера, удовлетворяющая некоторому условию симметрии. В частности, это уравнение охватывает интегро-дифференциальное уравнение и уравнение
m
— Au + p(x)u(x) — Pi(x)u(hi(x)) = pf,
i=1
где :П ^ П - измеримое отображение.
Ключевые слова: функция Грина; задача Пуассона; теорема Валле-Пуссена; спектр самосопряженного оператора
СПИСОК ЛИТЕРАТУРЫ
1. Labovskiy S., Getimane M. Poisson problem for a linear functional differential equation // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2016. Т. 21. Вып. 1. С. 76-81.
2. Adams R.A., Fournier J. Sobolev Spaces // Elsevier, 2003.
Поступила в редакцию 3 сентября 2017 г.
Лабовский Сергей Михайлович, Российский экономический университет им. Г. В. Плеханова, г. Москва, Российская Федерация, кандидат физико-математических наук, доцент кафедры высшей математики, e-mail: labovski@gmail.com
For citation: Labovskiy S.M. On positivity of the Green function for Poisson problem for a linear functional differential equation. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2017, vol. 22, no. 6, pp. 1229-1234. DOI: 10.20310/1810-0198-2017-22-6-1229-1234 (In Engl., Abstr. in Russian).
Для цитирования: Лабовский С.М. О положительности функции Грина для задачи Пуассона для линейного функционально-дифференциального уравнения // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2017. Т. 22. Вып. 6. С. 1229-1234. DOI: 10.20310/1810-0198-2017-22-6-1229-1234.
