The Dirichlet problem for the wave equation (*)

By DAVID W. Fox and CARLO PUCCl (a College Park)

Santo. - Si riesamina un classico esempio di problems • non ben posto ~), il problems di DIRICHL~T corrispondente at modetto fisico di una corda vibrante, fissa agli estremi e assumente posisioni note in du~ diversi istanti. Lo studio retat iw all' esiste#za ed unicit& della sqluzione ~ ricondotto a quelto di una equazions funzionale ed ~ strettaments con. hesse a questioni di teoria dei humeri Si esamina p o i l a dipendcnza della soluzionc dai dati e si discutc it problems da~ p u ~ o di vista delle sue applicazioni.

Summary.. We consider a vibrating string fixed at the ends for which the position is known at two different times. This corrisponds to a classical not well posed problem, the DIRI- CHLET problem for the wave equation, which we reconsider here ia order to determine under what conditions it is possible to obtain useful information about the physical phenomenon. This problem is related to a functional equation from which the principal results can be deduced.

We will reconsider here a classical (( not well posed ~ problem, the DIRICHLE~ problem for the wave equation in the plane. I t is known that for some regions in order to determine the solution it is sufficient to prescribe its values on only a part of the boundary, as in the ease of a rectangle with sides parallel to the characteristic lines. HADkl~XRD in [1], [2] investigated whether this happens for any region and constructed many interest ing exam- ples in which the DIRIC~LE~ data overdetermine the solution ~1}, or in which the data must satisfy conditions of trascendental character. On this basis he rejected the DIRICKLET problem as unsuitable for hyperbolic equations. BOURGI~ and DUFFII~ on the other hand, showed in [4] that for some rectangles the solution of the DIRTCHLET problem exists provided only that the data are sufficiently differentiabte (2~. In the same paper they observed that al though the DIRICHLE~ problem for the wave equation can correspond to a real physical situation, their analysis could not be of practicnt use because small variations

(*} This research was supported in part by the United States Air Force under contract No. AF ~9 (638~ 2"28 monitored by the Office of Scientific Research, Air Research and Development Command.

(~) This problem was studied also by A. HUB~R in [3] where he gave necessary con- ditions for the existence of a solution for more general regions in the plane.

(~) The same problem was considered later for the damped wave equation by BOURGIN in [5], and O. Q. OwEss considered the homogeneous irichlot problem for an inhomogeneous ultrahyperbolic equation in [6].

156 D . W . Fox - C. Pt 'ccI: Tl~e Dirichlet problem for the w(~ve.~ equation

in the length of the sides of the rectangle could cause a lack of uniqueness or existence of the solution.

It seems to us that the ¢ not well posed ~ problems must be investigated further since they can correspond to interesting physical situations, and in these cases, it is reasonable to believe that it must be possible to formulate a mathematical quest ion in a way in which useful information can be obtained (8}. Proceeding from this point of view our research shows that in a case chosen for its simplicity this belief is justified, at least in part.

We consider the case of a vibrating string, fixed at the ends, for which the position is known at two instants of time. This leads us to the D I ~ c ~ E ~ problem for the equat ion

(1) u ~ - - utt = 0,

in the rectangle R : 0 _~ x < : 7:, 0 ~ t ~ ~¢% with the boundary data

(~) u(o, t) = u(~:, t) = o,

(3} u(~, o ) = v(x), u(x, ~ : ) = tp(~).

BouR~IN and DUFFI~ considered the DIRIOHLET problem for the same region, but with more general boundary data, and for this they proved a uniqueness and existence theorem and gave an expression of the solution for irrational a.

In section 1 we obtain a different expression for the solution and neces- sary and sufficient conditions for its existence. This establishes a correspon- dence between our problem and the functional equation for an even continuous function E of period 27::

E(z + ~ ) - - E (x - - ~ ) = F(x) ,

where F is a prescribeii continuous odd periodic funct ion of period 27:. In section 2 we give some results for this equation. Precisely, for a rational we give the necessary and sufficient condition for the existence of a solution and its express ion; for a irrational we show the uniqueness of the solution and we give an existence theorem and an expression of the solution for a class of irrational u. I n s ec t ion 3 we consider our DIRIOnLET problem for the case a rational, giving existence theorems and expressions for the generalized and classical solutions. In section 4 we consider the case a irrational, and we prove a uniqueness theorem in the class of generalized solutions. We find the values of the solution if it exists on a set everywhere dense in and we improve the existence theorem of BouI~GIN and DUFFIN by means of a number- theore t ic theorem of HARDY and LIT~LEWOOD. In fact, our

(s) this has been done for the • not well posed, CAUCHY problems for parabolic and elliptic second order linear equations. See [7], [8], [9], [10], [11], [12].

D. W. Fox - C. PuccI : The Dirichlet problem ]or the waves equation 157

problem has several connections with quest ions in the theory of numbers. In section 5 we construct two examples which show that the solution does not depend cont inuously on the data % ,¢ or on a. On the other hand, we observe a feature pecul iar to our p r o b l e m : T h e solution depends continuously on the data on a subset of R.

In the early sections of this paper we have considered our problem from the classical point of view, i.e., uniqueness, existence, and expression of the solution. Start ing with section 6 we inquire what information about the solu- tion can be obtained when the data are known approximately. In this hypo. thesis we construct in section 7 an explicit approximation to the solution with a known error on a system of grids and in section 8 we find approxi- mations to the solution in all R where the error depends on a bound on the derivatives of" the solution. In section 9 we discuss the usefulness and validity of the assumption that the derivatives of the solution can be a priori, bounded, and we analyze there the theorems of section 8 from the point of view of applications. In section 10 we observe a formal expression of the solution suggested to us by Professor ~[AROEL RIESZ,

1. S o m e p r o p e r t i e s o f t h e s o l u t i o n .

Let R be the set of points (x, t} such that 0 _ < x _ < z:, 0 ~ ' t ~ ' a : ~ , with a positive constant. Let ~{x! and ,¢(~:~ be two continuous functions in the

interval [0, g]. W e consider the DIRICrtLET problem for the unknown function stated by :

(1) uxx ~ ~ t t t - - " 0 in R,

(2) u(0, t) = u(~, tt = O, 0 ~ , t <--:c%

{3~ u(z, 0) = ~0(z), u(x, ~ ) = ~(z), 0 <_ z <_ ~.

A function of class C c2) in R that satisfies the previous conditions will be called a solution of the considered DIRIC~rLET problem (4). ~ function st, that is the limit of a sequence, uniformly convergent in R of solutions in R of the differential equat ion (1), is called a generalized solution in R of the equation (1). Clearly u is continuous in R and it is well known that it can be wri t ten in the form

(4) u(~, 0 = f ( z + 0 + g(~ - t~

with 7 ~ and g continuous.

(4) Obviously if a solution exists ~ and ~b must be of class C ~2~ in [0, ~] and vanish with their second derivatives at the extremes of the interval

158 D. W. Fox - C. Pucc~: The Di~'ichiet problem for the wave.~ cquatio~,

A function u which is a generalized solution in t~ of equation (1) and satisfies the boundary conditions (21 and (31 is called a g,,neralized solution of the DIRICELET problem {1), t2), (3}.

In the study of our DIRICl~LE~ problem we will consider always the extended boundary data ~, .,~; that is, ~0 and ,~ will be considered defined on all the real axis as odd periodic functions of period 2~.

I. Necessary and sufficient conditions for the existence of a generalized solution of the Dirichlet problem (1p, (2}, 131 are that a continuous function E~rt exists such that E is even, periodic of period 2~, and satisfies the fun- ctional equation :

I f such a function E exists, a generalized solution is given by

1 ~6) u(x, t) = ~ [~tx + t) + ~(~ - - t) + ~ ( x + t~ - E(x - - t~].

Let us suppose that a generalized solutiou of the DII~C~LE~ problem (1), (2}, (3), u(x, t )= f(~-t-t) + g ( x - t) exists. In order to have a simple expres- sion of ~his solution taking into account the data, it is convenient to extend f and g to all the real axis.

We observe firs~ that f is defined in [0, (1-~-a)~:], and g in [--0¢~, 7:]. From t2} we have

(s)

We set

and

f ( : + ~) + g ( , : - - ~} = 0, 0 _<, ~ - ~ : .

f ( ~ = - - g ( - i), - - ~: <- i <- 0,

g({) = - - f ( - - f}, - - (1 + a)7: _~ ~ _~ - an.

By (81 it follows that ill the new intervals of definition we have

f (~ + 27:) = f ~ ,

g(~ + 27:) = giG).

Thus we can suppose f and g extended as periodic functions o[ period 2• to all the ~'eal axis. The equations (7) and (8) hold ~or all the values of ~. Now by (4) and (7j the solution u can be wri t ten

u(x, t) = f ( x + t) - - f ( t - x),

and u is thus extended to all the plane wi th periodicity 2T: in x and t. As

D. W. Fox - C. PuccI: The Dir ichle t prob lem for the waves equat ion 159

a consequence, considering the extended boundary data, we have u(w, 0 ) : --q0(x), u(x, ~7:)----'~(x) for all values of ~. By the first of (3) we have

- - f ( - - =

and so 1

= + E

where E is a continuous even function of period 27: defined by this re la t ion; thus u has the expression given by (6). By the second of (3 )we have the funct ional relation (4).

In order to complete the proof of the theorem we observe that if a fun- ction E(~) which satisfies the hypotheses of the theorem exists, the function u(~, t) given by (6) is a generalized solution of equation (1). u is of the form 14), and clearly satisfies the boundary condition (2) since ~ is odd, E is even and ~ and ~, are periodic of period 2~. Also u satisfies the boundary condition (3) since E is even and satisfies the condition (4).

OBSERVATION. - , The expression (6b gives an extension of a generalized solution u of the DIRICHLET problem (1), (2), (3), to all the plane. It is the unique extension that is a generalized solulion of equation (1) in the strip 0 <_ ~ <_ ~ and that satisfies the conditions u~O, t) = u(% t) - - 0 for all t; it is odd in x and periodic in x. and t of period 27:.

2. A f u n c t i o n a l e q u a t i o n .

The previous theorem leads us to the study of the following functional equation,

(9) E(x, ~ a~) - - E ( x - - ~rc) - - F(x,),

where F is an odd continuous funct ion of period 2~. This equation has been studied by NORLUND in [13] where he gave an expression of the solution in terms of Bernoulli polynomials when F is a polynomial. The results of NO~LUND are not of use here because we seek solutions E that are continuous even functions of period 27. Under these restrict ions on the class of admis- sible solutions our problem resembles more the study of the funct ional equation, first considered by CAUCHY,

f(~) + f t y ) = f(x + y).

We observe that if E is a solution of (9) E @ a, whore a is an arbi t rary constant, is also a solution. For this reason, without loss of generality, we will consider in the following

160 D. W. Fox - C. PvccI : The Dirichlet problem ]or the waves equation

Problem A : Determine the functions Etx), continuous, even, periodic of period 2r: that satisfy the condition E ( 0 ) = 0, and the functional equation t9}, where F is an odd continuous periodic function of period 2= (5).

By the identity

E(a: A- man) - - E(a~ - - ma=) -= ~v {E[x + (m - - 2 j )~=]--E[~c-4-(m--2j - - 2)~=] 1, j=o m - - 1, 2, . . . ,

it follows that any solution of problem (A) satisfies the relation

tit--1

(10t N(a: -t- man) - - E(x, - - ma=) = ~ F [ x -f- (m - - 2j - - 1)~=], m = 1, 2, ..., j=0

II. I f ot is irrational there is at most one solution of problem tA). It is sufficient to show that the only solution of problem {AI with F-=-0

is identically equal to zero. If F ~ - 0 and E is a solution we have by (101

(11) E(z + = E(x , m = 1, 2 , . . . .

But since E ( 0 ) = 0 and E is periodic of period 2% E vanishes on the set of the points I 2(ma -4- n)T: l with m -- 1, 2, ..., and n - - 0, ± 1, +--- 2 .... By the irrationali ty of a this set is everywhere dense and from the continuity of E it follows that E is identically zero.

I IL I f ~ P -- ~ , (p, q} "- 1, the class of the solutions of problem (A) with

F ~-- 0 is the olass of all continuous, even, periodic functions of period 27: -q -

Since (p, q) = 1 there are two integers m and n such that r a p - - nq -- I. Thus, if E is a solution of our problem, by the periodicity of E and by (10~ it follows that

E(~) -- E x + 2m ~ ~ - - 2n~ = E x "4--~ , q

and thus the solution must be periodic of period 2~. On the other hand, ob- q

viously any continuous even function E periodic of period 2~ is a solution. q

IV. I f ~ = p- , (p, q) --'-- i, a necessary and sufficient condition for the q

ea:istence o f a sol~ttion of problem (A~ is that

j=o

(6) The hypothes i s E even and F odd is not essential , but here it is introduced only for s impl ic i ty because this is the case w e w i l l need for our D~RICHLET problem.

(6) ~he notation (p, q) ~ 1 means, as nsua], that 10 and q are integers re la t ive ly prime.

D. W. Fox - C. PueeI : Tlte Dir ieh le t prob lem ]or the waves equat ion 161

If E is a solution, by the periodicity of E and by (10} with m - - q it follows that

,) j=0 q- ~: -- i=o ----q- p'" + 2j ~- u .

Since (p, q ) - - l , the set of numbers Ijp(modq) l when j - - 0 , 1, . . . , q - - 1 is the set of numbers {0, 1,. . . , q - - 1 } ; thus it follows thai

which, since x is arbitrary, is the same as {12t. Let /r(x} be a continuous, odd, periodic function of period 27: satisfying

the condition (12). We will p r o w now that if a = P-, {p, q) - - 1, there exists q

a solution E(x ) of Problem (A). We consider the formal FOURIER series of F in ( - -u , 7:):

00

(13} F(w) ~ Z b,~ Sin nx .

We note first that {12~ implies

(14) b,.q = 0, r = 1, 2, ....

In fact, we have

b,.. = -~l ]'sino'q~,F(x)dx--l~ ]'Sinlrq(x-~n)]E(~,~c-2J-~

thus, summing the first and the ]ast members with respect to j between 0 and q - - 1 , (14) follows by (12~.

Let FN be the ar i thmetic mean of the first N partial sums of the series (13}, that is

t¢ N - - n-~-- 1 (t5} FN(x) = Z ,=1 N b• Sin n x ,

Since /7' in continuous and periodic, by a theorem of FEZ¢,R ('} the sequence i Fir} is uniformly convergent on all the real axis to F.

We set

iv /V- - n -I- 1 0os nx (16) EN(x) = - Z - - b ,

,**rq S i n n ~:

(~) See G. SAI~SOSE [14] pg. 110, Th. 38.

,4nnali di Matematica 21

162 D . W . Fox - C. ]'~t'('I: Th'e D irh;hlet problem for ti~e wttl'e.~ rq~et~on

The function EN is analytic and even, thus in the interval i - - -

be developed in F o v R I ~ series

t n ~ 0

q , it can

and this series, which we will denote by SN(X), is uniformly convergent in

q , . Thus, the function E~. ~-- EN -- S.~. is continuou~ on the real axi~ and

(18) E~,~ (x) -- 0, t ,v I "<~ ~-" q

By {15) and (16) we have

EN(x + P r~)-- EN(x -- P- ~:) = FN(X,,

and by q17)

S~v(x + P~ 7:)-- SN(x --P-- ~ O.

Thus

(19)

Obviously EN is even and periodic with 2% and thus it is a solution of pro- blem (A) with respect to FN. W~e define

= % 7: , j = 0 , 1, .... q

By (18) the sequence I/~gl is uniformly convergent in Io. If I/~Nl is unifor- mly convergent in Ii_l it is also in I i by the unii'orm convergence of i FNI and by {19). Thus i/~Nt is uniformly convergent in Io, I1,..., Iq_l, and by the periodicity of EN also in their translations by multiples of 2=. Since these cover all the real axis, ~ENI is uniformly convergent to an even continuous periodic function E of period 2~:. Thus by (19) E is a solution of problem (A) with respect to the prescribed function F.

Before establishing an existence theorem in the case a irrational it is convenient to recall a well known classification of these numbers 18).

Let a be a fixed i r ra t ional ; if ~ is the upper bound of the positive numbers eo for which

lira' h ~" t Sin 2hart i : O, lh integers), h -.--*- OO

we say that a is of type I~. The set of the irrational numbers of the interval (0, 1) which belongs to some In, ~ < oo, is of measure one (gj.

(s) See for example KOKSMA [15] p. 27. (9) See E. BOR]~L [16] p. "27.

i). W. Fox - C. I'u(:cI: Tbc Dirichlet problem ]or the waves equation 163

V. Let ~¢ be an irrat ional number of type Is+~, s an integer and 0 < ~ < 1. I f F(x) is of class C( 8, ~*), with ) 2 > k, then a solution o f problem ~A) exists (~°).

The FOURIER coefficients of F in (13) satisfy the l imitations (~1)

b. ----- 0(n-'-x*);

thus for a convenient constant C

I G o s n ~ l 1 b. S i n nat: < C ns_ ~ ~, ] S i n n a ~ I ' n -- 1, 2, ....

By the hypc)thesis on a and by a theorem of H~.RDY and LIT~'LEWOOD (~) the series

1 Z

,~=1 n s+x* ] Sin n ~ !

is convergent, and thus we can write

co C0S n ~ t20) E ( ~ ) - - - Z b,, .

,~=1 2 Sin naT:

Since the series is absolutely and uniformly convergent, the function E is continuous, even, and periodic of period 2u. By the expression for the cosine of a sum

c o

Eq~ ~ ~r:) - - E(~v - - ~7:) -- Z b, Sin n~ - - F(:c)

and thus E is a solution of problem (A}.

We observe that since any irrational algebraic ~¢ is of the type I~ (13), it is sufficient to suppose in this case ttmt F is in the class C(l,x*) in order to have a solution of problem (A).

In the following we will need this corollary of Theorem IV:

P VI. I f a - - ~ , (p, q) -" 1, and F($) is an odd periodic function of period

2re, is of class C ('', and satisfies condition (12!, then there exists a solution E(x) o f problem tA) o f class C c~'.

The continuous odd function F"(x) of period 2n satisfies the condition

(10~ F ~ Ccs,),*~ means that F has s cont inuous der iva t ives and 1he der iva t ive of order s satisfies a TJipschitz condit ion of order k%

(ii) See for example G. SANSONE [14], p. 6.2, Th. 10. (12) See for example KOKSMA [15] p. 108. (t~) This was proved by K. F. ROTH. See [17] p. 1"2.

164 It. W. Fox - ('. l't'4'vl: T/,' DiriHr/t't i , ' ,b/rut for //,v n',r,'s cqmtlion.

{i2). thu~ by the~,ren IV there exists a continuous even periodic function E2(a~) of period 2r: which satisfies tho functi(mal equation

( (21) E~ x +

We define J3

y -- ~-~ E2(xldx, E~(x) -- [E2l~l--y]d~, Nix) = Ed~td~,

- - g 0 0

and note that E~ is an odd periodic function of period 27: and E is similarly an even periodic function. We add and subtract ? on the left side of (21); and integrating both sides twice with respect to z between 0 and x, we obtain

g i n + ~ ) - - g(a~:) - - g , ta~)x - - B l x - - a~) + g ( - - a~) +

+ E,(-- aTr)z = F(x) -- F'(0)x --/P(0~.

Since F(0) = 0, E(an) - - E ( - - an), and El(aT;) -" -- El(-- an} we havre

E(x + aT:) - - E(a~ -- a;:} - - 2Edau)a~ = F~x) - - F'iO)x.

But E and F are periodic of period 2n, and thus

2El(alr) - - F'{O),

and E is a solution of problem (A) and of class C ~.

3 T h e c a s e ~ r a t i o n a l .

We suppose throughout this paragraph :¢--~P with (p, q}--1 . It is q

known {,4) that the functions

Sin nqx Sin nqt, n = 1, 2, ...,

are solutions of the homogeneous DIRICHLET problem stated by (1), (2) and

(22} u(x, O) = u(x, ~r~) = O, 0 <' x ~__, 7r.

h classical tgeneralized) solution of the homogeneous DIRIOHLET problem not identically equal to zero will be called a classical {generalizedt eigenfunction.

VII. The generalized eigenfunctions of the Dirichlet problem [1), (2). t22) are those and ordy those whivh have the form

(23) u(•, t) -- 2 I- [g(x + t) - - E(a~ - - t)],

(t4) ~ e BOURGIN and DUFFIN [4:] and ~ . PICONE [18] pp. 679-683.

1). W. Fox - C. l?l;c(:z: The l)iri,hlct problem ]or the warc.~ cquatiou 165

where E i.~ at, arbitrary coJ~t~nuous even function of period 2...~ These eigen. q

functions are odd and periodiv in z and t of period z___~, and they vanish on q

n ~ n ~ each straight line $ - - - - and similarly on t - - - - , n - - O , 1, 2, ....

q q

If u(~, t} is an e igenfunct ion (23) holds by theorem I with E an even cont inuous funct ion of period 27: that satisfies the condit ion,

By. theorem t l i E is periodic of period 2_~_~, and by (23) the e igenfunct ion q

u is odd and periodic of period 2_~. On the other hand, if u is given by (23) q

and E is continuous, even, periodic of period 2~, then (24) holds by theorem q

III , and thus by theorem I u given by (23} is an eigenfunction. The specified propert ies of the e igenfunct ions follow from the evenness and periodici ty of E.

VII I . The classical eigenfanctions of the homogeneous Dirichtet problem are linear combinations of the functions

Sin nqx Sin nqt, n - - 1, 2, ....

If u is an eigenfunet ion, by theorem VII we have

1 [g(w n'- t) - - E(x - - t)], u{z , t} =

with E an even periodic function of class C "} of period 2~, - - .

q Thus

o o

E(~) = Y, b,, Cos nq~.,

and

E(~v -}- t) - - E(x - - t) - - -- 2 E b~ Sin nqx Sin nqt. n = l

We prove now two existence theorems, tks it is to be expected from the lack of uniqueness , this is possible provided the data satisfy a compatibi l i ty condition, a type or orthogonali ty to the eigenfunetions, as is amplif ied by a r emark at the end of this section.

~25)

IX. I f the ex, tended boundary data are vontinuous and satisfy the condition

~=OL , q= -]'' q.]

166 D. W. F, ",: . ~" ~_'~-,,,,T. T h e ! ~ i r i c h h : t t~r~blem, . t o t t ier w(tre.,~ t ' q u o t i o ~

(26) Ix nr:~ u/ , - ( ]=

and

then a generalized solution e~ists. All the solutions assume the same values on the straight lines

n ~ n ~ ~ = - - - , t = - - , n = 0 , 1, 2,.,.,

q q

and precisely, denoting with u one of these solutions, we have

,=o - - ~ + ~ ~= ~ " - - ~ + ~ P ) ,~=o, ~,...,

( n T : ) k(,)-I ( nT: 2 j ' ~ ) ~(,)-I ( nT: 2i \ (27) u \ - ~ , t - - Y. ~ l - - p r : - - E ~ t - - - - ..., ;=o -~ + ;=0 -~ +-~p~)' n o, 1,

where k{n), r(n} are the unique pair of integers that satisfy

(28) kp - - rq - - n,

under the restriction 0 ~ k ~ q - - 1. W e set

(29) F(~) = 2~p(x) - - ~(w + a~:) -- ~o($ -- an),

and we observe that F is continuous, odd, and periodic of period 2,~. By the periodicity of qo we have

i=0 \ q = + ~:)' and thus from (25} it follows that (12j holds. By theorem IV there exists a continuous, even, periodic function E of period 27: that satisfies (5), and by theorem I the existence o[ a solution of our problem is proved. Furthermore, a sohttion has the expression (6). Using the definition of kin) and r(n) given by (28} and the periodicity of E we have

and by (10) and (28)

By (6), (29~, and (30) we have

u ~ , - - = ~ @~-- ]=0

k(n)--I n= 2j+ 1 )

n 2 j + l _.]_1 1 q +~P= 2:PX+~- +icP~-~-

_ _ ~v ¢ p x - - - - - { - pr: - - ~ p~: 2 i=0 q 2 j=o -q + "

D. W. Fox - C. P~zcci: The Diriehlet problem ]or the v, aves equatiou 167

Observing that by {28}

( . nT:\ ( n~: 2k, n } - - 2 \ q -q

we obtain (26t. which must hold for any solution of our problem. By (6), the oddness of 9, and the evennes of E we have

and similarly we obtain (27).

X. I f the extended boundary data 9 and ~ are of class C c~ and satisfy the compatibility co,dillon (25) (hen there exists a classical solution of the Dirichlet problem (1), (2), (3).

By the previous theorem a generalized solution exists and is given by i6} where E is a solution of prGblem (A) with F of class C ~) given by {29}. By theorem VI, E is of class C c~) and thus by t6) u is also of class C c'-'~.

Theorems IX and X can be proved under slightly more restrictive hypotheses bv the technique of FOURIER series. In this paper we use the following notation :

c:~ cx~

{31} q~(~c}~ Z a,, S i n n x , ~ (x )~ Z A, S i n n x ; n = l n ~ i

a,~ and A,, are the usual FOUI~IER coefficients of 9, ~ respectively over the

interval [ - - u , u].

XI. I f the extended boundary data 9, ~ are of bounded variation and in the class C ~°, ).) and the coefficients of their trigonometrical expansions (31) satisfy the conditions

(32} a,,q --- (--1)!'~A,,q, n : 1, 2, ...,

theu a generalized solution u exists and has the form

a, S inn(P ~:--t) ~ - A~, Sin nt (33) u(x, t) -- Z Sin nx.

~ o Sin -n-p

For any pair of integers m and n. with n not a multiple of q we have that

[ n p - - mq [~_ 1.

Observing that for any n not a multiple of q we can find an integer m' such that

i n P - - m ' [ < 1 q 2 '

we have

168 D. W. Fox - C. P~cci : The Dirichlet problem for tlte u~are.~ equatiol~

By our hypothesis the series Z I a,, ] and VIA,, I are convergent tt~l and thus by (34} the series in the r i gh t hand side of (33) are uniformly convergent. Consequent ly the funct ion u given by (33~ is a cont inuous function, the uniform limit of classical solutions of (1), and therefore a generalized solution of (1). It is easy to see that u satisfies the boundary condit ions (2}, (3).

XII. I f the ex~tended boundary dala ~, ,~ are of class C (~, ~), their second derivatives are of bounded variation, and the Fourier coefficients of ¢~ and satisfy the conditions ~32), then a classical solution u exists and has the form (33).

By the previous theorem we have only to verify that the funct ion u g i v e n by (33) has cont inuous second derivatives, and this follow from the theorem of ZYa~UND previously quoted.

Remarks

la) The compatibi l i ty "condition (25~ is equivalent to the other, (32). In fact, by the periodici ty of ¢p and ,~, the left hand side of (25) is a periodic

funct ion of period - - , and its formal FOURIER series is q

co

q E [(-- 1)"~A ~,~-- a,q] Sin nqx,

and then (25~ implies aud is implied by (32).

(b) By theorems VII and VI I I it follows in the case a rational that if our DIRIC~LEI ' problem has a solution, it has inf ini tely many solutionsl But it is in teres t ing that all these solutions coincide on the straight lines

n ~ n ~ x------- , t - - - - b e c a u s e on these lines all the e igenfunct ions vanish (by

q q theorem VII). Consequent ly even in the case a rat ional there is a part ial uniqueness . We also observe that the solutions are de te rmined uniquely on a set that becomes dense as q approaches infinity, so there is a k ind of cont inuous passage from non-un iqueness in the rat ional case to uniqueness in the i r rat ional case, which will be proved in the next section.

4. T h e c a s e ~ i r r a t i o n a l .

We suppose throughout this section that ~ is i r r a t iona l By theorems I and I I it follows that

X l I I . If ~ is irrational the Dirichlet problem (1), (2}, (3) has at most one generalized solution. Remember ing the classif ication I., recal led in section 2 we have :

W) See ZYG~U~D [19] p. 186.

D. W. Fox - C. P[r(,(r~: Thc Dirichlet problem for the wares equation 169

XIV. Let ~ be a~z irratio~,al number of type 1~:.~,, k an, integer and 0 <_~ ). < l. I f the exleuded boundary data ~, "~ are of class C ~'- ~*~ ,with "~,* > k. t l~n a generalized solution of the Dirichlet problem {1), 12), 13) exists. I f % '~, are of class C c'~'';.*' a classical solution exists. I n either case the solution is

given by

. . . . . . . . . . . . . . . . . . . . . . . ~ J . . . . . 135) uix, t ) = ~ a,, Sin n ( ~ -- t ) + A .... r a n t S innx ,

where a . , A,, are defined in {31) I:').

Suppose the extended boundary data, % ,~ are in the class C ~,~*~, then

a . , A,, = 0 {n-h'-~*).

We have for a convenient constant C

a , Sin ntar: - -

By the hypothesis on a lhe

1 tl -1- A . Sin nt _<_ C nk+~. [ Sin narc 1"

series

1 Z

is eonverge~t a(~eording t,, a iheorem el' }tAI~.DY-LIq'TLEWOOI)I~71, and thus the, series i.'!~5} is ab,~olutely and uniformly convergent, lts sum u is a gene- ralized solution of {1t becaus,~ e~ch partial sum is chss ica l solution of tl}, al~d it is ea,~y to verify that u satisfies the boundary conditions. The result for the c]a,~sical solution follows analogously.

If ~: is an irrational algebraic number it belong~ to the class /1 and thus in this case it is sufficient to suppose the extended boundary data have first derivatives that satisfy a LIPsctIITZ condition in order to insure the existence of a, generalized solution.

XV. I f a generalized solution u(a, t) of the Dirichlet problem (l), 12}, 13} exists then it is given on the lines t : n~r:, x : n ~ , n - - 1, 2, ..., by

n--I n--1

136) u { x , n ~ } - - Z ~ [ ~ - ] - { 2 j - ] - l - - n ) ~ ] - - E ¢~[x+{2j--n):c~], n : l , 2 . . . , j=o j=l

n--1 n - - I

{37) u(na~:,t)-- Z ~ [ t + ( 2 j W l - - n ) a ~ : ] - - E ¢~[t-~(2i--n~a~:], n-- -O, l , 2 .... {"). 1=:o /:=o

I i~) This expression of the solution was obtained by BOURGIN and DUFFIN in [4]. Thele they proved the existence of a classical solution unde r slightly more restr ict ive hypothesis on the di f fercnt iabi l i ty of %o and '~.

(t7} See (12), (is) This express ion of the solution is equal ly val id for ~ rat ional and it is easy t#)

check that for ~ rat ional it coincides wi th the expression given by (26) and {27).

Annall di Mat~.atica 2~

170 D. W. Fox - C. P(Tcci: T/~e Dirivbtet problem for the waves equation

Accounting for the periodicity of u, the formulas (361 and (37)give the values of u on a set everywhere dense in R.

By theorem I there exists a continuous even periodic function E of period 27: that satisfies (5). From the expression (10} of E, where F is given by (29), and from the expression (6} for u, (36) and (37) follow.

5. D e p e n d e n c e o f t h e s o l u t i o n s o n t h e d a t a .

It is interesting to consider the dependence of the solution u(x, t) of our DIRICHLET problem on the data % ~ and a. We begin by giving an example which will clarify in part the situation.

Let a be a fixed irrational number and

1 ~-0, ~" - - ~ n S i n n ~ "

The DIRICHLET problem (1), (2), (3) corresponding to these data has by a previous theorem the unique solution

1 Sin nt Sinn~ u,,(~, t ) - Vn Sin na~

By a classical theorem

and therefore

(19} there is a sequence of integers p,,, q,, such that

a - - ~ < q,--~,

and thus

7~ I s i n i = [ S i n lq,, f < - -

Vq~ l uq,, (x, t) [ > --~- I Sin q,J Sin qnx t"

Thus, in any open set G in R the sequence of the suprema in G of the solutions uq,, diverges to infinity while the corresponding data converge uniformly to zero. From this it is clear that any arbitrarily small variation in the data ~, + of the DIRIC~[LE~ problem (1), (2), (3} can produce an arbi- trarily, large variation in the solution.

This example has a remarkable similarity to the classical example by which HADA~[ARD showed the instability of the solution of the C~ucHY problem for the LXPLACv, equation,

A similar example also shows that the solution of our DIRICHLET problem

(19) See PERRON [20] p. 128,

D. W. Fox - C. Pvccz: The DiricMet problem for the waves equation 171

is unstable with respect to :¢. Consider the DIRIGHLET problem (1}, (2), (3} corresponding to the data

¢p -~ 0, ~ ~ Sin q~,

n with ~ variable. For = - - ~ , n - - 1 , 2, ..., no solution exists since the compa.

tability condition (25} is not satisfied. For any other = a solution is given by

Sin qt Sin qx uric, t) - - Sin q~:r

n If = is rational, :¢ =~= ~ , the solution is indeterminate to the extent of additive

eigenfunctions. If = is irrational, this solution is unique and is divergent to

infinity as = approaches _n. q

These examples show that the solution of oar DIRIOHLET problem does not depend continuously on the data. In this there is a similarity with the CAUOHr problem for elliptic equations as was pointed out by HADAMARD {2o). But in our case there are some pecul iar features, in fact, whenever the solution exists, even if it is not unique, there is a :subset of R on which the solution is uniquely determined, and there it depends continuously on the data.

We consider ~ as fixed and define R~,~ as the set of points of R that belong to one of the straight lines

--n:¢% t - - n ~ , n ~ 1, 2, . . . , ~ ,

or to the straight lines obtained from these by translations of multiples of QO

2=. We will set R~- - U R~.N. N = I

For any line in R~ the solution is expressed by (36) or (37) as a finite l inear combination of the functions ~o and ~,, and therefore it depends conti. nuously on the data at each point of R~, which is a set dense in R if = is irrational. However, the dependence is uniform only on each R~,~, i.e., there exists a constant CAr such that

[ u l ~ C ~ v ( m a x ] q 0 [ q - m a x [ + [ ) in R~,~v.

For fixed continuous data % '~ and variable g the solution, if it exists, depends continuously on g in R~,N in the following s~nse: Let gx and :¢2 be two values of the ratio = and ux, u~, two corresponding solutions, then

l u~(m, n=~r:) - - u , ( z , n~,r:) I < D ~ I =~ - - =~ I, n = 1, 2, . . . , IV, and

I u,(no:~r:, t} - - uz(n=~=, t)[ < D~ I a~ - - =~ I, n -- 1, 2, ..., IV,

(so) See [9] pg. 26.

172 D. W. Fox - C. P~;cci: The Dirichlet problem ]or the v'at'es eq~tatio~

where D~v is a constant that depends on N and the rnoduli of continuity of and '~. This follows im~nediately from the expressions of the solution given

by (36~ and (37}.

6. S o m e p h y s i c a l a n d m a t h e m a t i c a l r e m a r k s .

As we noted in the introduction and as it has been observed by Boua~IN and DUFFI~, a physical realization of our DraICHLET problem is given by a vibrating string, fixed at the ends, for which the positions at two different times are known, as for example by two photographs. However, the knowledge of the times, and thus of a, ban~oaly be approximated. Consequently, BOURGIN and DUFF~ observed tha t s ince when a is rational the solution is undetermi- ned, the previous mathematical theory gives no useful information about this physical problem. This was confirmed in section 5 by the two examples which demonstrate that an arbitrarily small variation in the data can produce an arbitrarily large variation of the solution. As we have noted this behavior is similar to that in the case of the CAUCHY problem for the LAPLACE equation. On the grounds that in any experimental problem the data are known always with a certain error H/LDAV.[ARD considered such problems not well posed (~1}. Still these pr~oblems correspond to concrete physical situations and so it is reasonable ~o believe that it must be possible to formulate a mathematical question in a way in which useful information can be obtained. We will seo later that the introduction of another condition on the solutions, a condition with natural physical meaning, leads to the result that these solutions depend continuously on the data.

We observe now that according to the results of the previous paragraphs the solutions depend continuously on the data on a subset of R, and from this we will show in the next paragraph that with an approximate knowledge of the data ~o, ~ and a it is possible to compute the solution with a known error on a grid that can be arbitrarily fine in R. Furthermore, the computation of the solutions at such points is simple and thus there is no difficulty in practical calculation. However, the fact remains that an interpolation of the solution between these points cannot be justified without the introduction of aa additional hypothesis. How this can be done will be discussed later.

7. T h e D i r l c h l e t p r o b I e m w i t h a p p r o x i m a t e d d a t a .

Let ? and ~ be two continuous functions defined in the interval [0, ~] such that

~(0) = ~(0) = ~(~) = ~p(~:) = O.

(2,) See [2I] Chapter II .

I). W. Fox - C. PuccI : The Diriehlet problem ]or the waves equation 173

Let a, ~, -c be three f ixed positive numbers . We denote wi th F~,~ the class of the generalized solutions u of the equat ion (1) in { 0 _ < x < % 0 ~ t ~ , : l that satisfy the condit ions (2) and

where a is an arbi t rary real number such that t ~ - - a l < ~ . The study of the funct ions of class F~, ~ will settle the quest ion of the

solutions of our DmIcHIm~ problem with approximated data % ~, a. Fi rs t of all we will give for this case an existence theorem.

XVI. For any positive ~ and ~ the class F~, ~ is not empty.

We denote with /, an algebraic irrat ional that belongs to the interval [ a - :, a + ~], We denote with ~(x) and ~(x) two odd functions, periodic of period 2r:, of class C (~, and such that

I~ ~<~, I ' ~ - ~ l < ~ ,

in [0, ~:]. It is well known that such ~, % and ,~ exist. By theorem XIV a classical solut ion u of our DIRIC~tSE~ problem with data 0:, % q exists. Thus we have proved also that the subset of classical solutions of (1) conta ined in I ~ , , is not empty.

We now will prove a weak type of un iqueness theorem in the sense that two solutions of I '~, , differ little on certain subsets. We denote wi th ¢o(~) a majorant of the moduli of cont inui ty of the funct ions ~ and ~b. Let uz and u~ be any two funct ions of F s , , corresponding to the ratios ax and a~. We wilt denote wi th ~N the m a x i m u m of the two numbers

Sup

Sup

max{I ul(x, hair:) - - uz(x, n:~==) I ;

max {t ul(nalz:, t t - - u=(n:c2r:, t) t;

O ~ x < . ~:, n = O , 1, . . . , N } ,

0 < t < 2 ~ , n = 0 , 1 , . . . , N t .

XVII . inequality

We will call ~N the d iameter of the class Fz,¢ relative to the set R~,N.

The diameter ~N of the class r c,~ relative to R~, N satisfies the

N (41) ~2v<2 Z ~o{nr:z) + 4 N s ,

a n d thus ~N converges to zero when ~ and • approach ~ero.

Denote wi th ~¢i, ~ , and q~ (i = 1, 2) the data corresponding to a funct ion u~ in Fc,~, F rom the fact that

174 D. W. Fox - C. Puce i : The Dirichlet problem for the wares equation

and by (36) we have

~--1

1=1 n --I n--1

+ E ~(12j+1--nl~-)=4n~+2 ~ ~(j~-). j=o i=0

With analagous consideration of the difference uz{na~=, t) - - u~(n~7:, t) and by the definition of 8N the inequali ty (41) follows.

8. S o l u t i o n s w i t h b o u n d e d d e r i v a t i v e s .

We now consider for any % ~, and ~ the generalized solutions of our D~RICHLE~ problem tl}, (2}, (3), extended in the manner of theorem I to all the plane. We define I'L to be the class of these solutions which almost everywhere have first derivatives bounded in absolute value by L. For any function f{~) continuous in [0, ~] we define

I] f l l = max If(x) l x ~ [0, ~1

XVIII . For a --P-, (p, q) "- 1, any function u(w, t) which is of the class q

PL (and corresponds to the data ~, ~, :~) satisfies the limitation

(42) l u(~, t)I <_<_q( ]1 ~ II + tl ~ LI) + ~-. We first note that

(43) u( nT:, q , t)_< q(Ll~l[ +liq, tt), n--0, 1, 2,..., 0 < t < p - - _ _ q 7:.

This follows immediately from (27) and (28).

For any x in [0, ~z] there is a point of the form n__~_rc which is distant from q

a~ by no more than ~ . Thus by (43) and the bound on the derivatives of u

the] inequal i ty (42) follows. We denote by d(a) the maximum in R of the absolute values of the

functions u of the Class I~L, which are generalized solutions of the homo- geneous DIRICHLET problem (1), (2~, (3), with q0 _ ~ ~ 0 for a given :~. We have

d(a) ~ - 0, ~ irrat ional ;

~L p (2~i " d(~) = ~ , ~ =q-, (p, q)= 1

(~2) The func t ion d(,,) is a we l l k n o w n example in the t heo ry of func t ions of a rea l va r i ab le .

D. W. F o x - C. P u e c I : TI~e Dirichlet problem ]or the waves equation 175

For the case a i r ra t ional this follows from Theorem X I I I , for the ease

a - - P - we have by the previous theorem q

=L d(a) < ~ .

That the equal i ty can hold follows by Theorem VI1 and by set t ing

E ( z ) = ~ z - - , 0 _ < ~ < ~ ,

E ( z ) = - ~ ~ + , - -~ _ ,

with E a periodic funct ion of period 2__~. q

XIX. For any irrational ~ a function F~(8) is defined for 8 > 0 with the properties that

(44} lira E~(8) = F(0 ) - - 0, 0 </7'~(8) < L n , ~ - - ~ 0

and such that any function u of class PL (corresponding to the data % ,~,, ~) satisfies in R the limitation

(45} !nix, t ) !< i%t it ~ il + il '~ ii).

Let F~'~} be the sup remum of the max ima in R of l u l for u in the class FL, and u a generalized solution of our DIRICHLET problem with ii ~P ]I-4-]i ? !i .< 8. Clearly F~{8) is non-nega t ive and is loss than L~ by {2) and the boundedness of the derivat ives of u. F~ (0} vanishes by Theorem XII I . In order to complete the proof of the Theorem it remains only to show that we arr ive to a contradict ion if we suppose that there exists a sequence ~ 8, t converging to zero such that

(46) l im F~(~,,) - - a > 0.

In fact, by the defini t ion of F~ there is a funct ion u,,{x, t) in I'L, a genera- lized solut ion of the DIRICRLET problem (1), (2), and

such that a

(471 F~(8.) < max [u.(x, t) t q- ~ , (x, t) ~ R

a n d

(48) II ~. li + II % tl ~ 8. .

176 D. W. Fox - (7. P~:ccx: The Diric'hlet problem, for the u'at'es equation

Let ~'~ be nn integer such that the set of points

fm I m ~ - - [ 2 - ~ - ] . n , m - - 0 , 1 2 N~ (.,8)

part i t ions the interval [0, 27:] into sub- intervals such that each subinterval a

has a length less than~-~ . The funct ion u,, on the lines

x = m ~ - - ~ - 2r:, m = 0 , 1, 2 , . . . , N~.

is bounded by N~,( It %, II -{- II '~,, ti) as follows by {37) and the periodicity of u,,(x, t). By the bound on the derivatives of u and the definit ion of N~

By (47) and (48)

a I t)T <N (Hv. if+ tf ,+. 1 t )+3 .

No is independent of n, and thus N~8,~ converges to zero as n approaches infini ty so that we arr ive to a contradict ion to (46).

In the theorem we have jus t proved we have shown that for ~ irrat ional the solutions of our DIRIOHLET problem in the class FL depend cont inuously on the data .% ~; in the corresponding theorem X V I I I for ~ rational the

~:L resul t is weakened by the term ~-~ due to the presence of eigenfunctions.

We will now show that in the class I'L the solutions depend in a way cont inuously on ~ {precisely with the same type of cont inui ty as the function d{a} previously defined}.

XX. Let us(x, t) (i -- 1, 2) be two functions o f the class FL corresponding

to the data ¢~s, ~ , as, (i - - 1, 2). I f a~ -- P- we have the limitation q

(49) I u , ( x , t) - - us(x , t) l ~ q(II ~ , - - ¢Ps II + lI '~, - ~ II + L i a , - - a s 1 ) + L n . q

The funct ion u t - - u s is of the class r2L; by the bound on the derivatives of u~ we have

l u,(m, ~ ) - - u,(~c, a,u) l <_ 11 9~- - 9s ]l + L]a , - % [,

and by X V I I I follows {49).

(sa} W e denote, as usual , w i th [x] the la rges t in teger contained in x.

D. W. Fox - C. Put t1 : Thc Diriehlet problem for the waves equation 177

XXL Let ~, % ~* be three positive numbers and q~*(w), ~*{:e) two continuous functions defined on [0, 7:]. Let F* be the subclass of FL which consists of solutions of the Dirichlet problem {1), [2), (3}, with data ~, % q~. that satisfy the inequalities

(50)

I f in the interval [a*--z,

with q - - 1 -~- 4~ ~- 2L'c

(51) i

Let ~ = u~

~* + ":] there is a rational number a : P-- {p, q) : 1, q'

, then for any two functions u , , u, in F*:

t} - - u.,{w, t) t ~-- 3\~ Lr~(2L: + 4~). {24)

- - u ~ . Then v is in the class I:~,~ and satisfies (1), (2) and

v(x, ol ;1 <- it v z, rc II N 2t +

By Theorem XVIII , L ~

I viw, t) l <-- ~4~ + 2L'Oq + , q

and by the definition of q

W L~: _ 4~ + 2 L ~ q - - ~ , 0 ~ . ~ < I ,

and thus

I v4x, t), <_ 2 VLrc~4~ q- 2L~i + ~ Y L~(4¢ q- 2 L J .

From this (51) follows.

9. A d d i t i o n a l p h y s i c a l c o n s i d e r a t i o n s .

In many applications of partial differential equations to physical problems more information is known a priori about the solutions than appears as data in the usual formulation. This additional information may be used as legiti- mately as the usual data in the determination of the solution. When it is the case that approximated data cannot yield the solution with sufficient accuracy, it is natural to try to use other information about the solution.

In many cases in which the one dimensional wave equation {1} arises, i% is reasonable to suppose an a priori bound on the derivative with respect to x of the solution. In fact, in the derivation of equation (1} for the vibrating

(~4) The value of q indicated is obtained by rninimiz~n~ the second number of ('49) with respect to q.

Annali di Matematica 23

178 D. W. Fox - C. Puc(:t: The Diriehlet problem for the ware,~ eqttation

string or the oscillating gas in a pipe the assumption that Ux is small is necessary in order for the equation 11} to describe the physical process.

It is easy to show that a solution of our DIr~ICHLET problem satisfies the relation

ut(x, t + x} = u~(x, t + z} - - u~(O, t),

and from this it follows that a bound on ux implies a bound on u t . Consequently the consideration of solutions in the class FL, as done in the previous paragraph, is natural.

W e suppose at f irst that :¢ is known exactly. If a is irrational then by theorem XIX it follows that the solution of our DIRiCI-ILET problem depends continuously on the data % ~. If there exists a solution corresponding to approximated data 7, qb, then it is possible to approximate this solution by using the expressions (36) and (37) to establish its approximate value on a sufficiently dense but finite family of straight lines and then interpolate using the bound on the derivatives. The accuracy of this approximation unfortunately depends strongly on a (ca which the integer N.~ in the proof of theorem XIX depends}. If a is rational then by q26t and {27} it follows that the solution depends continuously on ~ and ~ on the straight lines

}~rc x - - n r : a n d t - - - - - - . ( n - - 0 , 1, .. and ~---Pt . If there exists a .solutiol, (;or.

q q q responding to approximated data :p and ,~, the accuracy of the approximation

L= of this solution cannot be insured to be bet ter t h a n - - a s follows by theorem

q XVIIL This depends on the presence of eigenfuncfions.

The hypothesis that a is exactly known is unnatural and for this reason we have established theorems XX and XXI which deal with the dependence of the solutions also on a. Theorem XX shows the dependence of the variation

~u on the variation ~ for a rational ( a - - P } . The bound given by (49) can q

be very poor for q too large or too small. This shows that the dependence of ~u on 8a is not uniform with respect to a.

We suppose that the data % ~ can be approximated with an error tess than a known ~ and that :¢ can be measured with an error less than a known ~, and the bound L on the derivatives of the solution is known. If we suppose that the possible range of a is chosen in a convenient way, as indicated in theorem XX[, then the solution u can be approximated rather well, depending only on the magnitudes of ~, ~, and L. An approximate solution u~ if a

solution exists, can be computed by (26), (27j (using :¢ P --" ~, p and q defined

in the theorem XXI), and by the usual interpolation. Then (51) gives a bound for the error of approximation.

D. W. Fox - C. PvccI : The Dirich~et problem for the waves equation 179

In this section and in the previous one we have always made the hypo- thesis that a solution of our DIRmHr, E~ problem that satisfies the data approximately exists in the class FL. Clearly if we remove the condition that the solutions most belong to the class PL there are infinitely many solutions. But with this condition the existence is not always insured for arbi t rary %

and a. We now will discuss the meaning of an existence theorem from the point

of view of the physical application, aside froln its possible mathemat ical interest. If in a mathematical problem that arises from a physical question no solution exists, this means that the mathemat ical model of the physical situation is inadequate or mistaken. In this way mathematical considerations can provide a check on the reasonableness of physical assumptions. Bat this check is only partial, because the fact that the mathematical problem is soluble does not guarantee that it is a correct translat ion of the physical situation. Thus, in mathematical problems that arise from physical questions to suppose the existence of a solution is no more than to assume the cor- rectness of the translation to the mathematical problem (2s).

When our DIRrCELET problem corresponds to a physical situation for which the existence of a solution with bounded derivatives is known, formulas (36) and (37) provide a method for the approximation of the solution, and theorem XX[ can furnish a bound Oil the error of approximation of the solution, taking into account the approximation of the data.

10. A f o r m a l e x p r e s s i o n o f t h e s o l u t i o n .

We have considered two different expressions of the solutions ~)f our DI~tICI~LET problem, the first given by (6) depending on the funct ion E, the second given by (33) or (35) as a tr igonometrical series.

We will indicate now another expression of the solution that may have some interest for fur ther investigations (28).

For a fixed irrational a we denote with G(x) the formal series.

(53) ,o Sin nw

,~=1 Sin nay:

(~) ~ practical interest in an existence theorem can be that if its proof is costructive the proof furnishes a method of approximation of the solution. But this is not essential, because under the assumption that the solution exists it is often possibte to obtain procedures for the approximation of the solution.

(~8) We Wish to thank Professor MARCEL R~ESZ for suggesting this expression of tl~e solution to us.

180 D. W. Fox - C. l 'ucc~: The Dirichlet problem for the wam~.~ cq~atio~

and by G¢-*'(~v, the series obtained from (53t by k formal term by integrations, that is

(54) Sin nx ~ Cos n~

G~-~a'(x) ~ ~ (-- 1)~ n "~* Sin n:¢~:' = -n ~*+~ Sin na~: " ~ I ~ 1

term

We consider now an irrational :¢ and an integer k so large such that the series G C-a~ is uniformly and absolutely convergent (for example this is true for a¢ irrational algebraic and k - ' : 3). If ~p and .~ have continuous derivatives of order k, and these derivatives have a development in FOURIER series which is uniformly and absolutely convergent, the funct ion

2Tr

(55) u(x, ti= ~ / [ ~p'*'(~) - ¢p'*'(.~- a~:i][G~-"(x-l-t---:,-I--(--li*G(-*;(x--t.,:)] d% 0

is a generalized solution of the DmmHL~T problem (1), {2}, (3). To verify this is very easy. Suppose for example k even, k -- 2s. We have

a n d

on

G~-*"(x~ -t - t - - "O - G ( - 2 " i ~ - - t + ~) ~ 2 ~. {--1) ' n = l

Sin nx Cos n(t -- ":) n ~* Sin naT:

~c~, , (~) _ ¢p,~,,(~ _ ar~)~ ~ m~'{-- I)'[A,, Sin m': - - am Sin m(': - - a~:)]. m ~ l

The product of these two series is a uniformly convergent series, and integra- flag term by term we find for u the expression (55~.

FQrmally integrating (55) by parts k times we find the expression

(561 27T

u(x, t l ~ ~;: 0

which is a formal solution of our problem. We observe now the relation between the expression {56) of the solution

of our DIRICHLET problem and the expression of the solution of the CAUOHY problem stated by (1), (2} and

{51) u ( ~ , o)=~(x), u,(x, o)=g(x),

where gtx,) is an odd periodic function of period '2u. We have observed in section 1 that it is always possible to extend the solution of our DIRIC]tLET problem to all the plane as a solution of equation (1) so that it vanishes on the straight lines x -- 0 and x - - ~. Thus, using this extension, if we consider

l). W. l",~x - ('. l't~'('~: 7'he Iliri,hh't problem fro" !he wam:s ('qmttiop 1.81

a fami ly of DIRICHLET problems for d i f fe ren t £ s wi th cor responding da ta ,~,, ~ such that

lira ~ { x ) - - ~ { x ) _ g(x),

it can be expected that the solut ions u~, if they exist , converge as ~t approaches zero to the solut ion of the CaUCHr problem (1t, ~2), (57). This is not a lways true, bu t it is fo rma l ly t rue t ak ing in cons idera t ion the express ion of u~ given by (56~.

In fact, a lways formal ly , x-I- t

Gix 1

a n d

Observing that

we h a v e

xq-t 2rt

1 / " ;*~"~,--~0(,-- ~)O,(~__~:)d,~d~" t - - x 0

l i m ff¢lx) - - :p(; - - ar:) _ _ ~ ' ( x ) "4- g ( '0

oo l im aQ'(~ - - x) ~ )3 l im n~t ~ - . 0 n~--~l u.---~O

Cos n(~ - - ~) ~ _ 1 E Cos n{~ - - x) ~ ~(~ - - x) Sin n ~ : = ~: .,,=I = ~2::'

where ~ is tlle DIRAC funct ion. We then obta in

tq-x fin

1i,.o..,o, , , i f / : _,_ t.--x 0

T ak ing into account the fact that

~ ( t - - x i - - - - :~(x - - tl

and tha t by the oddness of g

x+t ~ + t

t - - x x - - t

we have as a formal l imit express ion of u~ as a approaches zero:

x+t

u(x, t) -- ~(x q- t) q'2 ~ ( x - t) _}_ 21 f gl~ld~. x - - t

t--x

182 D. W. F o x - C. P u c c I : The', l)irichlct problem )or the irocc.~ cqtmtio~,

BIBLIOGRAPHY

[I] J. ttADAMAaD, 0a Some Topic s Connected with Linear Partial Differential Equations. Proceedings, • Benares Mathematical Society ,, Vol. 3, {1921), pp. 39-48.

[~] 5. HADA_MARD, Equations aux Derivees Partielles, • L'Enseignment Mathematiquo ,, Vol. 35, {1936), pp. 5.42.

[3] A. HUBER, Die Erste Randwertaufgabe fuer geschlossene Bereiche bei der Gteichung u~y-- f(x, y), • Monatshofte fuer Mathematik und Physik ,, Vol. 39, (1932), pp. 79.100.

[4] D. G. BOtraGIN and R. DUFFI~r, The Dtrichlet problem for the Vibrating Stri~,g Equa. tien~ eBull. Am. Math. See. , 45, (1~5) pp. 851.858.

[5] D. @. BouRGi~,The Dirichlet Problem for the Damped Wave Equation, • Duke Math. Journal ,, Vet. 7, (1940).

[6] O. G. OWESS, Homogeneous Dirichlet Problem for Inhomogeneous Ultrahyperbolic Equa. lion, , Am. Journal Math. • 74, (1952}, pp. 307-3t6

[7] C. PtrCcl , Studio col metodo dell¢ difference di un problema di Cauchy relative ad equa. zioni parziali di ripe parabolico, ~Annali Seuola ~'ormalo Sup. di Pisa ,, (1953).

[8] C. PuccI, Sui problemi di Cauehy non ben posli, c Rend. Ace. ~az. Lincei ,, 8, 18, (1955), pp. 473.477.

[9] C. Puccx, Discussions del problema di Ca'uehy per le equazioni a derivate parziali di ripe ellittieo, • knnali di Mat. pura App1. *, 11958).

[10] F. JOHn, A note on • improper, problems in partial differential equations, • Comm. Pure Applied Math. , 8, (1955}, pp. 591.594.

[11] F. JOHN, Differential equation with approximate and improper data, Lectures, New York University, (1954.55).

[12] M. M LAVRENT/'EV, On the Cauehy problem for the Laplace equation, • Isvestia Akad. ~auk U S S R , 20, {1956}, pp. 819.8~2.

[13] N. E. NoaLuNo, Memoire sur les polynomes de Bernoulli, , Aeta Math. • 43, {1922}, pp. 1"21- 196.

[14] G. SANSONE, Moderna teoria delle funzioni di variabile reole, Parte Seconda, Zaniehelli, Bologna, 1946.

[15] J. ~. KOKSMA, Diophantisehe Approximationen, ~ Ergebnisse der Math.. , Vo]. 4, Springer, Berlin, 1986,

[16] E BERYL, Lemons sur la Th~orie des Fonetions, @aathier-Villars, Paris, 1928. [17] TH. SCHNmDm~, Einfuehrung in die Transzendenten Zahlen, Springer, Berlin, 1957. [18] M. PlCOl~E, Appunti di Analisi Superiore, Bondinetla, lgapoli, 1940. [19] A. ZYGMUND~ Trigonometrical S~rios, ¢ Monografje Matematyezne ~, V:ol. 5, Warszawa-

Lwow, 1935. l~0] O, PERRON, Irrationatgahlen, Chelsea, New York, 1948. [~1] J. HADAMARD~ Locl~,ur~ on Cauchy's problem, Yale University Press, New Haven, 1923.