SCIENTIFIC ABSTRACT URALTSEVA, N.N. - URANCVA, L.A.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001858010020-6
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: April 3, 2001
• Sequence Number: 20
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

87391
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On the Variation Problem and Quasilinear Elliptic Equations With
Mt;ltiple Independent Variables
L-1 (u) =_ ajj(x,u,uxt) UX4 + a(X,U,ux,,) = 0
assume that it belongs to the class 0 3 and satisfies
(2) u1, = 'P(s) .
For aij(xspu9Pk), a(x U Pk)4E 01(dl x El X Ed let (B) and
(7) ~ ( 1u 1) (P2+ 1),12-1 0, a j M ju 1) (p2+1)m/2-1
be satisfied for 2 ij(X'U'Pk) 1i Jj u
1. Then the author estimates %x I I
by max I uJand J((jC , if the oscillation of u(x) is small in
a 2,o
and S belongs to C2,o'
Theorem 2: If the conditions of theorem 1 are satisfied-except
those for S and then max I u I is estimated by max I u I for
every (V C 0, A X;
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On the Variation Problem and quasilinear Elliptic Equations With
Multiple Independent Variables
Theorem 3s Modification of theorem 1 under renunciation of the small
oscillation of u(x).
Theorem 4 and 5 give similar statements on the estimations of the
norms of solutions for the equation
ra
(4) M, (u) -E -,~- (a,(x,u,ux,,)) + a(x,u,ux 0
xi K
where in theorem 4 the author assumes that
(9) ai (x'u I Pk) Pi >r V ( 1u I ) Pm' P >> 1-
� 2. Theorem 6 is the statement of existence for the problem
(10) Mm (u) -C Mj L u 3 + (1- -U-) M0(u) = 0 , u
where
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On the Variation Problem and Quasilinear Elliptic Equations With
Multiple Independent Variables
PO - F 0, Fo(x,l u2 + 1 ) m/2+, -7,
M0(U) xi Uxi U 1,Ux X,
Theorem 7: For (3) let (B) and (7) be satisfied-for n = 2, where
m = 2 is assumed without restriction of generality. Let
a(x 'u'Pk) 1u 1) (p2+ 1)1- F-~ , E > 0 be instead of (6).
Then the problem L., (u) =- rLl(u) + (1--u)( A u - U) = 0,
u 1, = -u ~(s) possesses at least one solution u(x,"C,) from
C2,cK (,fl ) /-, C3 9 -x (~ 1) for all -U t [ 0, 1 ] , if the values U(X, r
are uniformly bounded for all such possible solutions u(x,'C ), The
functions aijo a must be belong to C T 6 C 2, o ' S -"~- C 2,
A homeomorphic to the circle.
� 3. The variation problem
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87391
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On the Variation Problem and Quasilinear Elliptic Equations With
Multiple Independent Variables
(1) inf I(u) = inf S F(x,u,u ) dx , x = x1,...,, X
.XL XK n
is considered under the condition (2). Assume that F(x,u,p,)
has the order of growth m > 1 in p and that-every differentiation of
F to pkreduce this order at least by 1, while the order does not
increase by differentiation with respect to xnand u. Let
F(x, u'Pk) Z vl(lul) Pm
m-2
(xvu~Pk) ~i Sj _Z Y 2(1 ul)(p2+1) 2 .71a
F (x9u9Pk) Pi Y Oul) pM , p >>
Theorem 8: Let u be a generalized solution from W of the
"conditional" variation problem (1) - (2), i. e. Rf the problem
completed by the condition that all comparison functions do not
exceed a certain constant: M I max Jul. The solution u belongs
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On the Variation Problem and Quasilinear Elliptic Equations With
Multiple Independent Variables
to C0,0( '(dl), if F C1 and if the conditions
(~-(Iul)pm P (X'u,
(12) Pk) Pi 2_: V ( lu 1) PM I P>>
Fu (x'u I Pk) lu I ) 'D"
are satisfied. Under the same assumptions for F every bounded
function u 6 W1 m (,Q), for which ~ I(u) - 0, belongs to C 099<
If A satisfies the condition (A) and if CPE C, , then u C- Co'oe
Theorem 9. Under the conditions for F formulated at the beginning
of � 3 every bounded generalized solution u(x) of the variation
problem (1) - (2) from the class W1 (SL) belongs to C "< (A ), if
F E C , kv> 3 and A I(u) . I(u+ M) - I(u) > 0 for 4ery sufficient-
k,K "?
ly small local variation ~(x). If, however, SE C1,cK (P(C- C1,0(
2 4 1 ='- k, then u E C1,0( 0) n Ck, c~
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On the Variation Problem and Quasilinear Elliptic Equations With
Multiple Independent Variables
Finally the author gives two lemmata generalizing the lemma due to
E. de Giorgi (Ref-4)-
S. N. Bernshteyn is mentioned by the author.
There are 4 references: 2 Soviet, 1 Italian and 1 American.
[Abstracter's note: (Ref.1) is the book of C., Mirandas Partial
Differential Equations of Elliptic Type :1 -
ASSOCIATION: Leningradskiy gosudarstvennyy universitet imeni A. -A.
Zhdanova (Leningrad State University imeni A. A.
Zhda.nov)
PRESENTED: June 10, 1960, by V. J. Smirnov, Academician
SUBMITTED: June 2, 1960
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AUTHORSs Ladyzhouskaya, 0. A., Uralitseva, N. N.
TITLE: Quasilinear elliptic equations and variational
problems with several independent variables
PERIODICALt Uspekhi matematicheakikh nauko Y. 16, no. 1, 1961,
19-90
TEXTs The paper is a general lecture which was given on November 24,
1959 on the occasion of the 80th birthday of S. N. Bernshteyn at the
Leningrad Mathematical Society. The new resultEr were represented
in the seminaries of V. J. Smirnov (Leningrad) and J. G. Petrovskiy
(Moscow) at the end of 1959.
Two problems are considered: 1.) the first boundary value problem
for quasilinear elliptic equations
a (Xvutu ) u + A(XjUjU 0
ij 2k xi Xk
i9J-1
1)
and 2.) the differential properties of the generalized solutions
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22407
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Quasilinear elliptic equations ... C Ill/ C 333
U(X n ) of the regular variational problem concerning the
sini;u*a**o,fx
IN) F(X,U,U Xk ) dxI#o. din
under the condition U/S - %P(S).
Let be a bounded domain of the x xn) in the Euclidean
En; -- strictly interior subdomain of Cljo(fj ) the set of
all functions u(x) which are continuous with respect to xk in the
open R together with the 1 first derivatives; lot
4
1U. I C (.f,) -2 max li)lu(X)
1'o KUOXF-A
be the norm. Let C (ft) be the set of all functions from C
for which 1 POG l'o
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Quasilinear elliptic equations ... C III C 333
ID'u(x+h) - Dlu "x) Q4 l
max D u
xf-I'th E: A h
hi > 0
04
is bounded. The norm is; 1u1C 1p( (41) - lulc 1,0 (j1)+ 6 D'u. Let CO(Q
be the set of all functions continuous in ft Jul C A . max lu(x)l
1 01 0 X F- f L
Let WM(IR) and Wm (A) be defined as usual (see V. J. Smirnov (Ref.2:
Kurs vysshey matematiki ~Course in higher mathematics I t. IV, M.,
Fizmatgiz, 1959)). Mf Jul x)j for u E Wm-(fl) is defined to be vrai
~t Ju(x)J. Let Dl(" be the class of the functions u(x) which in
possess 1 - 1 derivatives with respect to xkt and for which the
derivatives Dl-lu possess a differential in every point of fl-Let
01(Q) be the class of the v(ylt.*49 y.) e D,(Q)t the 1-th derivatives
of which are bounded in every bounded domain of the y,, .... ym.
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Quasilinear elliptic equations ... C 111/ C 333
Let 0 (1) be the class of the functions measurable and
bounded in every finite domain of the y1, ... P Ym. The statement
11 the norm 1-1 is estimated by the data of the problem" zeans that
the estimation is possible by the constants which occur in the
conditions which are fulfilled by the problem. tL,(jul) denotes
positive nondecreasing and vk(Jul) positive nonincreasing functions
of ju I defined on LO , ODDand finite for all finite Jul. The state-
ment "the function f(x,, ... 9 X 1, ul PrI 1 .... Pd, XL% has the
order of growth 15-~ m in p 2 says that max
Itp k 1 f (x,u - Pk)
2 m/2 I Xe SL
&'OUMP +1) The boundary S possesses the property (A), if
there are a > 0, 0 < 0 1 such that for every sphere Ky with
center on S and radius it holds
mes EK (3) n (3) mes. K(5
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Quasilinear elliptic equations ... 0 111/ C 333
S belongs to C 1, t.". 6,- , 0, if it can be covered by a finite number
of open pieces, equations of which belong to C1,CA.*
Theorem I. Let u(x) be a bounded generalized solution of
NJ (u) = a (&,(X,U,U )) + a(x,ugu . 0 (29)
'9 xi Xk xk
i. e. u GW (fQ,juj.!:-~ I and u(x) is assused to satisfy the.inequality
~'[ai(xfuqu 'k )%i- a(xPu#U Xk dx 0 (30)
A
for arbitrary 9 (x) G W021 (SI). Let furthermore Mx I ux K1,
ai(x'u'pk) G 01(fL>eEI-x Fn) and &(x,u,pk) e0o(ft.)CE, ~Cx ). Let
Da,(x+Thp v, Y Xk n 2
I 1 7?' ' V1 (1 9 2 (IV T
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Quasilinear elliptic equations ... C 111~ C 333
for v(x) - (l - T) u(x) + % u(x + h), T 6 E Ot 13 , x, x + h
The norm lu IC (11'), o(, > 0, for arbitrary Sj,' C 11- is then
estimated by Jul, 1'o(S1')* If, moreover, S P, C 2,o and 'f(s)
W U/S C- C2,o (a), then I U 1(; 1 , ~;dm )is estimated by luic Ito (n) an d
I'FIC 2,o(S) * If ai and a belong as functions of their arguments to
C1- 1 Ot- (1',> 2) or to C on every compact, while 8 and 'P(s) belong
to C 1,0C then lul, 1, '~a is estimated by 1u1C,',('(L) and by the
data of the problem.
The equation (29) is said to belong to the class (-])), if it satisfies
for arbitrary lit ... P the conditions
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Quasilinear elliptic equations ... C 111/ C 333
a-2 n
2 2 2
VIOUI )(p + 1) ai4 (x u 9'pk) bk 1
a-2 n 2
(p2 + 1) 2 (16)
&(X'Uppk) &2 (1 u P' + t'U3 (1 u (17)
and for large p
ai(3c'U'Pk) Pi >' V1(IUI) pa (a 1>1 1 (31)
2 n 2
where.p . 7- Pi
Jul
Theorem II. For an arbitrary equation (29) of the class the first
boundary value problem with the boundary condition u/ S . (~s) has at
eard, 7/45

22407
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Quasilinear elliptic equations ... C III/ C 333
least one solution in the class C 2p'- UL ) ) ' 'f the N&Xin&
of the absolute values of the solutions u(x"r of the boundary
value problems
N z (u) -= (I - T ) No ku) +TNI (u) - 0, U/S Eo' 11
are uniformly bounded, where M (U)=-2- FU0 (U)u - Po(UOU ) and
0 0 X. 'zi Xk u xk
I
Fo(u,pk) - (1+p 2)z/2 + U2 . The coefficients a i(x'u'Pk) and a(x,u, Pk)
must belong to C 2,&- and Cl,,,-respectively as functions of their
arguments on every compact. The boundary 9 and Y ts) must belong to
C2 , '
Theorem III is a special case of theorem II.
Theorem IV. The propositions of theorem II are maintained, if all
conditions except (31) are satisfied and if moreover the orders
of growth in p of the functions
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Quasilinear elliptic equations ... C 111/ 0 333
,a2ai(XIU$Pk) 2 ai(xtuvpk). and 9a(XIu4 are not greater
apj ~)U 'a U2 *?u
than ni and m - E , where 0 is arbitrary.
Theorem V. Let u(x) e Wal(-C)-) be, one of the generalized dolutions
of the variational problem
inf I(u) - inf S f(XFUPU xk ) dx, dx - dxl..* dxn (2)
U/S :IU q(s) (3)
with the additional condition that all comparison functions are in
the absolute value not greater than- a constant *~~ max Jul . This
solution belongs to C O'ji-D, c/-->0, if S
F (x'u'pk) e C 1 (ft X N E.)
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Quasilinear elliptic equations ... C III/ C 333
fPi kxlulpk) Pi (Jul) pm for p
and
n
+ F (x, u, pk)
p Pi(X'U'pk) u (p0 + 1)
iml
Under the same assumptions on F, every bounded u(x) which
gives I a stationary value belongs to U090(-kn). If, moreover, the
boundary of -CL satisfies the condition kA), and if fks' can be
continued in -CL so that kf (x) e 0 1 then in both cases i t hol ds
u (X) (- C
0 ,
Theorem VI. If only the natural restrictions 1,) - 4.) are satilfied
for F(XU,P K) , then every bounded generalized solution u(x)C-- t~'Cft)
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of the variational-problen (2), (3) belongs to Ck,oL('a)' OL-> 0, if
F(Xtuvpk ) as funct:~on of its arguments belongs to Ck :cw, k ~r 3 on
every compact. If9 moreovery-S e C an d C c'L 2 1!~- 1 -.~ k,
then u(z) belongs to Cl,-e-(~j ) too. As natural restrictions for
F(x,u, Pk) there are denoteds
1.) ViOUMP 2 + 1)m/2 t6 F(x,u,p.) ::5 (!yjuj)(p2 + 1)a/2
2.) The Euler equation for F(x 'u'Pk) is unifornaly elliptic.
((l) is called uniformly elliptic, if (16) holds).
3-) F is sufficiently smooth, where the differentiation of F and of
its partial derivatives with respect to Pk reduces the order of growth
of F and of the derivatives mentioned at least by 1, whila the
differexti&tion iith respect to irk and u does not increaso those orders
of growth.
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Quasilinear elliptic equationa ... C ill/ C 333
For all sufficiently large p it holds
Fpi (X'U'pk) Pi >1 \)2(lul) P'
The given theorems are the main results of the paper; 25 theorems
and 11 lemmata are proved.
The author mentiont V. J. Kazimirov, A. G. Sigalov, A. J. Koshelev,
G. J. Shilova, S. L. Sobolev, V. J. Plotnikov, A. D. Alsksandrov,
A. V. Pogorelov, Ye. P. Soulkin, J. Ya. Bakellman.
There are 16 Soviet-bloc and 25 non-Soviet-bloc references. The
four mQst recent references to English-language.fublications read as
follows: L. Nirenberg, Estimates and existence . solutions of
elliptic equations, Commun. Pure and Appl. Math..2, 30956), 509-
531;
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J. Nash, Continuity of solutions of parabolic and elliptic equations,
Amer. Journ. Math. 809 No. 4 (1958), 931-954; R. Finn aftd D. Cilbarg,
Three-dimensional subsonic flows, and asymptotic estimates for
elliptic partial differential equations, Acts, math. 98 (1957),
265-296; C. B. Morrey, Second order elliptic equations in several
variables and Hblder Continuity, Math. Z. 72 (1959), 146-164.
SUBMITTED: July 12, 1960
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23799
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C Ill/ 0 222
AUTHORSi Ladyzbeuskaya, 0., k. sad Uralytseva, N. X.
'~~-bounded gemerali2ed solu-
TITLF,s Diffely-ential proper,~;es
%ions to n-dimensional quasilinear elliptic
equaVions aud variation probleag
PERIODICALs Akademlya itauk SSSR. Dokladys v.. 158, no. 1, 1961,
29-32
TEXT& The authors investigat-e the equation
(T~,.X,u B.(x,u: U 0
-xi - x
Wkere a ail-d a are measur4ble funatione salisfying
i
a (XI)UPP P+~%(X,U.p P)m
1 (2)
ai(JCsUVP u C u
ar a 1 /C

2.3 '? 9 1)
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Differential propertleR-o,f j 0 Ili/ 0 222
wbere jR and p 2 Let besAS3 the condition
p
j
2 x-2-
!U
1 :2
t P P JD(.
7,
be satisfied incidentally, where (t) 19 monotone nar--inax-eaalng,
(40.(t) -- monotone noza-decreasang, t) and t) > 0, t a. T,
A function u(x) e'-..W' (P) for which
(X".I,u a(x,a,u 'I - 0 (4)
I(U'
x /xi x
holdB for every bounded function Of W1 ia called a generalized
m
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Differential properties of C ill/ C 222
solution of Mu
Lemma iz For the bounded generalized solution u(x).of (1) there hold
+.he inequalities
u ~ 'd-.x