On some limit properties of the solution of the Riquier problem for the iterated Helmholtz equation in the half-plane

ANNALES SOCIETAT1S MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVII (1987) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVII (1987)

Eu g e n iu s z Wa c h n ic k i (Krakôw)

On some limit properties of the solution of the Riquier problem for the iterated Helmholtz equation in the half-plane

1. Let R+ = {(x, y) y > 0 } . We consider in R 2h the following function (1) u ( f , g ; x , y )

1

2k

f(s)[2cr 1 y K ^ c ^ + ^ y K o i c r f t d s - — \g(s)yK0{cr)ds,

where r2 = (x — s)2 + y 2, c is a positive number, K v is the MacDonald function of index v ([4]) and / , g are given functions defined in the set R of all real numbers.

In paper [6] it has been proved that the function и is a solution of the equation

(2⁾ ( A - c 2)2u(x, y) = 0

in R 2+ satisfying the following boundary conditions u(x, ÿ)\y=0 = /( x ) , Au(x, y)\y=0 =g(x), where /, g satisfy some assumptions.

Let us denote by L p, 1 ^ p ^ oo, the class of functions with a finite norm defined as follows

I I / I Ip =

( J |/ ( s r r f s ) 1"’

R

sup ess I/ (s)|

for 1 ^ p < OO, for p = oo.

In the present paper we shall prove a theorem concerning an estimation of М П /, g;y) = ||u(f, g; • , y)-f{o)\\p, M 2(f, g\y) = \\Au(f, g; o, y ) - g ( o)||p according to properties of the functions / , g. Moreover, for the case g = 0 in R we prove an converse theorem (in a sense).

This problem for the biharmonic equation was investigated in paper [2].

2. Let* eu denote a function of the type of the second modulus of

smootheness ([5]). Let Щ L p denote the set of all functions / e Lp for which II/ (o — s) — 2f (o ) + /( o + s)||p ^ со(|s|).

We prove

Theorem 1. I f / еЩ Ь р, g e L p, then (a) u e C ^ i R l ) ,

(b) (A — c2)2u(x, у) = 0 in the domain R+, 00

2 у f со (s)

(c) M x(f, g;y) ^ - c o ( y ) + - - j - d s + ky for у > 0,

n к J s

' У

where k = %c\\f\\p +~\ \g\\p.

P ro o f. Applying the Holder-Minkowski inequality, it is easy to verify that the following integrals

f / (5) Щ D; (yK0 (cr)) is, I f (5) Щ Щ (y r - ‘ K , (cr)) ds, j ÿ (s) D™ D«s (yK 0 (cr)) ds

R R R

are almost uniformly convergent in R \ for arbitrary non-negative integers m, n.

Hence, by the formulas ([4])

(3) 2v

K,+ M = j K , + 1 (t)+KM, =

we get (a) and (b).

We shall prove that (c) is true. In view of the properties of the convolution ([1]) the function и (/, g ; о, y) e Lp for every y > 0. Moreover,

\ \ § f ( s ) y K 0(cr)ds\\p ^ \\f\\p j y K 0(cQ)ds,

R R

Il f g (s) y K 0{cr) ds\\p ^ Hfifllp f y K 0 (cq) ds,

k R

where g2 = s2 + y 2.

By the formula ([7])

(4) t 2m+1 (t 2 + z2) - n/2 K„ (at2 + z2) dt _{—m —1 n m i v r>}

0

where m > — 1, a > 0, z > 0, and n = 0, 1, . we have

Solution o f the Riquier problem 193

y K 0{cQ)ds = — ye cy, y > 0.

Hence

and

1

2n c2 y ( s ) y K 0(cr)ds

R

1 2 71

Further, by (4), we obtain 1 n and

g( s) yK0(cr)ds ^ 2 ^ У 1 рУ> У > ° -

cyr 1 K 1 (cr)ds — e cy, у > 0,

(/, 9 \ У) < u ( f , g ; o , y ) ~ - If (o)cyr 1 K ^ c ^ d s

R

^ - 1

К U ( s ) - f ( o Y \ c y r 1 K 1 (cr) ds + 2k

+ \\Л\Р\е~су- Ц

f(s) y K 0(cr) ds

R

1

2tu g(s) y K 0(cr)ds + \ \ f \ \ P \ e - c y - l \

< A + [ic\ \ f\ \ p + -^\\g\\p )y, у > 0,

■2c where

A — — n 1 f

[ / (s) - / (o)] cyr К ! (cr) ds We remark that

A — —

n U ( o - s ) - f { o ) ~ ] c y Q 1 K 1 (cg)ds

U (o - s) - 2 / (o) + / (o + s)] eye 1 /Ci (eg) ds

13 — Prace Matematyczne 27.1

By Holder-Minkowski inequality we get

A - I II/ (o ■- s) - 2f (O) + / (O + 5)11, ye - 1 K t (CQ) ds 0

^ - c

тс a>(s)yg 1 K 1(cg)ds, у > 0.

Using the formula ([4])

(5) ^{^ ,_4} 2vT ( v + 1/2) j‘ cos zt ^

K v(z) /— J i2y + 1/2 dt

2 ^{ч / TC} we have K^^icg) ^ (eg) ; hence

A € у j‘co(s) у 'co(s) у <u(s) ds ^

TC J Q1 о

ds — I —y- ds, у > 0.

TC J s У

From the inequality

(6)

we get

cu(As) ^ (Л + l) 2co(s)

^(s) , ^ Г (s+y):

0

ds ^

} ф 2 + У2)ds(o(y) ^ 2<u(y) and

/1 ^ - со ( у ) 4—

тс

у С со (s) ds

TC J S' y and finally (c).

Theorem 2. I f f e L p, g e H ^ L p , then (a) н е С » (Я 2+),

(b) (A — c2)2u(x, y) = 0 in /? + , 00

(c) M 2(f, g; У) < - ю ( у ) + - U ^ d s + ^ y , y > 0 ,

TC TC J s

y

wlier^ fcj = i c 3 ||/|L + tsl|W

Solution o f the Riquier problem 195

Pr oof . From formulas (3) it follows that u(f, g; x, y) = I f ( s ) c 4 y K 0(cr)ds

2k

+ 2k 1 yK t ( cr) ~ c2yK o(cr)']ds,

hence, by the proof of Theorem 1 we get the assertion of Theorem 2.

3. Now we consider the case where g(s) = 0 for every s e R . Let u(f; x, y) = u(f, 0; x, y) = 1

2k f {s )[2 c r 1 y K ^ c ^ + c2 yKoicrftds.

From Theorem 1 we get

T^heorem 3. I f f e H 2 L p, then

|u (/; o, y) - f{ o) \ \p ^ -co{y) + - j ° ^ d s + i c \ \ f \ \ py

K K s

f o r y > 0.

In the sequel we consider (in a sense) the inverse problem to the problem of Theorem 3. First we prove a theorem of the Hardy-Littlewood type.

Theorem 4. I f f eH 2 L p, then

d2u(f; o, y) dx"

^ ^ { y ) л

< 5 - 2 - , у > 0.

У

Pr o o f . Using formulés (3), we get d2u ( f ; x , y ) 1

dx2 2k ' f ( s^ 2c y (x ~ s> r К з (cr) + c У ( x - s f r 2 К 2{cr)

— 2c2yr 2K 2(cr) — c2yr 1 K l (cr)]ds

f ( x - s ) [2 c 3 ys2 Q- 2 K 3 (CQ) + c4 ys2 Q “ 2 К2 (CQ) 2k

- 2 c2yg 2 K 2( c g ) - c 2yg 1 K ^ c g f t d s .

It is easy to verify, by formula (4), that

j [2c3 ys2 g~3K 3 (eg) + c4ys2 g ~2K 2(cg)- 2c2yg ”2 K 2(cg) R

— c3 y g~ 1 К i (eg)] (is = 0, hence

00

d2u( f ; x, y) _ = 1 Г [ / ( x + s ) _ y w + / ( x _ s ) ] [²c 3 y s 2 e - 3 K 3 < c e )

OX 2 л J

0

+ c4 ys2 e “ 2 K 2( c e ) - 2 c 2.ye"*K2( c e ) - c 3y e “ 1K 1(ce)]</s.

Further, by Holder-Minkowski inequality and (6) we get

< ^ J <»(■*) [|2c3 ys2 ^ 3 K 3(cg) + c4 ys2g 2 K 2{cg) о

— 2c2y g ~ 2K 2(cg) — c3y g ~ 1 (c^)|]ds co(y)

"" 2яу2 о

+2c2y g ~ 2К 2(cg) + c3у д ”1 K t (cgjjds.

Applying formula (4) and (3) we can calculate that

GO

J (s + y)2 [2c3 ys2 g~3K 3 (eg) + c4 ys2 g~ 2 K 2 (eg) о

+ 2c2 y g ~ 2 K 2(cg) + c3 y g ” 1 K 1 (cgf\ds = ке~су(6 + 4еу + c2 у 2)

+ 6с2 у 2 К 2(cy).

It follows from formula (5 ) that c2 y 2 K 2(cy) ^ 2 for у > 0, c > 0. Since e~cy(6 + 4cy + c2 y 2) ^ 6 for у > 0, c > 0, then

d2u(f; o, y) ax2 which ends the proof of Theorem 4.

Using the results of paper [3], by Theorem 4 we get

Th e o r e m 5. Let f e L p. I f

I N /; °> k)-/(o)llp < a>(y),' у > о,

then l

co2(f, t) ^ M t 2

j

j (s + y)2 [2c3ys2£ 3 K 3(cg) + c4ys2 g 2K 2(cg) d2u(f; o, y)

ôx2

Solution o f the Riquier problem 197

where co2{f, t) denotes the second modulus o f smoot heness o f the function f and M is a positive number independent o f t.

References

[1] P. L. B u tz e r , R. J. N e s s el, Fourier analysis and approximation, vol. I, New York and London 1971.

[2] V. I. G o r b a ic u k , Conditions of solvability of the Riquier problem for the biharmonic equation in the half-plane and limiting properties o f its solution (in Russian), Dokl. Akad.

Nauk Ukrain. SSR Ser. A, no. 7, Kiev 1983.

[3] —, Direct and converse theorems o f approximation by solutions o f boundary problems for some elliptic equations (in Russian), Approximation Theory of Functions, Proc. of Int.

Conf., Moscow 1977.

[4] N. N. L e b e d e v , Special functions and their applications (in Polish), PWN, Warsaw 1957.

[5] A. F. T im an, Approximation theory o f functions o f a real variable (in Russian), Moscow 1962.

[6] E. W a c h n ic k i, On the Riquier problem for the equation (A — c2)2u(x, у) = 0 in the half­

space, Rctcznik Naukowo-Dydakt. WSP w Krakowie, fasc. 51, Krakow 1974.

[7] G. W a ts o n , A treatise on the theory o f Bessel functions, Cambridge 1962.