5 C2 со (nt) ^ n2co(r), (A — c2)u(x, у) = 0 in the half-plane On the limit properties of the solution of the Dirichlet problem for the equation

ROCZNIK1 PO LS Kl EGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

Eu g e n i u s z W a c h n i c k i

(Krakôw)

On the limit properties o f the solution o f the Dirichlet problem for the equation (A — c2) u( x, у) = 0 in the half-plane 1. In paper [6] a solution of the Dirichlet problem for the equation {A —c2) u( x , у) = 0, c > 0, in the half-plane Æ+ = {(x, y): |x| < oo, y > 0] has . been given. This solution is of the form

u(x, y) = /(/; x, y) = —^ = I f ( x - s ) H ( s , y)ds = f * H ( -, y)(x),

where

12 1

H (s, y) — - c y-

71 \/},2 + s

2 K 1 (C\/y2+S2)

and K v is the MacDonald function of index v ([7]).

The purpose of this paper is to present some limit property of the function / (/;x, y) as y 0+. Such a problem in the case of Laplace equation was investigated in papers [1 ]— [3], [5].

2. Let

be a function defined in

If (i) co(0) = 0,

(ii) со is a continuous and non-decreasing function in .[0,

(iii) for every t ^ 0 and every positive integer n

со (nt) ^ n2co(r),

then we call со a function of the type of 2-nd modulus of smoothness.

It is easy to show that there exist positive constants C1?

l

(1) Cj

t2

0 < t < j . t

Moreover, there exists a function со* satisfying conditions (i)— (iii) such that

5 Co(f) ^ co*(f) ^ co(f),

<

and the function

is a non-increasing one in (0,

180 E . W a c h n i c k i

Let Щ L„ denote the set of all functions f e L p(R) (1 ^ p < oo) for which

\\f(- + h ) - 2 f ( ' ) + f ( - - h ) \ \ p * : a > m,

where со is a function of the type of 2-nd modulus of smoothness satisfying the condition

m o

h ^

J ^*

о We prove

Th e o r e m 1.

I f f е Щ Ь р

p

oo), then there exist real numbers M > 0 and a ^ 1 such that

(2) II/(/; % y)—f (* )llp < M ^ ° ^ ~ d t + y J ^ - d t + y j.

о у

Proof. It is known ([6]) that I ( \ ; x, y) = exp( — cy), hence

(3) II/(/; -, y)-/(-)llp < »/(/; -, l; -, y)llp+

+ll/(i;% y)/(-)-/(-)llp

= ll/(/; -, y ) - H i;-, y)/(-)llp+ll/llp|exp(-cy)-i|

<11 /(/;-, y ) - / ( i; - , y)/(-)llp+c3y,

where C3 is a positive constant.

The kernel H(s, y) of the singular integral /(/; x, y) is a positive one and it is an even function with respect to s. Hence, by Holder-Minkowski inequality ([2]) we obtain

(4) l|/(/; -, •, y)f(')\\p l r

UJ (/ (x - s )- / (x ))# (s , y)ds

p \i/p dx

^/bz J ^/271 (/ (x - s) - 2/ (x) +/ (x + s)) H(s, y) ds

p \i/p

dx j

< U Iu ^, . | / (x -s )-2 / (x )+ / (x + s)|pdx ^y/p H{s,y)ds

J l K

J

J /

a)(s)H(s, y)ds.

We have the following ([6])

H( s, y) < ~y{ y2 + s2) 1, J > 0, |s| <

. V n

Since ([7])

К 1 (z) ~ /— exp( — z) asz->oo,

thus there exists a constant a ^ 1 such that

H(s, y) ^ M t ys~3/2exp( — cs) for s ^ a , where is a positive constant.

Hence

1 f , u f y^(s) 1 f yw(s)

J b . J л J ^s2+y2 n ] s 2 + y2

О О у

+ M 2y со (s) s 3/2 exp ( — cs) ds < — f ds+—

71 J S 71

О у

'co(s) ds +

+ M 2y co(s)s 3/2 exp ( — cs)ds,

where M 2 is a positive constant.

It is easy to show that 00

Jco(s)s~3/2exp( — cs)ds ^ C4, .*

a

where C4 is a positive constant,

Hence, by (3), (4) and (5) we obtain (2).

From Theorem 1, in the case where to(h) = Khx, 0 < a ^ 1, К is а positive constant, we have the following

Co r o l l a r y.

I f f e L p(R)

^ p <

and

then

11/(4- h ) - 2 f ( • ) + / ( - h)\\p = 0(\h\% 0 < a ^ 1,

II/(/;*, y)-/(-)llp =

0 (y l 0(y\\ny\)

for 0 < a < 1,

for

= 1

as у — ■» 0+.

182 E . W a c h n i c k i

3. Now we consider (in a way) the inverse problem to the problem from Theorem 1. First, we prove an inequality of the Bernstein type for the integral /(/; x, y).

Le m m a

1. I f f e L p(R), 1 ^ p < oo, then ( 6 )

dx2 I if', x, y) < MAf\\Py~\

where M 3 is a positive number.

Proof. It follows from the results of [ 6 ^{] that}

dx2 /(/; x, y) = f (s ) к % R

d_2_

dx2 H( x — s, y)ds.

Applying the formula ([7])

y -z v K v(z) z~v К

(z) we get

(7) ^ / ( / ; x, y) = — \ f { x - s ) s 2{cQ) 5K 3{cg)ds c6y

сАУ f { x - s ) { c e ) K 2(cQ)ds,

where

= s 2 ^{+ y 2.}

In view of properties of the convolution ([2]) and (7) we have d2

dx 2/( / ;

)

с у

^ Wf Wp^- ^i C

|s 20 3K 3{cg)ds + Ip 2

ds cz у

^ 2||/||_ —I c s2

3 K 3(cg)ds +

2K 2(cg)ds By the formula ([7])

00

J t2m+1(t2 + z2) ~n/2K n{ a J t 2+ z 2)dt = K „ - m-i(az)

о

for m > — 1, a > 0, z > 0 and n = 1, 2, we obtain i l

ôx2 i (/; y) ^ C 5\\f\\py - l>2K 3l2(cy), where C5 is a positive constant.

Since ([7])

K 3/2(z) = J ^ { l + l / z ) e x p { - z ) , z > 0,

then (6) is true.

We shall prove

T

2. Let f e L p(R), 1 ^ p < oo. I f

(8) \\I ( f- , y) - f ( - ) \\P ^co(y),

where co{y) (y > 0) is a function of the type of 2-nd modulus of smoothness in L p metric, then 2-nd modulus of smoothness co2(f, t) of the function f in L p metric satisfies the inequality

l

(9)

r)=SC6r2 0 < f < 1/2,

t * where C6 is a positive constant.

Proof. We apply the Bernstein method. We have m- 1

f ( x ) = /(/; x, i ) + X [/(/; x, 1/2j +l ) - I ( f ; x, l/ 2 ')]+ / (x )-/ (/ ; x, 1/2")

for every integer m ^ 2.

Hence

f (x + h) — 2f (x) + / (x — h) = (x, h) + A 2(x, h) + A3(x, h), where

A i (x, h) = /(/; x + h , % ) - 2 I ( f ; x, i) + /(/; x - h , {),

m— 1

A2( x , h ) = z {[/ (/ ; * + &, 1/2J + l) — I ( f ' , x + h, 1 ^/ 2 ^{J)] —}

— 2 [/ (/ ; x, i/2j +i ) — I ( f ; x, 1/2-0] + + [/(/; * - A , 1/2J + x) —/(/; x - A , 1/20]}, Л3(х, A) = [/ (x + A) — / (/; x +A, 1/2")]- 2 [/ (x )- / (/ ; x, 1/2-)] +

+ [/ (x — A) — / (/; x — A, 1/2")].

184 E . W a c h n i c k i

We notice that

/(/; x + h, i ) - 2 /(/; x, !) + /(/; x-/i, i) ft

ft

^ / ( / ; x + f i + t j , $)dtl dt2.

— ft/2 —h/2

Hence, by Holder-Minkowski inequality and by Lemma 1 we get

(10) \\AA',h)\\P -

«/ v

■h/2 -h/2

d2

dx 2I { f ‘> x + tx + t2, -j)dt! dt2

< h 2 x, i) ^ C 7 h2,

where C7 is a positive constant.

From the assumption and Hôlder-Minkowski inequality if follows that

(И ) 1Из(‘> h)\\p ^ 4co(l/2m).

In order to estime \\Л2(-, h)\\p let

(12) m„ (x) = /(/; x, 1/2n+1) —/(/; x, 1/2"), n = 1, 2, ...

From the Fubini’s theorem it follows that

/(/(/; *, 1/2"); X, l/2"+1) = /(/(/; X, l/2"+1); x, 1/2").

Hence, by the properties of the convolution and by (12) we get (13) u„(x)

= / ((/ (• )-/ (/ ; -, 1/2")); x, l/2"+ * )- / ((/ ( )- / ( / ; -, l/2"+1)); x, 1/2").

Further, we find

h/2 h/2

u„ (x + h) - 2 un (x) + un( x - h ) = A 2

dx 2un(x + t1+ t 2)dtl dt2,

hence, by (6) and (13) m- 1

И2(%

h)\\P

+ h) - 2uj { ' ) + uJ(--h)\\l j= 1

m-l ( Д2

h2 I \ g p / ((/ (• )-/ (/ ; % i/2J)); x, i/2J+1) +

+ A 2 _

dx2

m— 1

< M 3h2 I {! ! / ( • ) - / ( / ; -, s-1

+11 / (• )- / (/ ; -, i/2J+')llp22j! . By assumption and by properties of the function to we have

m— 1

(14) \\A2(-, h)U, ^ M 3h2 £ (<u( l/2j) 22° + u + w ( 1/2-7 + *)22J)

j = i m - 1

^ 5 M 3/i2 X w ( 1/2') 22

j = i m— 1

= 10M3 h2 £ ûi(l/20(l/20-3(l/ ÿ +1)

i = i

1/2

^ C 8 h 2

J

where C8 is a positive constant.

Therefore it follows from (10), (11) and (14) that

1/2

\\f(- + h ) - 2 f ( - ) + f ( - - h ) \ \ p ^ C 1h2 + 4co(l/2m) + Cgh2 J ( o( z) z~4z

2~m

for every integer m ^ 2 and real / 1 ^.

If we assume 0 < t < 1/2 and choose m ^ 2 such that the inequality 2~m

< t < 2~m+i holds, then by properties of the function o> we get

1/2

ll/ (‘ + ^) —2/(•)+/(• —/i)||p ^ C7 h2 + 4(o(t) + C8 h2 J m (z)z-3rfz.

t/2

Consequently

1

(o2(f, 0 < C7 f2 + 4a>(f) + C9f2 Jcu(z)z~3dz t

and by (1) we have (9).

In the case where co(h) = Khx, 0 < a < 1 from Theorem 2 it follows

Co r o l l a r y.

I f f e L p(R),

p

and

ll/(/;% y)-f(-)\\P = o ( f ) , y > 0 ^,

then \\f(- + h ) - 2 f ( - ) + f ( - - h ) \ \ p = 0(\h\*) as Л - 0 .

We notice that all the results of the present paper remain true if p = оc .

The proofs are similar, but more elementary.

186 E . W a c h n i c k i

References

[1 ] P. L. B u tz e r , W . K o b le , R. J. N e s s e l, Approximation by functions harmonic in a strip, Arch. Rat. Mech. Anal. 4, 5 (1972), 329-336.

[2 ] P. L. B u tz e r , R. J. N e s s e l, Fourier Analysis and Approximation, vol. I, New York and London 1971.

[3 ] V. I. G o r b a jc z u k , P. V. Z a d i e r j e j , Svojstva reszenij zadaczi Dirichle dla kruga i polosy, Mat. Sbor., Inst. Mat. USSR, K ijev 1976, 64-68.

[4 ] A. F. T im a n , Tieoria priblizenija funkcji diejstvietielnovo pierjemiennovo, M oscow 1962.

[5 ] M. F. T im a n , D A N SSSR (1962), 145.

[6 ] E. W a c h n ic k i, Solutions de certains problèmes aux limites pour l'équation A u (x , y) — c 2 u (x , y) = 0, Demonstr. Math. 10.2 (1977), 417-442.

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