SCIENTIFIC ABSTRACT VISHIK, M.I. - VISHKAREV, A.F.

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• Collection: Scientific Abstracts
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• Release Decision: RIF
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• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: September 1, 2001
• Sequence Number: 1
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

Accr-ssicK lift: Ap4ozP946
AWHOH: -ViAbUl I.; Eakin.. G. I.
TITLEt Boundary value problems for geaerul singular cqwA:0.r,,: j~ EX:
S01MCE. A14 SSSR. Doklady*, v. 155, no. 1, 1964, 24-27
TMC TAGS: ainpxLar integral, integral equation, Ongular
value problem, elliptia equationj, eaneJvais, interrdidiffemritic-1 thnory
of functions
d
~IADSTVACT; An equation of tLe type
Kfp Kfp
TT K- (X, X (Y) "Y + T
11,where x e G in examined In a bo-urAed domain GMD vfth bc~zdarr
r . In this paxticular caac.. K.(x,z) and T(x,z) am genorallzr-e'
1: reapect to z. smoothly.depe-,ndmt upon x,, and the in (1)
T
in the nence of the theory of &meraUzed fuzetious. IN 17o.Uawirip, cc,
Ss sumad: (a.) V (x, and (b) that the nualogue of allipticit*y cewilt1w,
(X, :1, 0 Vor aU 1--pEOP X'c G U to is fulfilled& Wh-WI cl 0., the equation
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ACCESSXON NRs AP4236709 8/0020/64/156/002/0243/0246
IAMHORs Vishik, M. X.; Eskin,'G. X.
I
ITITLE% Singular elliptic equations and systems of.variable series
SOURCE: AN SSSR; DoklMy*;-V. '156, no. 2, 1964, 243-246
TOPIC TWS: closed manifold, singular elliptic equation, variable series, space
functions boundary value, finite region, complementary potentialp Fourier conversion
ABSTRACT: The authors studied equations representing a closed manifold and the
apace function of a variable series of evenness. The boundary value problem for
singular elliptic equations in a finite region was examined, where the finite region
was equated by 0 C An having a smooth boundary, Problems with complementary
potentials were investigated using
L (x'))- f(x)t z Coo x1cr (8)
a(X) (U(X) +
Orig. art. has: 12 equations and 5 theorems,
Card 112

-A=SSION M~.-'AP4067'60
ASSOCIATIONt none
SUBMMEDs Ilian64 DATS ACQ: 037un64 ENCL: 00
SUB CODS: MA NO RV SOVs 003 t 000
Card. 2/2
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eqjation or oraer /n. V S r RMUSL n rl 0 r1 CT I "I
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19, (~K ~~) ~ a ~ - = T . . !~,. ~ 4 1, 1 ~ - - - T, , , , , .-~ I %. m i ( 2 )

VISqIK, M,I.; ESKIN' G.I.
Singular elliptic equatiGns and sy5tems of variable order.
Dokl. AN 559R 156 no, 21243-246 I/w 164. (~ffRA 17:7)
1. Prodstavleno akademikom I.G.Petrovskim.

VISHIK, M.I. (Moskva); ESKIN, G.I. (Vororiczh)
Equations in ccnvoI,..:t-4cr-s in a boi;rded regicn in 5paces vit,-~
weight norms. Mat. sbor.. 69 no-1:65-110 Ja 166.
(MIRA 19: 111
1. Submitted July 12, 1965.

_YIT~~.K, M.I.j ESKIN, G.I.
Convolution equations in a bounded region. Usp. mat. nauk 20 no-3:
89-152 MY-Je 165. (MIRA 18t6)

VISHIK, M.I.; NOVIKOV, S.P.; POSTNIKOV, M.M.
Meetings of the Leningmd Mathematical Society. IIOPO MP.*,.*
19 no.6s229-236 N-D 164 NIRA 18:2)
Gorkly Mathematica.1 Seminar on Homotopic Topolo". ]bid. s237-238

0 AGRANOVIGH, M.S.; VISHIK, 14.1.
Elliptic problems w-',.h a Fi-ram--ter &nd Trarabal!c prob2e--- of
a general t7pe. Usp. nat. nauk 19 no.3:531-161 !.17-Je 164.
(MIRA 17:10)

z r a G

W~
VISHIK, M.I.; Mflfi, Gol.
General borundar-I value problem!-, wft,~, bour."48--y
conditionn. Do'KI. AN SMi 158 rio.1:25-28 8-0 164
(KITa 17:8)
1. Pr(idstavltmc akademikom T.G. Pet-rovskim.

VISHIK, M.I.; FSKIN, G.I.
Boundary value problems for general singular equations In a bounded
region. Dokl. All SS-SR 155 no.1:24-27 Mr 164. (MIRA 17:4)
1. Predstavleno akademikom I.G.PetrovsVim.

VISHIKP M.I.
......
Quaal-linear-strongly elliptic syste.xs.of differential equations
of divergent fom. Trudy Pfb~k. mat. ob-va l2tl25-184 163.
(MM 16: U)

VISHIK, M.I. (Moskva)
Solvabi~ity of boundary value problems for quasi-linear parabolic
equations of higher order. Mat. sbor. 59 (dop.):289-325 162.
(MIRA 16:6)
(Boundary value problems)
(Differential equations, Linear)

VIS!H~~~I.
Solvability of the first boundary value problem for quasi-Aimar
equations with rapidly increasing coefficients in Or3icz classes.
Dokl. AN SSSR 151 no.4:758-761 Ag 163. (HIRA 16-8)
1. Predstavleno akademikam I.G.Petrovskim.
(Boundary value problems) -(U*ar equations)

VISM4, M.I.
Solvability of the first boundary value problem for certain
nonlinear elliptic systems of differential equations Trudy
NEI no.42:3-17 162. ~MIRA 16:7)
(Boundary value problems) (Differential equations)

AGRANOVICH, M.S.;~VISHIKI M.I.-
Elliptic bou6dao value problems depending an a parameter. DAL
AN SSSR 149 no.2:223-226 Mr 063. (MM 16:3)
1. Predstavleno akademikom I*G.Patrovskim.
(Boundary value problems)

VISHIKX M.I.
qwi-linsar elliptic systems of equations containing zubordirzated
terms. Dokl.AN SSSR I" no.1 s13--16 W 162. WRA 15:5)
1. Predstavleno akademikom S.L.Sobolevym,
(Differential equations, Linear)

VISHIK, 411. 1.
Boundary value 9rcble,:-.!3 for qi-,asi-linear paraboli c a-ysT,:-_-x
of equations and Gattchyls problem for hyperbolic*equations.
Dok-1. All SSSR 140 no.5:998-1001 0 161. (MIRA 15~2)
1. Predstavleno akademikom S.L.Sobolevym.
(Boundary value problems)
(Differential equations)

AUTHOR: Vishik, 14. 1.
S/020/62/144/001/001/024
11112/11102
TITLE: Quasilinear elliptic systems of equations containing
subordinated terms
PERIODICAL: Akademiya nauk SSSR. Doklady, v. 144, no. 1, 1962, 13-16
TEXT: The boundary value problem
L(u) a 7-- (-, P, DaA (x,D^fu) - h, x8G, (1)
jai 'em
U!, - Vo( X1 ... j1puir - %.)(X)), x1er, 16JI-em - 1 (2)
11
is considered under certain conditions concerning A = (A1 ...,A where
a a Cx
a . (a .... an ). It is demonstrated that the boundary value problem (1)
and (2) can be solved unambiguously. This result is generalized for
operator3 M(u) with subordinatdd terms.
PRESENTED: December 29, 1961, by S. L. Sobolev, Academician
SUBMITTED: December 26, 1961
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20,109
-3 ~-oO S/02o/61/140/005/004/022
0111/C222
AUTHOR: Vishik, Mj.
TITLE: Boundary value problems for quasilinear parabolic systems
of equations and Cauchy's problem for hyperbolic
equatione
PERIODICAM Akademiya nauk SSSR. Doklady, v. 140, no. 5, 1961,
998-1001
TEXT: The author investigates the mixed boundary value problem for
quasilinear parabolic systems of equations;
MU ~ ~)u+ (-1) D `~u + L(u)=h, (1)
at 1.41,1 X I-< ,,~(x, t, Dr) `~~ -j t
ult, - Y(x), u1r- (f(`t) j..*9 D(OuIr -fCo (xl,t), (2)
where x (X19 ... YX n), U . (UIj... 9 0)9 h - (h19 .... hN
D./ = aI&L ~X,,t ... 9 x ,,, , ICL xf
Card 1/4~ 1 14 6 r, I (J) M-1

2Y109
S/020/61/140/005/004/022
Boundary value problems for . . . C111/C222
F- boundary of a region G, 0 -, t < T, and L(u) for all t, O-Cr < T.
is a strongly elliptic operator in the sense of (Ref. 1, M~J. Vishik,
DAN 138, no. 3, 1961). Here in contrary to (Ref.1) instead of the
defiaiteness of L(u) the author demands only the semiboundedness of
its variation. It is shown (theorem 1) that if (2) and h(x,t) satisfy
certain conditions of smoothness then the problem (1), (2) has a uni-
que solution In a corresponding space,
In the second part of the paper the author considers the hyperbolic
equaLion
a(u) = h (12)
of the order m+1 being normal in the sense of J. G. Petrovskiy
(Ref.4, Matm sborn., 2(44), 50937)), where the coefficient of
M+1u/ 9t M+1 equals one. Choosing the initial conditions
u1t=O , Of ... I ~mu/at MIt-0 , 0 (13)
then, according to Petrovskiy (Ref-4) it suffices to prove the existen-
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S/0-0/61/140/005/004/022
Boundary value problems for . . . C111/C222
ce of a solution of (12), (13) for periodic boundary conditions
Dpulx=o = D i~~j x=2JY' 1 P 1 m - (14)
The existence of a solution of (12), (13), (14) is proved with the
aid of a method similar to the method of Galerkin by seeking the
approximate solution u r(x,t ) =Y- C, r(t) z1 (x), where a6 (-~-J' -, , , - -vn),
1-6 1 -4- r, r = 1, 2, . ~ ~ , zOC(x) - exp i (,~ x 1 ++ oL nxn The coeffi-
cients cotr are determined so that it holds
14 t)a(ur b(z X(t)dl(t))] =Ecf(t)h(x,t), b(zl(x)d (15)
where b(u) Da(u)/~D tu, CP(t) = e- Xt _e-,A T(O _, t-I- T), >~ -- suffi-
ciently large number, d~t) -- smooth functions which for t = 0
satisfy the conditions (13). Fqr Cejbr(t), the condition (15) leads
to a boundary value problem having a unique solution for sufficiently
large A (lemma 2). It is shown that the sequence of approximations
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2M9
S/020/61/140/005/004/022
Boundary value problems for C111/C222
u (X,t) obtained in this way converges to the solution u(x,t) of
(52) - (14) (theorem 2).
There are 4 Soviet-bloc and 1 non-Soviet-bloc reference, The roferance
to the English-language publication reads as follows: J, Leray,
Lectures on hyperbolic equations with variable coefficients, Princeton.
1952.
PRESENTED; May 22, 1961, by S. L. 3obolev, Academician
SUBMITTED: May !6, 1961
Cara 4/4

21553
Iro L) 8/02o/61/137/003/001/030
CIII/C222
AUTHORt Vishikp' X.I.
TITLEt Solution to a system of quasilinear equations having a
divergent form under periodic boundary 6onditions
PERIODICAM Akademii nauk SSSR. Doklady, Vol-137,no-3,1961, 502-505
TEXT: The author proves theorems of existence and un4queness for
solutions of systems of quasilinear equations the principal terms of
which are written in a divergent form. The given boundary conditions
are periodic.
Given the system
2: (-1)1' D A (xtugD u)+B(x,u,D u) - h(x),
1CO-9-m, JPI*m OL cc A 13
lat W1 ZX 0~1, C&
where x (X 3) 9xi
lto**#Xn n O,O, fi is a monotone function.
Condition P. For a certaion p >1, f 2 and f 3 it holds
13 (U)S V( 7- 11 AOC(x, U, ~u) I P+ R B p :!~ f2(11(u))+f3 (l 2(u))' (10)
I I fam
where 11 kp Iv where for JKJ. iu and a certain p, >I ;Lt holds
Gard 3/5

21553
5/020/61/137/003/001/030
Solution to a system... 0111/C222
11 DWI lip, !~ f4(11(u))+f5(l 2(u))-
Generally (3) follows from R and P.
Theorem lt If R9P and (3) are satisfied then for every h(x)C- t (cf.Ref.2:
X.I.Vishik, DAN 1340 no-4 (1960); Ref-59 F.Lax, Com.Pure and A551.Math,
8, 615 (1'955)1 Ref.6s J.L.Lions, Acta Math., 98, 1-2:(13 (1955 there
exists at least one solution u(x) of the problem (1) 2). Here the norms
at the left-hand of (10) and (101) are finite, and for every
v ew(m )(G), 1/q+l/p - 1 it holds the relation
q
K(u,v) - (h,v) (11)
(cf.notation (Ref-5))-
The proof is performed by a limiting process in (5).
Theorem 2: If for a certainip.;"I there holds the estimation
RZA x,k+ NIB( Va x. Aplr--f6(12(u))+f7( 11(u)) (18)
for arbitrary w6 P 2m then the conatructed solution u has-finite norms
of the derivatives given at the left-hand sida.
Theorem 3 asserts that the constructed solution is unique in the cliws
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S/020/61/137/003/001/030
Solution to a system... 0111/0222
of functions with finite forms 11(u) and 12(u)'
The author mentions S.L.Sobolev. There are 8 Soviot-lboo and 2 nou-
Soviet-bloo references. The reference to the English-lanau4e publication
reads as followes P.Lax, Com. Pare apd Appl.Xath.t 8, 615 (1955).
ASSOCIATIONsMookovski onergetioho8kiy inatitut (I(oscow Power Engineering
lnxtitutd~
PRIBINTEDe Ootob,4r 28p 1960, by X.G.Petrovskiyp Aosdomioian
SUMaTTZDi October 27, 1960
Card

AWXSANDROY, P.S.; VIMIX, )LI,,; SAULIYEV, V.X.; RLISGOLITS, L.M.
Lazar' Aronovich Llusternik; on his 60th birthday* Uspe =at*
nauk 15 'no.2:215-230 W-AP 160- (KIRA 130)
(Liusternik, Lazar' Aronovich, 1899-)

VISHIK. M. I.; LTUSTFJWIK, Im A.
Asymptotic behavior of solutions to linear differential equations
with large and rapidly varying coefficients and boundary condi-
tiona. Uspe.mat. nauk 15 no.4:27-95 Jl-Ag 160.
(MIRA 13:9)
(Differential equations, Linear)

S/042/60/015/04/01/007
C111/G222
AUTHORSs Vishik, 1A.I., and Lyusternik, L.A.
_T"- 16
TITLEt Asymptutic Behavior of the Solutions of Linear Differential Equations
With Large or Quickly Vitriable Coefficients and Boundary Conditions
PERIODICALs Uspekbi matematicbeskikh nauk, 1960, Vol. 15, No. 4,
PP. 27 - 95
TEM._ The---Suthor-a-consider.boundary value problems-depending on the pax&-
meter C , where for C-310 the coefficients of the equations or of the
boundary conditions tend to co. In chapter I the authors investigate equations
defined in-the whole space, where the coefficients are finite in a subdomain Q~
while in the complement 'q for 6 ---!,0 they increase infinitely. In chapter II
the authors consider problems in which the coefficients of the equation in-
crease unboundedly in an infinitely thin layer Tc around a "singular manifold
I . In chapter III the authors investigate boundary value problems in a domain
Q on the boundary F of which there is an oscillation (problems of the type
of the Skin - effect). In all these problems the solution u = u(xc )
di:tinguisbes by the fact that it has a singularity for 6 - 0 . Similar
Ca d 1/2

Asymptotic Behavior of the Solutions of Linear S/042J60/015/04/01/007
Differential Equations With Large or quickly C111/C222
Variable Coefficients and Boundary CondItions
quest-Ions were treated by the authors in (Ref. 2) and now the methods de-
veloped in (Ref. 2) are applied again. Besides the authors use arrangements
of M.A. Leontovich (Ref- 4) and O.A. Oleyniki(Ref. 8,9) The principal re-
sults of the paper are announced in (Ref. 3). The authors mention A.L.
Golldenveyzer.
There are 23 references : 21 Soviet and 2 American.
SUBMITTED: December 1, 1959
Card 2/2

-VISHIK. X.I.; LYUSWR 1K, L.A.
Initial jump for nonlinear differential equations containing a
small parameter. Dokl.AN SSSR 132 no,6:1242-1245 Je 160.
(NIRL 13:6)
1. Chlen-korrespondent AN SSSR (for Lyusternik)p
Rifferential equations)

VISHIK M.I.- LYUSTERNIK, L.A.
Solution of some perturbation problems in the case of matrices
and self-adjoint and non-ealf-adjoint differential equations.
UBP-mat-naulc 15 no-3:3-80 MY-Je 160. OAMA 13:10)
(Differential equations)

66399
_bs 0 0 3/020/60/134/004/025/036XX
CIII/C333
AUTRORj Vishik# X.I.
TITLEt On the Solubility of the Firit Boundary Value Problem for Non-
linear Systems of'Elliptic Differential Equations
,v
PERIODICALs Doklady Akademii nauk SSSR, 1960, Vol. 134# No- 4,
PP-749 - T52
TXXTs Let the system
0) L(u)u U + B U )Uh
dx (X, U) 7- + C(x,u
(Aik(x'u) t I X.
ilk-I k i-I i
where A,B,C are matrices of order N ; u (Uig ... quN h - h(x)
- (h 1,...,hd ; xe D Fiv the boundary of D. Assume that the boundary
condition is
(2) U11- - 0
Suppositions : I. For all w(x)f u(x)6 Co 1)(D) (space of all continuously
Card 1/ 5

86399
On th,% Solubility of the First Boundary Value Problem 3/02 60/134/004/025/OX61X
for Nonlinear Systems of Zlliptic Differential C111YC333
Equations
differentiable functions satisfying (2)) let
(3) (L(,w)u,u) Z 11(wl u,u) Z 7 (Aik(xlw) 3 u ~U
u
+ u , U) + (C(x,w)u,u);, /J u u 02 uu. y 2
7 X.,
X, 4. 7xi Xi It2
where c2> 0 does not depend on u,w .
11. For every matrix A B there exist inver"Able "majoring" matrices
IV 1.0 ik' i
Aik' Bi so that
(4) 1 'A" -' (Xtu)l < M , I ^B' -'(x,u)l < X 'A' -'kA X 'B -1*B #1 0 (p:~2) and all uE C(l)(D) it is
0
Card 2/5

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On the Solubility of the First Boundary Value Problem 3/02 60/134/004/025/036XX
for Nonlinear Systems of Elliptic Differential C111YC333
Equations
(5) z ik (x'u) '?Xu + Bi( X'u) '? u + 11 C(X,U)U 11 Opp<
k Opp orp