SCIENTIFIC ABSTRACT VAYNBERG, B.R. - VAYNBERG, D.V.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001859110012-3
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: September 1, 2001
• Sequence Number: 12
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

32805
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The existence and uniqueness C111/C444
I er I :> k M(c4 ) is the plane (Y- k for 61,1-- kand 'r= -OLk for
el1> k. Let lad bo choson so small that the roots of the polynomit].
P(s,, 6) for which Im. s, has different signs, lie at different side,---,
2
of M(04 ). Let M 1 be the set M(c4 ) for the chosen c>t , if Ot > 0, and
M2 be the corresponding set, if ck 0.
Theorem 1: The integrals
E e ixI31 +iys2dH eixsl+iYB2 dHi, i=1,2, (4)
i(X,Y) P(sj' s 2) s 2n + s
H i H 1
exist and give two fundamental solutions for (2). -
Theorem 2: The fundamental solutions
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slo2ol6211421OC110011021
The existence and uiaiqueness . . . 0111/0444
00 ixs +iys
Et(x,.Y) e e dff, d 62 (6)
P(elf 6 2) +
for the operators
a ) + (5)
P (j P (i 5-Y
e ax ay ax
converge to El(x,y) for 6 0, and to E 2(x'y) for F, -4 + 0.
.It is put coo (arc tg sin (arc +.g
ZX) 1 02 x
Theorem 38'For all from the given interval there holds
ap+I&
axpay"
jk + oj~_-j)-j.-f (OR-7i + 0;~=,
OXP I S -1 L - - +wpl, (7)
CA (jk)P+1 +2f*
n-IL-
0 022n-:1
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The existenoe,,and uniqueness . . . C111/C444
where
P+1 1+1
01,91) 2n-1 J92 12n-1
W e. 1 > (8)
P, I 2n N 2,
(r 101 a21 2n-
From the formulas (7) and (6) the following conditions for the
existenoe and for the uniquenees-of the solution of (2) are obtainedi
,Theorem 0 It is u C- W, if
u (9)
r1/21 e1 02 12n-1
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The existence and unioueness . . . Clll/C444
1
Ole, 2n 1
2) 3pu U ap-lu e, '92 (10)
laxp 1 lxp-l 2n
(.-1e, 21 2n-1 A
1
3) apu ap-lu cio,eXn-l
gyp W2 W-1 2n
(rie 10212n-17
where p lp2l ... , 2n-1; A being an arbitrary number between and
+ and either
2n
2n 2n 2n
iik( 01 'n-1 + ~ 2n -11 2,, 2n-' or bei-ik 2n-T +
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The existence and uniqueness . . .
2n 1 1
+ 02 2n-1) 2n 2n-1 1,2.
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C111/C444
The author mentionsi J. N. Vekua, B. P. Paneyakh and V. P. Palamodov.
The author thanks P. P. Mosolov for advice.
There are 5 Soviet-bloc references.
ASSOCfATION: Moskovskiy gosudarstvennyy universitet im. M. V. Lomonosova
(Moscow State University im. M. V. Lomonosov)
PRESENTEDs July 26, 1961, by J. G. Petrovskiy, Academician
SUBMITTEDi. July 22, 1961
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VAYNBEFL, B.R.~.
Asymptotic behavior of Green's function for Sobolev-Gallpern
equations. DoklAll SSSR 136 n0.5:1015-1018 F 161. (PUl 1425)
1. Moskovskiy gosudarstvennyy universitet iji. M.V.Lomonosovs.
Predstavleno akademikom I.G.Petrovskim.
(Groups, Theory of) (Functional analysis)

203,4,3,
S/020
,/61/136/005/001/032
6111/C222
LUTHORs Vaynberg, B.R.
TITM The Asymptotic Behavior of the Green's Function for Sobolev-
Gallpern Equations
PERIODICALs Doklady Akad6mii nauk SSSR, 1961, Vol. 136, No- 5,
PP- 1015 - 1018
TEXTz The author gives asymptotic formulas (f03~ I X I --P CID and t- oonst>,O)
for the solution (Green's function G(x,t)) of the Cauohy Problem for the
equation
(1) P t, i ID I'D X) U - 0
with the initial conditions
Glt-o .-')GAt It-0.- ... .j I- 2G/,a tl-2 Itwo -0; G/1? ti- 1It-0 ,RX)
Here P(A 90) is a polynomial of two variables with constant coefficients
a - 6"+ 17,- ; and it is assumed that the condiblon
(2) Re 0, j 192,,91
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20343,
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C111%222
The Asymptotic Behavior of the Green's Funotion for Sobolev Gail4srlk
Equations
is satisfied, where (e) are roote of the equation
(3) P(A s) Pl(8)x + Pl_1(s) A'-' + ... + Po(s) It.0 ~0
At first the special oaBe
(4) Q(i ? /D X) a U)~) tP(i X)U
in considered. The asymptotic behavior of the Green's funotion of (4) is
determined by the behavior of the quotient P(s)/Q(s) in the neLghborhood
of the real poles, ihe non-real poles, and for lsl---> co, and his three
summandet
(9) + G
2(x't) + G3(x't)
Rere G2(x't) is given by n +2 1
. n fCj,., 2nj+2t 2nj+2 ex n1+1
(13) B-02(xlt)- P_ 1, XG J11+0 xi
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ClII/C222
The Asymptotic.,Behavior of the Green's Function for Sobolev-Gallpern
Equations
where a - 6' + i is a complex root of Q(s) - 0 an-d nis the
multiplicity of this root (the;author only considers roots for which
x Zi,,8, simplifications are possible in the design of the plate it-
self which are based on neglectiriLf the discrete character Of the
distribution of the reactions of the ribs. The reaction forces are
supposed to be uniformly distributed on the surface of the plate.
If the number of ribs is large (k >,,.12) the plate with ribs can be i
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S/124/,61/000/009/028/0-58
Hethods of designing... D234/D.303
considered as orthotropic in construction.-Comparison of the re-
sults of calculating deflections according to a third method and
experimental results for the case L-.,= 16 is given. For cases iffien
the centers of gravity of the rib sections are Situated in the
middle surface of the plate, a design'based on methods of the theory-
of disturbances.is offered. Results of design according to this
method are given for k 4 ''and k S. Z-~bstracter's note: Com-
pletetranslatiort-T.
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