Mathematical Notes, Vol. 63, No. 6, 1998

S m o o t h n e s s o f G e n e r a l i z e d Solutions o f B o u n d a r y V a l u e P r o b l e m s for C e r t a i n D e g e n e r a t e N o n l i n e a r Ordinary Differential E q u a t i o n s Yu. D. Salmanov

UDC 517.927.4

ABSTRACT. The variational method is applied to the study of a boundary value problem of the first kind for a class of nonlinear ordinary differential equations of order 2r with strong degeneracy at the endpoints of the interval (a, b). An inequality is obtained in which the norm of the solution U of the problem under study in the sense of W~,a(a , b) is estimated from above by the norms of the given functions ~ ( z ) and F ( z ) . KEY WORDS: variational method, boundary value problem of the first kind, nonlinear ordinary differential equation, Sobolev space, coerciveness inequality.

We study the differential properties of generalized solutions up to the boundary in relation to the differential properties of the coefficients of the equation (Theorem 3). We shall prove an inequality of the coerciveness type for the solution (see inequalities (42), (43)). Let Wp, a = W~,~.(a, b) (A = (a, b)) be the space of functions f(x) for which the finite norm has a meaning (see [1, Chap. 10]):

Ilfllw..(,,) = Ilfll,.,, + -7lip.,, < +co' where r > 1 is an integer, a , p ( 1 < p < +co) are real numbers, II 9 II,,~ = II 9 p

fa),

= p(x) = min{x

f(k) = f(k)(z ) = -

a, b - x}

Let 0
d~I(=) ~ ,

k < r,

Vx e A = (a, b).

1

(1)

P

and so is a natural n u m b e r such that

1 so-l
r ~
(2)

Under conditions (1) and (2), let us define the subspace (see [1, Chap. 10])

Wo = l~rp,a = {f E W;,,~(a, b): f(k)(a) = f(k)(b) = 0, k = 0, 1 , . . . , so - 1}. Let us choose a function (I)(x) belonging to W;,•(a, b) and by W,~ denote the set of functions f(x) belonging to W;,~(a, b) such that f ( x ) - ~(z) E Wo. Under conditions (1) and (2), the following r

1e

inequalities are valid (see [2-4]): f(k) I .~-ggT~,-kII t~

lip,&

LLoo+,_~ II

p,A

<-

co f - ~ P

Vf e W'o, k<__r, p,A

___c, Ilfllw;.o(A)

vf e w;,.(A),

J" k, / so,

k_
/%

k > so, k < So,

k_
(3) (4)

Translated from Matemalicheskie Zamelki, Vol. 63, No. 6, pp. 882-890, June, 1998. Original article submitted September 20, 1995. 0001-4346/98/6356-0777 $20.00

(~1998 Plenum Publishing Corporation

777

To each nonnegative integer k (k < r) we assign a measurable function ak(z) on (a, b) such that C2

p.,~+~-,.)(=)

C3

<_ak(z)<_ p . . ( ~ + ~ - ~ . ) ( = ) '

2<_pk
k=O,1,...,r-1,

pr=p,

(5)

where c2 and c3 are positive constants independent of x E A = (a, b). Let us consider the integral

E(f)

=./b_~ __1 ak(~)lf(k)(x)lp J~

h dx,

/~_
which is obviously finite for f E W~,~(a, b) under conditions (1), (2), and (5). Set / t ( f ) = E ( f ) - ( F , f ) , where (F, f ) is the inner product of the functions F(x) and f ( x ) by A and the function F(x) satisfies

[Ip~+~-~~

< +oo,

1

1

P

q

- + - = 1.

(6)

Under conditions (1), (2), (5), and (6), the functional H ( f ) is bounded below in the class W,. Set

inf /-I(f) : A. fEW,a T h e o r e m 1. Under conditions (1), (2), (5), and (6), there exists a unique/'unction U(x) belonging to W,~ satis[ying the relations

(7)

rain H ( f ) = H(U) = A fEW,~

~b(t~
F(x)v(x)) dx = O

(s)

v f ~ wo.

A functie". U belonging to W,~ and satisfying relation (8) for any v E W0, will be called the generalized solution of the boundary value problem where

L U = ~">A--1)k(akCx)luCk)c=)l'h--2uCk)(~:)) -'-" --/~<_r

ok).

T h e o r e m 2. The solution U of the variational problem (7) belonging to W,~ satiszqes the inequality

IIuII~,~(A) __
(9)

A),

where

70 =

p, min~pk,

II~llw;.~ >__ x, [[~[[wAo(a ) <_ 1, and

c4 is a constant independent of q~ and F. Inequality (9) is proved by using identity (8). T h e o r e m 3. Suppose that conditions (1), (2), and (6) are satisfied, the functions ak(x), where k _< r/2, k = r, have continuous derivatives on the interval (a, b) up to order k inclusive such that s

C5

[a )(x)l < pp~(r+~_flk)+s(z) , 778

xE(a,b),

s
2_
r

k < ~,

pr--p,

(10)

and the function U belonging to W, satistles the relation (a(z) = a,(z) )

f

f

@=)lUC')(=)l"-'uC~)(=)"cr)(~)

+ ~

.,,(=)lVC")(=)l,'~-~vck)(=)vc")(:O - F(=),,(=)) d= = 0

(II)

k_<,-/2

= .(x)lu(*)(.)l,-*v(~)(.)

for all v 9 Wo. Then for the function r

__

the foUowing inequalities hold:

p--1

lip"ell,,,. < c, llVll,,,,-;...(,,), Ao IIf+~+"~176162 < c,(llUIIw;,~ + IIf+~-~176 where

IlUllw;..c~)k

Ao = ~ p - 1, ( minkpk--1,

1,

IIUllw;,o(a)
1 /~

(12)

(13) 1

Pk--+--=l'qk

and c~, c7 are constants independent of U and F , U, respectively.

P r o o f . Let a, a', a" 9 (a, b) be intervals centered at x0 9 (a, b), whose radii are equal to 4-1p(xo),

2-~o(zo), p(xo), respectively, where p(zo) = min{xo - a, b - xo }. For any function v(x) belonging to Wo and finite in a ' , we shall write v(x) 9 Wo(a'). Using (Ii), we obtain (h r 0)

+

A'F(X)v(x)~h" ] dx

~ VA" (~(':)lvc~)C~)?-'vc~)(~))r

=0

(14)

k<,-12

for M1 v 9 Wo(a'), where A ' f ( x ) = A~,/(z) is the sth-order difference of the function f ( x ) at the point x with step h. Let us consider a function ~(x) belonging to C(r+1)(R) and finite in a ' , with support A , C a', possessing the foUowing properties: 1) rt(x) = 1 on a;

2) o _< ,s(*) _< 1, 9 9 a; 3) I,Tck)(,)ff _< ~ ( x ) , k _< ,- + 1. Set (for sufficiently small h ~- O) 1 ~ (z - t)r-lT/(t) A , ( x ) - F(r -- 1) o

sign

dr,

z 9 a'.

We have

(15)

(16)

IIA, II,,~' << h-'(q-l) A,r q/p ,,A," Further,

s IIA8r

j=0 << p

--or

L'

j=o \JA, I o--~(~+-hj~ P/q

(xo)llUllw;,o(~,).

(17)

The second inequality in (17) follows from inequality (10) for k = r, s = 0 (fir = r) and from the obvious

fact that p(x) ~ p(x0) for any x 9 ~' (see [2, p. 101]). 779

It follows from (16), (17) that

IIA.IIp:, << Ihl-~('-%-~q/'(*o)lluIIw;,.(.,). In a similar way (k < r), we obtain

IIA?) llp,~'<< lhl-~(~-%-~'/'(~:o)llUllw;, o (~,,)o

A,(x) for a given h ( h # v(x) = ~}(x)A.(x). Then we obtain

Therefore, the function we can set

x(~',~) =

j(

0) belongs to the class W0(a'); therefore, in relation (14)

As./, q ,~(~) ~-'*1 d~ = Jl(~',~)+ I

6(~',~) + s3(~',~),

(18)

where r--1

ffl(O",3) =-EC~

/~ks~(sX)A!J)(~)~(r-J)(~)d2: ,

j=o

~(~',~)= - ~ ~, ~-(,,~(~)IUC~)(~)I"-~uC~)(x))(A,(~),I(~)) (~)d~, I,_<~/2

The following inequalities ( j < r, r < r + 1) can be easily proved:

IIA?),7(r)llp,:, << X,/r,(a,, s),

llr

<
--r

(19)

p/q (xo)llUllw;.o(~,).

(20)

Now let us estimate the integrals d,(tr', 1), d2(a', 1), dn(a', 1) (s = 1). For the integrals appearing in the expression for J l ( a ' , 1), we have

<< IIr

<< p-~(x0)llVll~.t~,)XX/'(r

1).

(21)

The equality in (21) is written in view of the finiteness of the function t/(z) ; the first inequality is valid on the strength of HSlder's inequality and inequality (5) from [2, p. 115] (in what follows, we often use this inequality without mentioning it each time); the last inequality in (21) follows from (19), (20). For the integrals appearing in the expression for J2(a', 1), we have ( k + 1 < + 1 < r)

r/2

A I~, -h (at(x)lU(k)(x)[P*-2U(k)(x))(Alrl)(t)dxl

<__IlaklU(k) l'*--'ll,~,~,ll(A,'7)(k+')ll,,,~'

<< p-'-~+~*(~o)llUtlwh, o(,,)xl/,(~ ', 1).

(22)

For J3(~', 1), we have IJa(a',l)l = ~ ,

F-~(A,71)dx < IIFIIa,~,II(A,T/)'Hp,~,

<< p-'-~176176 780

', 1).

(23)

In view of (18) (for s = 1) and (21)-(23), we obtain ,

w ; , , . ( . , ) + IlPr+a-'~

'

- S(~').

(24)

t_
Hence, using the fact that if(x) = 1 on or, we see that

Replacing (7 by ~i in this inequality, we obtain (25) Let ,r C (a, b) be an arbitrary fixed neighborhood of the point x0 E (a, b) such that (7 C *rl C ~r'. Let us prove that tl

. - t i t

(26) !

To this end, consider a system of concentric intervals ,rr,err,..., al, ~rl centered at the point =0 E (a, b) and for which the following strong inclusions axe valid: O" ----OfT C

t

O"r C

"'" C

O'I C

t

OJ

(27)

O"1 -----

Suppose that the following inequalities have been established: ht I1,,,, ~ s(~'),

l-- 1,..., m,

(28)

where m is a natural n u m b e r such that 1 < m < r. Let us show that under these conditions we have the inequality h---~-&-V~-I

<< S ( a ' ) ,

(29)

I qTO'm+l

and hence inequality (26) is also valid. Inequality (28) for l = 1 is proved (see (25)). For each I ( l = 1 , . . . , r), let us consider a function r}t(x) belonging to C(~+I)(R) mad finite in ,~, with support A~t such that 1) w ( x ) = 1 on *t; 2) 0 _< ~,(~) _ 1, 9 e R;

3) [W(k)(=)12 < crtt(~), k < r + 1, where c is a constant independent of the point x E R. In (15) let us replace ,7(x), q, q' by rtl(x), qt, cry, respectively. Now, in (18) let us set s = 1 = m + 1 and estimate the integrals di(a~, 1), i -- 1,2, 3. In view of (19) and (27), (28), the integrals appearing in the expression for Jl(~r~, 1) can be expressed as A~h(AIDr/(~-D)dx _< A ' - l ~ b

,

At-hk

<<(I[I ~

I

~

q,^,,

q,crt_tX1/p(~ l) << S(o")xl/p(o'~, t).

(30)

781

Let us estimate the integrals appearing in the expression for J2(a~, l). First, consider the case I < k:

~(ak(x)lUr162

E <

p-~-"+#*(~o)

h

p,,
,

IL,.,r

w,, (31)

The equality in the chain of relations (31) is valid in view of the finitenessof 7/t(x); the firstinequality is obvious; the second one follows from inequality (10) and the fact that p(x) ,~ p(xo) for any x E a~ [2, p. 101]; the last inequality follows from (3) and (19). Let k < l < r. T h e n

E << I[(aklU(t)[Ph-2U(k))(k)l[.~,~,~[l(Atrlt)(t)[[p~:,~ k

<< ~ :-"(~+~-#')-~+"(=o)ll(Iu(~)"-=u (~)'("~

x~/"~' O.

(32)

n~O

The last inequality (32) follows from inequalities (10), (19), and also from inequalities (1) from [2, p. 101]. Now let us obtain an estimate for the integral (k) ph-2

llE.ll-ll(Iu I To do this,let us estimate the

Lq~ (a~)-norm

u

(k) (.)

) [l,,,.,-

of each summand appearing in the following relations"

2j-s E~i = IU<')(-)I"-~U+')(2:_~CksU(k'l'2J-s)(x)(u(k+l)(T,))s(u(k)(x)) j-1

)

-F E dksU(k+'(i-s))(x)(U(t+2)(z))S(U(k)(x))2J-s

'

2 _< 2j < k,

(33)

s----I /2(/-1)

E21-I = [U(t)(x)l p'-2j ( ~

c~.U(k+21-l-~)(x)(U(t+l)(x))'(U(t)(x)) 2(j-1)-~

j--1

X

~-E dksU(k+2(J-a)-I)(~)(U(k+2)(x))a(V(k)(x))2(j-l)-') '

1 <_2j -- 1 <_k,

(34)

J

s-I

where ck,, dts, c~s, d~s are constants whose exact values are not essential. Let us use inequalities (I) from [2, p. 101] for estimating the summands of the first sum in (33):

IIiv<~)l,,-~-.(u
, :+~_/~,+,j_.

(35)

qie ,0"~

where Vl = (Pk - 2 - s)(v + a - ~k) + s(r + a -/3k+~) + r + a - / ~ k + 2 j - s 9 Let s = O, Pk > 2. Applying HSlder's inequality with exponents AI = pk/qk(Pk - 2), ,~2 = Pk/qk, we obtain U(k) Q <- pr+,~-~

p~-2 U(k+2J) [ , :+'~-~-----~J I

U p~-I << " NWL o(~D"

The last inequaiity in (36) is a consequence of (3). If Pk = 2, then qt = 2. Therefore, U(k+2J)

]l

Q= II pr+(~--~k--~2jlI2,c; 782

<<

IlUllw;,<<).

(38)

Let s > 1. I f p k - 2 - s = 0 ,

then p ~ - 2 > l .

Q<

Hence

p~+--~--~-~_~+~~,.; f+~-~+'~-" p~,~; <<

(37)

wh,~

If pk -- s -- 2 > 0, then, using Hhlder's inequality with exponents A~ =

Pk

q~,(pk - 2 - s)'

A2 = p~

A3 = P---k-~,

sq~'

qk

we obtain

q-< [[p-+~-p, p,,~;

f+--g=~'-~+=p,,~; f+=-p,+,i-,

e,,~; <<

wh.~

In view of (35)-(38), we obtain

IIIU<*>l"-'-'(~r)'U<~+2J-'~ll,,,.; ~ f'(=o)llUIl~,~l=(.;).

(30)

I n a s i m i l a r way, we e s t i m a t e t h e s u m m a n d s o f t h e second s u m i n (33) as w e l l as t h e s u m m a n d s o f the

first a n d the second s u m in (34) a n d E0. For them, we have the same estimates as (39) except t h a t the exponents vl m u s t be replaced by v2, va, v4, and vs, respectively: //2 = (Pk -- 2 - s)Cr + a -- ~ , ) + , ( r + a - ilk+2) + r + ~ -- ~k+2Cj-o),

//4 = ( P k - - 2 - - S ) ( T "~ (~ - - ~ k ) "~- s i r

"3L ~ - - ~ k + 2 )

J r r J~- ol - - ~ k + 2 ( j - - s ) - - l ,

//s = (P~ - 1 ) ( r -t- a - / ~ k ) .

It fonows from the definition of/~k (see (4)) that /~k, --/~k" > k' - k"

Vk', k",

k' < k" < r.

Therefore, -pk(r + a -/~)

- k + n + / / j > - r - a +/~k - k,

j = 1, 2, 3, 4.

Now, in view of (31), (32), and (39) for J2(a;, l), we obtain the following estimate:

IJ2(O.t,,l)[<< ~ k_<~/2

p-~-~+~'-~(=o)llUIl~'oXV'(~;,z).

It now remains to find an estimate for the integral

(40)

J3(a;,/):

IJ~( OJt, l)l = f ~ ; /---~--A,r/, x~r d~ = ~ F~-~A(Atzl,)dx[ < ]lr]lq,.;ll(A,Tll)(t)llp,.; <
l).

(41)

It follows from relations (18) for a' = a~, T/(x) -- ~},(x), (30), (40), (41) t h a t

/2'

q,cr~ 783

Since rit(x ) = 1 on or, we now have

I} h'x' llII.,., <
l=m+l,

l
Inequality (29) and hence inequality (26) are proved. In view of inequalities (8), (9) from [2, p. 116], it follows from inequality (26) that (see (24) for S(a'))

Ih//')[[,.., < p-'-~(~0) ( liutl~'~~.

+ ~

k
,,J'u''p'-',wh,.(..,) + IIp~+~-'~

Let us multiply this inequality by pr+a-lll~ where 10 = maxk qk, and then raise it to the qth power and integrate over x0 E A = (a, b). If we raise the result obtained to the (1/q)th power and apply Troisfs theorem to each term, then we obtain the inequality (see [2, p. 103])

IIP'+~+('~176

p--I

< IIV'llw;,.(~) + ~

k
,,"u .w;,,=(a) ''p'-'

+

..f+~-'oF,,a. II II

Since 2 < pk < p, the last inequality implies inequality (13). In a similar way, using (20), we obtain inequality (12). [] Corollary.

Under the assumptions of Theorem 3, the following inequalities hold:

IIf+~+('~

q ~0 .,_r+=-.o~.q lip ~,)3~,,, << II'~lJw;..(A) + ,p .,. , , , A , ,_r+a-,0 --,,q,A, w,q ~x0/p9 << IIP'+~-'~ + (II~II~;,.(A) + I/"

Inequality (42) follows from (9), (12) and inequality (43) from (9), (13).

(42)

(43)

[]

References l. S. M. Nikoltskii, Approximation of Functions of Several Variables and the Embedding Theorem [in Russian], 2nd edition, Nauka, Moscow (1977). 2. P.I. Lizorkin and S. M. Nikol~skii, "The coercive property of elliptic equations with degeneracy. The variational method, ~ Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 157, 90-118 (1981). 3. Yu. D. Salmanov, "Weighted function classes with degeneracy on manifolds of any dimension,'* Dok/. Akad. Nauk SSSR [Soviet Math./)ok/.], 294, No. 3, 539-542 (1987). 4. Yu. D. Salmanov, "Traces of functions from weighted classes on manifolds and inverse embedding theorems, n Dokl. Akad. Nauk SSSR [Soviet Math. DokL], 319, No. 4, 823-826 (1991). N. TusI AZERBAIDZHAN STATE UNIVERSITY Translated by N. K. Kulman

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