RICE UNIVERSITY
Three Circle Theorems in Partial
Differential Equations and Applications
to Improperly Posed Problems
t>y
Keith Miller
A THESIS SUBMITTED
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Houston, Texas
May, 1962
To Barbara
Acknowledgements
The author wishes to express his sincere appreciation
to Professor Carlo Pucci, who, visiting Rice University
from the University of Rome in the fall of 1961, first
stimulated his interest in improperly posed problems,
and whose suggestions in many valuable conversations led
to the preliminary three circle results of section 2.
The author is deeply indebted to Professor Jim
Douglas, Jr., who introduced him to the field of partial
differential equations, and whose guidance in the writing
of this thesis led in many instances to clearer and more
interesting results.
Table of Contents
1.
Introduction
1
2.
Three circle theorems for harmonic
functions
4
Three line theorems for periodic
solutions of the heat equation
18
Three line theorems for periodic
solutions of the Cauchy problem
for Laplace's equation
24
Harmonic continuation, data given
on a whole circle
37
6.
Some auxiliary three circle bounds
43
7.
Harmonic continuation, data given
at a finite set of points
48
8.
Analytic continuation
57
9.
Backward solution of the heat equation
on a rectangle
61
3.
4.
5.
10. The Cauchy problem for Laplace's
equation on a rectangle
64
Three Circle Theorems in Partial
Differential Equations and Applications
to Improperly Posed Problems
1,
Introductiono
For complex valued functions analytic and
single valued on an annulus, we know by Hadamard's three circle
theorem [16] that the modulus on an intermediate circle must go
to zero as we let the bound on the inner circle go to zero, the
bound on the outer circle being fixed.
Suppose f(z) is analytic
and single valued on the annulus a ^ Izl < 1, continuous on the
closure, and
If (z) I - m = a1^
,
Izl = a ,
If(z)1*1,
Izl = 1 o
Then, Hadamard's result is that, for a - r — l
log r
I f (z) I * m'*'0^
a
= r0*
,
IzI = r .
Further, this is an optimum bound; for when <x
, defined by m = a.°*
is an integer, the power function z^ satisfies the hypotheses and
assumes the bound.
This theorem establishes the Holder continuous
dependence of f(z) on the data for analytic continuation from
boundary data on the inner circle, provided f(z) has a prescribed
bound on the outer circle.
In this paper analogous three circle theorems are found for
harmonic functions on a disc, but with respect to
weaker bounds on the inner and outer circles.
or even
In fact, it is
sufficient that the function correspond to a Schwartz distribution
on the outer circle.
Analogous three line theorems are found for
periodic solutions of the heat equation and for periodic solutions
of the Cauchy problem for Laplace's equation.
In the process, a
very general and flexible method is presented for finding uniform
bounds on the solution and its partial derivatives in terms of
bounds on the data.
Moreover, these bounds are very precise; in
2
certain cases they are, except for a factor of at most two, the
best possible bounds.
The method involves reduction of the problem
to a consideration of the Fourier coefficients of the solution and
maximimization of linear functionals of the coefficients with
respect to quadratic constraints.
These results lead to numerical methods for three of the
classical improperly posed problems in partial differential equations:
harmonic and analytic continuation, backward solution of the heat
equation, and the Cauchy problem for Laplace’s equation.
See F. John
[ll] for a discussion of the wide class of improperly posed problems
for which continuous dependence on data can be restored by restricting
attention to those solutions satisfying a prescribed bound.
also pages
226-231 of [
3
] .
See
If data for harmonic continuation
is given on a whole circle, the method involves nothing more than
the computation of a certain number of Fourier coefficients.
If
data is given only at a finite number of interior points, the
method involves the solution of a set of linear inequalities.
Both
methods extend immediately to analytic continuation, in which case
the solution is found in Taylor series form.
Completely analogous
methods are found for backward solution of the heat equation on a
rectangle, and the Cauchy problem for Laplace’s equation on a
rectangle„
For the sake of notational simplicity,all three circle results
will be derived on the unit disc and with the prescribed bound on
the outer circle normalized.
Similarly, for the three line results
the period will be 2% and the prescribed bound will be normalized.
This is sufficient, since our equations are linear, with constant
coefficients.
The following simple result will be used repeatedly.
Maximimization lemma. The maximum of the linear functional
K
F lx | =
IT
x A
l ni
n n
o
with respect to the quadratic constraint
3
s
E
W” z
o
is
(1.1)
Further, the same result is true for E = oo if (l.l) is convergento
If (l.l) is divergent, then the supremum of such F jx^j is Infinity.
Proof?
If E is finite or (l.l) is convergent, write
K
E
n
(x
n
n
o
V
o
As
A
B
f
n
•2
n
’X' V
satisfies the constraint and F jx^ j assumes the bound (l.l), the
result follows from the Schwarz inequality.
If E = CD
and (l.l) is divergent, consider the maximum of F
over all sequences jx^ ^ which terminate after N terms and which
satisfy the constraint.
This maximum was shown to be
C
which diverges as N increases to infinity.
proved.
9
Hence, the lemma is
4
2.
Three circle theorems for harmonic, functions<>
For
harmonic functions on the annulus, we see that no theorem similar
to Hadamard's is true.
In fact, if we demand harmonicity merely on
the annulus we do not even have unique dependence upon the data on
the inner circle, much less continuous dependence.
For example,
the data zero has solutions u = 0 or u = ar”"*" cos © - a~^r cos © .
We therefore consider only functions harmonic on the whole open unit
disc.
A three circle theorem for the disc follows from Hadamard's
theorem.
If u is small
on the inner disc and
bounded on the outer
disc, then its harmonic conjugate can be made small on the disc of
radius a - e and bounded on the disc of radius 1 - e.
Thus we may
apply Hadamard's theorem to get bounds on u on intermediate circles.
However, these are not the best possible bounds, and they are not
clean cut.
Before treating three-circle theorems, let us consider some
preliminaries.
Harmonic functions on the open unit disc have there
the Fourier expansion
®
(2.1) u(r,©) = — + /_
(a
cos n© + b sin n0) rn,1
Y2" ^—
n
n
r
«<• 1 .
Denote the uniform norm and the L^ norm on the circle of radius r
as follows:
lul r =
sup lu(r,©)I ,
©
(2.2)
llul,
r =
f
2»
2%
J
I
dQ
u2 r Q
( > )
)
0
I
with the convention that b
o =0.
r ^ 1 ,
Notice that we have written the
5
constant term in (2.1) as
a
— , rather than the usual Fourier series
a
*2
notation
, in order that Hull
may be in the simple fofm (2.2).
Each term of the sum in the norm is an increasing function of r.
Hence, if
Hull
is bounded for r
1, the sum converges for r = 1,
and we may define
Hull
(2.3)
=
•
lim Hull
rfl
= sup Hull
r
r^l
-
-
+ b 2 21
= (| Z ( n
n >)
=
r
oo
2
a
•
0
For an integer N — 0 let the corresponding low and high order
parts of u be
a
*
uM(r,9) = ~1 +
XM
Y2
(a
cos n© + b
^
^
sin n©)rn,
(2.4)
(3D
uN°° (r, 9) =
(an cos n© + b^ sin n©)rn,
^
r
N+l
For the real number
that [cx] - c<
cK
, let [cK] denote the greatest integer such
.
Let us begin now the consideration of three circle theorems
by deriving the optimum L^ bound for L^ constraints.
Theorem 1.
If u is harmonic for r ^
Hull
cl
- m =
a*^
1, and if
,
Hull
then, for a - r - 1 ,
log r
(2.5)
Hull
l0
r
^ m S
a
=
.
1 „
6
Further, this is an optimum hound, since u = ~f? r
the hypotheses and assumes the bound when
Proof:
cos eXQ satisfies
is an integer.
Recall that
QD
o
Let
A
n
= ~(a2 + b2)
2 n
n'
and consider the analytic function
CD
F(z)
=
A
51
z2n
n
»
IzI ^
1 .
o
Since A^ - 0, the maximum of F on the complex circle Izl = r is
assumed on the real axis; therefore
Hull l = F(r) =
r
-
max
IzUr
lF(z) I .
-
The result follows immediately from Hadamard's theorem,
Lg bounds are often less interesting for applications than
uniform bounds, and we shall turn to the study of uniform bounds.
It will not be necessary to prescribe uniform bounds on the
constraint circles; somewhat
weaker bounds will be sufficient.
The result will be based on lemmas bounding the low and high order
parts.
Lemma 1,
Let L|j = ^u^ ; llu|j|la -
L(H,r) =
mjj and let
sup
U
N
e
•
L
n
Then,
U 2n\1?
(2.6)
L(N,r)=m(l+2 21 (§)
1
J
.
7
Proof:
contains the rotations and the negatives of its
LJJ
member functions«
Thus,
LJJ.
Therefore 1(11,r) is the supremum of uN(r,0) over
the problem is to maximize the linear functional
N
Ujj(r,0) =
a
n
r'
n
1
with respect to the quadratic constraint
N
o
l|un" a
,
2 , 2v 2n ^
(an + bJa
= 2
“
m
0
It is sufficient to maximize u^(r,0) over the subclass for which
b^ = 0»
Therefore, by the maximization lemma,
1
2n \ 2
N
max
U
H
e
u^(r
,0) =
(f)
%
Lemma 2.
Let
HJJ
=
ju® : llu®ll ^ - 1 J
and let
H(N,r) =
sup
00
U
N
TJ
luS? |
■N 'r
e
Then,
CD
(2.7)
2n
H(N,r)
1
2
N+l
Proof: Exactly as in the previous lemma,
the problem reduces to the
maximization of the linear functional
^
uN
oo
(r,0) =
N+l
with respect to the quadratic constraint
an r11
8
nuju
co
l
1
2
a
a
2
n
ar+i
^ 1
1
•
Therefore, by the maximization lemma,
max
u“ (r,0) =
\ 1
2n | 2
( 2
CD
H+l
IH
%
I
Let C"7 be the class of harmonic functions u(r,9) on
Theorem 2.
r < 1 which satisfy
(a)
(2.8)
Hulla -
m
- 1,
.
(b)
Hull ! * 1 .
For a - r < 1,
let
M =
sup lul .
u e/ Then,
(2.9)
max (L,H) - Mr - L + H ,
where
/
^ 2n\2\
L = L( [°<],r) = m^l + 2 21
J
1
oo
(2 o 10)
H = H([<*],r)=
(2
X
21
[<*+l]
and where °< is defined by m = a
.
Moreover,
(2.11)
L + H *
mXog”a
[ (2 ^
0
+
/1_2_)
1
~
r
2
]
9
Proof: Consider u e P .
Let
N - 0 and write
00
u
uN + u.
N
We notice that both the low and high order parts of u are also in
P
? since
N
2
(al) llu„ll =§
(bL)
Z0
2N
+
2 _ 1
N' 1 “ 2
<an2
I u-
(2.12)
+
a
2n
= m = a
bn2) ^ 1 ;
00
(aH)
lu”l
l=|
/ 2 , 2x 2n
2
2c*
(an + bn )a
- m = a
,
N+l
oo
0Q| 2 1
(bH) lu
N 1 “ 2
(an2
+
bn2) 5 1
N+l
By (2.12 aL), u^- is in the class
oo
N+1
U
is
in
class
%
N
of
1N
N
lemma
2.
of lemma 1.
By (2.12 aH)
We therefore have* for any N,
Mr * L(N*r) + H(N,r) .
In particular this is true for N = [©(]* so the upper bound of our
theorem has been obtained.
We consider the lower bound.
Let
fbe
the class of low
order harmonic functions satisfying (2.12 aL) and (2.12 bL)* and let
r7
be the class of high order harmonic functions satisfying
(2.12 aH) and (2.12 bH).
Let
and lu®lr for all functions in
and h® be the supremums of lu|jlr
P^ and
and P® are subclasses of P j
hence*
r7® respectively.
P^
10
1N
h
n
M
£ Mr »
r
00
N
or,
(2.13)
Evaluation of
max C^jph® ) - Mr .
^ or h® would involve the maximization of a
linear functional qf the coefficients with respect to two quadratic
constraints.
Fortunately, for the proper choice of N one of the
constraints implies the other, so we need maximize only with
respect to one constraint.
In (2.12 aL) and (2.12 aH) we bring the sT
to the left and have
N
(al)
|
21
(an2 +
b
n2>
<an2
+
* 1 ,
0
(2.14)
00
(aH)
\
Z
'»a2)a2(*«0 * 1 .
N+l
For N - °(
, all the terms
n
a2(
“*0O in (aL) are greater than one;
hence, the inner constraint (aL) implies the outer constraint (bl),
and coincides with
(2el5)
of lemma 1.
Therefore,
^ = l(N,r), N^*
For N+l - o(
f
n_
ail the terms a^ ^
(aH) are less than one;
hence, the outer constraint (bH) implies the inner constraint (aH)
Therefore,
(2.16)
Since [<*] from (2.13),
h®
= H(N,r),
N+l
o<
^ [e< + l], the lower bound of our theorem follows
(2.15), and (2.16) for the case N = [c<] .
11
Finally, we manipulate the bounds (2.10) into more convenient
form (2.11).
log r.
log
a
Recall that m
= r * .
From (2.10),
M
L ^
m ( 2
P
0
1
c<
2(n-[rf])
2
= r
0
Also,
r2M =
e
The bound (2.11) follows, and the proof is completed.
It follows from (2.9) that
(2.17)
i.e.,
| (L + H) £ Mr * (L + H)
;
1 + H is, except for a factor of at most two,
the best possible
bound.
Suppose we had assumed uniform rather than Lp constraints.
is,
suppose (2.12 a^ replaced by
1
by IuI- 1.
Let M
class.
is an integer,
If
lula - m = a
be the supremum of
then u = r
lul
That
and (2.12 b) replaced
over the corresponding
cos cK9
satisfies the
log r
0(
constraints; hence, M ' must be at least as great as r
=
l0£C £1
m ®
12
Therefore the ratio M /M '
is no greater than the value of the
bracket
from (2.11),
We see that this bracket tends essentially to
1
1
2
(2(log m/log a) + 2)2
(2 <*+ 2)
(2.18a)
as r decreases to a,
is bounded by
1
1
2
2
(2.18b)
(a/r)‘
independently of m for a ^ r ^ 1»
and tends to infinity like
1
(2.18c)
as r tends to
1.
Therefore, we conclude roughly that the bound for
Lg constraints differs greatly from the bound for uniform constraints
only near the boundary r =
1
.
We may obtain three circle theorems for more general constraints
on the outer circle.
In fact,
it is sufficient that the boundary
values of u correspond to a Schwartz distribution.
and 797 of
[5]
.
Notice that
,a
n'
Thus,
/~'
9
is contained in
in the following theorem.
Hull
^ -
1
implies
lbJ ’ “ ^
CQ,
See pages 795
n ~ 1 .
where the general class C^. is defined
Notice that
CQ
also includes harmonic
functions with delta functions for boundary values, for example,
the kernel of Poisson's integral on the unit disc.
13
Theorem 3.
For an integer k, let
be the class of harmonic
functions on r < 1 which satisfy
(a) Hull
* m = a*^ 1 ,
a
(2.19)
(b)
la.
n
> 'V
12
, -
nk,
n -
1,
M
z0
1
.
and a - r <
log r
alog a
r,
[>
Moreover.
2n\ 2
(?)
)
+
2
®
Z0
for k >■ 0
log r
ml0S
lulr
(2.21)
a
(«+l)k
Bk(r)
where
00
(l+n)k :
Bk(r) =
n=Q
i
Por k = 0 or k = - 1,
(2.22)
lulr <
log r
log a
m
"
M
z
/
_(2
ft)
1
2n \ 2
1 - r
0
]
Por k - -2 ,
(2.23)
lulr<
m
log
log
°
r
1
2n \ 2
M
a
Z
(5)"]"
+
rJ-
0
Proof:
We write
00
u = u
M
4*
U
M
It follows from (2.19 a) and lemma 1, as in theorem 2, that
lu^-j lr is bounded by the first term in (2.20).
Prom (2.19 b),
I
14
00
CD
(2.24)
lu^-j (r90)l =
(a„ cos n © + b
|
u
n
sin n ©)rn |
[<*+l]
CD
CD
/ 2 , - 2x2 n
(an + ^ ) r
2nk r11
[<*+1]
[<*+!]
CD
,[<X+l]
nk
n
~[c<+l]
r
[<*+1]
CD
r
Therefore,
(2.20) is proved.
Then (2.21),
2
/
(n + [=<+l])k r11
2
^
(2.22), and (2.23) follow from (2.20).
1
2n
2n \ 2
( Z ft) )
GO
(n + [*+l]k rn J
2
+
0
0
oo
1
5
,2
(2^+l])k
For k >■ 0 ,
+ 2[«+l]k
21
(1
0
+
n
Vs
ITT)
wn
rn
CD
(1 + n)k rn ;
4(*+l)k
0
hence,
(2.21) follows.
In like mammer,
(2,22) follows.
(2.23) is an immediate result of (2.20) and the identity
OD
n
and the theorem is proved
o
£.
=
6
Finally,
15
Not only u, but also all of its partial derivatives,
tend to zero on intermediate circles as we let the inner
constraint approach zero.
Lemma 5.
log r
(2.25)
F
If u is in the class
ai+3 U
^
log a
m
of theorem 2, then
. .1
(<X+1) 2
4
dr1 d©3
B.
.(r) ,
3
where
B
i.j,(r)
=
yL
(l+n)2<i+J>
( 2
2n
r
)
2
^
co,
0
a - r < 1,
Proof:
Again, it is sufficient to find a bound at the point
(r,0).
Differentiating the series (2.1) for u, we have,
a1^ u
(r,0) =
dr1 3©*i
CD
rs
y
An n^(n)(n-l) ... (n-i+1) r11^
,
1
where A^ is either a^, b^, -an, or -b^, as o(mod 4) is either
0,1,2, or
Suppose,for example, that j(mod 4) is 0. Then
An s an and
(2.26)
,i+d u
dr a© 3
M
(r»0)|
CD
laj n1+3 r11
16
Consider the constraints (2.8) for
f
in the form (2.12)„
The
maximum of the first term of (2„26) with respect to
[«]
(2.27)
a2n
(aL)
* 2 m2
1
is
1
(2.28)
z
m
n
2
2(i+j)
I
1
*M->) 2
1
r
log r
.
m
. I
1
+D+2
l0
<
i
S a
M
The maximum of the second term of (2.26) with respect to
CD
(2.29)
(bH)
a
n2
- 2
E*+l]
is
1
CD
n2(i+j)
(2.30)
(
^■(2
r
X
(n+[*+l])2fl+3) r2“)2
H|c\J
o
,
[ +1]1+S
mT§fi
\
r
Finally,
r2n)
z
[<*+l]
2
r
2
2
oo
X
U+n)2(i+3) r2n
G
(2.25) follows easily from (2.28) and (2.50)
17
The unbounded character of lulr in theorem 2 as r
approaches one, as shown by (2C18 c), points out the fact that
Lg bounds imply good uniform bounds only where the function is
smooth.
We can impose this smoothness by prescribing a bound on
the I<2 norm of either the
tangential or the radial derivative
of u on the outer circle.
Notice that
CD
(2.31)
llurll
Lemma 4.
±
= l!uQl|
If u is
±
*2 <an2 + bn2))
= (
harmonic for r < 1* and if
(a) Hull
Q
a
^ m = a** 1 ,
(2.32)
(b) liurJ| ^ = IIUQII
then, for a - r
lulr
(2.33)
1
- 1 ,
1 ,
log r
l0
S m « a
1
2n \ 2
M
r
IS
151
0
Proof: The bound for lu{p<]lr follows from (2.32
a) as in theorem
CD
2.
We find a bound for luj^-jlr with respect to (2.32 b), using
the formula (2.31) <>
co
lu
M'r
■£
log X
mlog a
The desired bound,
x
1_
\ _
2 I n >
?
19E..21
log
m
m
follows immediately, and the lemma is proved
a
%_
6
,
18
3°
Three line theorems for periodic solutions of the heat
equation,,
We consider solutions of the heat equation on the
upper half plane which are periodic in x with period 2%c
That is,
consider functions u(x,t) satisfying
(3.1)
(a)
u(x,t) e C2,
(b)
u
(c)
u(x,t) is periodic in x with period 2® o
XX
t ^ 0 ,
= u^,
t >■ 0 ,
The functions of this class have the representation
a
®
2
u(x,t) = — + >>
(a cos n x + b sin n x)e“
, t > 0,
n
n
YT
(3.2 a)
where
j a^, b^ J
is any sequence such that
Z
00
,
(3.2 b)
2 ,
(an
2.
-2n2t
+\)e
<
00
t >
The proof of this representation is as follows„
satisfying (3.2 b), then the sum (3.2
0
Given
|"a , b j
a) is a solution of (3.1),
since it converges subuniformly on t >0 and its individual terms
are solutions of (3.1).
2
Conversely, given a solution u(x,t) of (3.1), let a
e~n
2
nj c
an<
*
-h e
b
n,e
e
e
e
denote the Fourier coefficients of u(x,e) for some
9
> Go
Since u(x,e) is in C , its Fourier series converges
uniformly; hence,
a
®
u (x,t) = -°s-e + /
(a
cos n x + b
sin n x)e
z "/"2~
z
2,
*
is a solution of the initial value problem on t ~ e for the data
u(x,e)0
By the uniqueness for this problem ,
u (x,t) = u(x,t),
e
t - e .
19
Clearly, the coefficients
a
n
= a
n,
c1
b
n
=
n,
e‘
are independent of e; for if 0 < e1 <
e,
then
u»(x,e) = u(x,e) = u (x,e) .
w
w
Thus (3.2 a) is proved*
Finally, define the L2 norm on the
horizontal line at time t
r\
1
_
2n
(3.3)
Hull ^
~
u2(x,t) dx)
,
o
®
/I
= (2 Z_
/
2
K
p \2
2v -2mt r
,
+
b
n )e
)
,
t > 0 *
0
Since u is continuous, this integral is finite for t
>
0; hence
(3.2 h) is established and "the representation is proved.
For such periodic solutions of the heat equation we will derive
three line theorems which correspond exactly to the three circle
theorems of section 2.
Only the series representation has been
2
changed (e“n
of this section corresponds to r11 of section 2),
and in most cases we will merely point out the similarity to the
corresponding proof of section 2.
Let lul^ denote the uniform norm on the horizontal line at
time to
We may again define
GD
(3.4)
f ^3 X
Hull/") = lim Hull. —
. . u .
t 10
■ ■
^
Q
(a2 + b2))
n
n/
.CD
We again write u = uK, + uK.
, ,,
,
n
N
N , where these low and high order parts
are defined analogously to (2.4).
We have first the theorem corresponding to theorem 1.
Theorem 4* If u(x,t) is a solution of the heat equation for t > 0,
periodic in x with period 2K, and if
20
—8.
m = e
Hull
iiuii
Q
-
i
> «,
- 1 ,
,
then for 0 - t ~ a ,
(3.5) Hull .
«c.— m
ma = et
-o£
Further, this is an optimum hound, since u = V2 e‘' t* cos <?<x
satisfies the hypotheses and assumes the bound when ©< is an
integer.
The proof follows exactly as the proof of theorem 1, except
that we let Izl = e“
Lemma 5.
Let L^ = ^ u^s
instead of IzI = r
llu^ll
L(N,r) =
~ mj , and let
&
sup
lujJ.
uH e %
Then,
If
21
L(N,r) = m(l + 2
(3.6)
e2*1^3-^) )
The proof is completely analogous to the proof of lemma 1,
Lemma 6.
Let % = | u^? j |lu^°
H(N,r) =
IIQ
sup
OD
U
N
- 1 j-
, and let
lu® I
.
TT
e
%
Then,
z
CD
(3.7)
H(N,r) =
(2
e
2, \ 2.
-2n t
N+l
The proof is completely analogous to the proof of lemma 2,
21
Theorem 5.
let
F
he the class of solutions u(x,t) of the heat
equation on t > 0 which are periodic in x with period 2u, and
which satisfy
(a) Hull
CL
(3.8)
(b) iiuii
For 0 < t
-
™ m " 1 ,
-
i .
uSUPr
lulr
Q
a s, let
Mt =
Then,
max (L,H) £
(3.9)
Mt £ 1 + H ,
where
M
2
1 = L([e*],t) = m (l +
2
e2n
(3.10)
oo
2
H = H([o<],t) =
( 2
e ~2n
21
t '
)
[*+l]
and where
o(
is defined by m = e
-o<2 a
Moreover,
1
(3.11)
L + H <
a
m
^(2
22
e“
2n 2 a
0
♦ («
i
2
- ( “-'t))<
1
2
1
\ 2
(a~^) j
22
The proof for (3*9) and (3.10) is completely analogous to the
proof for (2.9) and (2.10) of theorem 2.
Again* for N = [o<], the
low order inner constraint (aL) implies the outer constraint (hi),
and the high order outer constraint (bH) implies the inner
constraint (aH).
Thus* in each case* we need maximize with respect
to only one quadratic constraint.
Finally*
(3.11) follows easily
from (3.9) and (3.10).
Theorem 6 .
For an integer k, let
u(x*t) of the heat equation on t
>
be the class of solutions
0 which are periodic in x with
period 2n, and which satisfy
(a) Hull
((3.12)
(b) lanl,
Then* for u e C,
m = e“*2a
^
3.
lbnl
i2
-
£ 1
nk*
n * 1
and 0 ^ t - a
1
2
M
t
e
0
GO
2
+
0
Moreover*
for k > 0
t
(3.14)
lul^ <
a
a
m
(o<+l)k Bk(t)
where
GO
0 < t - a
Bk(t) = 4
n=0
For k=0ork=-l*
23
±
(3.15)
a
r/
lult<m [(2
[<*]
00
?
X
e~2n
2
(a-t)j
+
£
0
?
e-n
t]
For k - ~2 ,
M
(3O16)
lul^ <
m'
z
-2n 2(a-t)f
+
*£]
0
The proof of this theorem is completely analogous to the
proof of theorem 3 in section 2„
One may also easily obtain the results analogous to
lemma 3 and lemma 4.
24
4.
Three line theorems for periodic solutions of the Cauchy
problem for Laplace^ equation.
We consider functions u(x9y)
<
harmonic on an infinite strip 0
period 2 it.
and periodic in x with
The functions of this class have the representation
+
(4.1 a)
y < a
u(x,y) =
c
„ y
“
i2
00
(an cosh n y + °n sinh n y) cos n x
oo
+
(bn cosh n y + dQ sinh n y) sin n
1
where
| aQ? b^,
d^ j-
(4.1 b)
U
nfy
is any sequence such that
0
+ B
nfy> * “ >
<
y < a
and where A„
and B
denote the sine and cosine coefficients
nsy
n,y
A
n,y
=
a
n
cosh
n
y + cn
sinh n y
*
n ^ 1
9
(4.1 c)
Bn>y = bn cosh n y + dn sinh n y ,
n - 1
The proof of this representation follows in like manner to the
proof of the representation in section
more complicated.
being slightly
In the proof we use the inequality
,
(4.2)
although
^An»y* ~ ^n^y-j^
6
-n(y~yn)
n
+ IAn»y le
, 2
(y?”y)
,
Y1< Y < Y2 J
25
which follows from the identity
sinh n(y2 - y)
A
(4.5')
n,y
— A
sinh n(y2 - y.L)
n,y1
sinh n(y + A.
sinh n (yg - y1)
*
Notice that if u satisfies a certain 12 constraint, then so do
(ao + c y)/ 2 and u = (aQ + c y)/ 2„
Therefore,
without loss of
generality, we may assume that the mean value term (a
zero.
+ cQy) is
That is, we consider only harmonic functions of the form
u(x,y) =
( 4 o 4 e 3-)
(a
cosh n y + CQ sinh n y) cos n x
CD
+
/
(b
cosh n y + d
sinh n y) sin n x,
0
y < a;
1
(4.4 b)
A
B
= an cosh n y + cn sinh n y ,
= b
J
cosh n y + d
II
II
sinh n y ;
CD
(4.4 c)
Hull y = (
We see that A
2
2 ^2
Z
(A
n"y
+
“ ’ 0 < y < a „
2
+ „ is a positive convex function, which
n,y n,y
therefore takes on its maximum at the end points 0 or a0
If Hull
is bounded on 0 < y
a, the sum converges uniformly to a continuous
convex function on 0 - y - a,
Thus, we may define
26
H|OJ H|c\j
z
CD
-(\
+
i
(4.5)
00
yf
(A
Z
={\
a
nfa
+
Notice that, formally
GO
y
u(x,0) =
(a
cos n x + b
sin. n x),
i
00
(ncn cos n x + ndn sin n x).
Uy(x90) =
1
00
y
*-
u (x,0) a
x
(~n a
sin n x
n
+ n b
xi cos n x)
i
Therefore
1
oo
Ml0-(i
21
‘an2
+
2A2
0)
00
14.&)
Iloilo-(I
E
(n2 on2 + n2 dn2))
00
'M 6 = (I
21
(n2 a 2 +
n
“2 bn2>)2
It turns out to be better, for the sake of greater
generality and symmetry, to hypothesize
00
constraints
on
Z
00
(c
2
+ d
2
)
i
instead of on
(n2 c
+ n0 d
2
).
i
Now, if u is of the form (4.4),
let D~^ u
v
x integral of u^.,
2
denote the termwise
27
(4»7 a)
D'”1 uy(x,y) =
00
21
<an
sinh n
(b
sinh n y +
7
+ c
n
cosh n y) sin n x
1
z
cosh n y) cos n x,
0 ^ y < a
which we immediately recognize as the harmonic conjugate of -u»
Thus D”1 Uy is also harmonic on the strip 0 < y < a, and as in
(4,5) we may write
OP
(4.7 b)
HD’1 uyll
If I ID"1 Uyll
0
0
=£lim
HD*1
Uyll £
= (§ X
1
(0 2 +
n
O)2
•
is finite, we may say that the initial values of the
harmonic conjugate of u corresponds to an L^ function, or (more
meaningfully for the Cauchy problem) that the initial values of Uy
correspond to a Schwartz distribution which is the formal x
derivative of
an
function.
Now, the norm IID"1 u II
See [3]9 pages 795 and 797»
for u
Q
is in the more general and
symmetric form desired, but its representation (4,7) merely in
terms of the Fourier coefficients c
and d
is not always useful.
However, suppose Uy(x,0) = f(x), where f(x) is at least an 1^
function.
Then f(x) has a Fourier series
(An cos n x + B
f(x) — 21
sin n x)
1
and by Fatou's theorem [10] its termwise integral
A sin n x - B cos n x
n
n
n
D”1 f(x)
1
converges uniformly to the continuous function
28
x
(4.8 a)
J
D^fCx)
-
2n
x
J J
f (t) dt - |-
0
0
f(t) dt dx ,
0
in other words, to that indefinite integral of f(x) which itself
has mean value zero. Since indefinite integrals differ at most
by a constant and this particular one has a zero constant term
in its series, it has a smaller L2 norm than any other indefinite
integral. Therefore
x
_1
(4.8 b)
llB fll *
II
f
f(t) at II
c
for an arbitrary constant c. Also, IlD
constant multiple of the
norm of f,.
*1
flj may be bounded by a
Let
215
iiifiii
1
2®
=
Then
IAu
lf(x)l
dx .
2%
'V -ii
s
/
f(x) cos n x dx | ^
2lllfIII ,
0
'V
n
2%
o
1
f(x)
sin n x dx |
~ 2l|lf III .
29
Now, let us find a uniform bound for the Cauchy problem for
I<2 constraints on the data»
Theorem 7.
(4.9)
If u(x$,y) is of the form (4.4)s and if
* m = e^a £ 1
(a) Hull
Q
(b) IlD
uyli
(c) Hull
3,
0
- 1
- m
9
then, for 0 < y < a
iJL
lul
<
[*-i]
a
m
-2ny\ 2
'|
+
(4ol0)
4^
Proof:
1
2
m a
1 - e -2y
1 - e
2
-2(a-y)
1
2
■VI
Since the translations and negatives of such functions also
satisfy the constraints, the problem reduces to finding a bound
for the linear functional
li
0
o
CD
z
i
CD
A
n,y
=
21
1
(an
cosh n y
c
+
with respect to the quadratic constraints
CD
(a)
CD
z
-
Z
1
I
00
(4.11)
(b)
Z
c
2
- 2m2 ,
n
’
1
CD
(c)
z
1
A
2
n,a
~ 2 o
a
n2
*
2
“2
sinh n y
n
•
^
30
Using (4o2)p we write
M
(4.12) u(0j,y)
z
an cosh n y +
Z
cn sinh n y
1
1
CD
00
+
n
sinh n(a~y)
sinh n a
^
+
n?a
sinh n y
sinh n a
[<*+l ]
[<*+!]
Then we apply the maximization lemma.
The maximum of the first
term with respect to (4.11 a) is
1
M
2
m ^2
cosh
n y j
2
[*-1]
<
-2ny \ 2
me My
m
1-*
a
,
1
fx-i]
f2 z
2
-2ny
The maximum of the second term with respect to (4»11 b) is
1
[<*]
m
*
m
2
2
sinh'
ij;
M
(I Z
\2
n yj
-2ny
2
1
The maximum of the third term with respect to (4.11 a) is
31
1
m
2
sinh nCa-y) \
sinh n a /
(
2 j 2
[ +1]
The maximum of the fourth term with respect to (4.11 c) is
1
GD
2
sinh n y \
sinh n a /
[ +1]
1
e-2n(a-y)
<
2
1
2
<
6
The conclusion then follows hy addition.
The I2 constraints (4*9) imply the termwise constraints
(4.13)
(a)
Ianl,
lbnl
(b)
lo l
ld
(c)
Theorem 8 .
n>
V2 m 9
<
n'
Ka'-X,
a
I
i2
m ,
V2
If u(x»y) is of the form (4.4)? and if its
coefficients satisfy (4.13)9 then* for 0
y < a ,
32
1JL
(4.14)
m
luly<
r-
a
a
[^-1]
V
2?
0
Proof: ¥e divide u(x,y) into four
in (4.12).
parts as we divided u(0,y)
The results then follow as in the previous proof.
Notice that the second terms in (4.10) and (4.14) tend slowly
to infinity as we approach the data line y = 0.
emphasizes the fact that
Again,
this
bounds imply good uniform bounds only
where the function is smooth.
In theorem 9 we will impose this
smoothness at the data line by requiring that the tangential
derivative u , as well as the normal derivative u ,
y
x
Payne
[13] has imposed similar conditions.
problem,
In terms of the Cauchy
this requires that we approximate u
the usual u and u .
be small,
closely, as well as
In theorem 10 we will obtain this smoothness
y
more naturally by letting the data curve be embedded within the
solution domain, as was the case in theorem
2.
Theorem 9.
and if
If u(x,y) is of the form (4.4),
(a) l| Igrad ul IIn
then,
for
Hull
&
m = e
'"O'a
<c
2_
0
(4.15)
(b)
-
* 1
9
0 < y < a ,
1
(4.16)
Proof s
lul
-<
m
By (4.6),
1
II Igrad ul |lQ
m
1
1
33
Then the problem reduces to finding a bound for the linear
functional u(0,y) with respect to the quadratic constraints
oo
2
(a)
a
2
+
n
^
c
2
•
n'
“
»
1
(4.17)
oo
A
>
5
nfa
We write u(0,y) in the form (4.12).
Then the maximum of the
first two terms of (4.12) with respect to (4.17 a) is
[°0
2
-
sinh
^
(
m
2
\
n y
\
n2
e2uy
(*z
1_a
n y + cosh
z
1
M
m
2
I
^2
1
1_ \ 2
[*]
( 2
n
2
The maximum of the third term of (4.12)
with respect to (4. 17 a)
is
1
GD
m
1
2
2
[<X+l]
n
CD
<
m
2
[ +i]
2
n
/sinh n(a-y) 'j
sinh n a )
[
2
x2
V
J
34
Therefore, the sum of the first three terms is less than
1
<
m
Finally, the hound for the fourth term of (4.12) follows from
(4.17 h) exactly as in theorem 7> and (4.16) as proved by
addition.
We now derive results analogous to lemma 1, lemma 2, and
theorem 2 of section 2.
We will now be considering functions u
which are harmonic on the whole symmetric strip, -a < y < a.
For u of the form (4.4) on -a < y < a we write u =
+ u^ ,
where these low and high order parts are defined analogously
to (2.4)
Lemma 7 .
Let LJJ denote the class of
(a)
lluNll
(b>
iiD-hu^yio s m
Q
which satisfy
- m ,
and let
Then
1
sinh
1
2
n yl
IT
+
1
35
Proof: The problem reduces to the maximization of
N
uN(0,y) =
(an cosh n y + c^ sinh n y)
I
i
with respect to
N
2
a
(aj
<
2
0
n
“
2m
f
°n2
5
2“2
>
(4.20)
N
(b)
£
1
Then (4.19) follows by the maximization lemma.
Lemma 8.
let EL. be the class of uS?
N
N
(4.21)
"
l * K
11
which satisfy
-a
4
2
>
and let
H(N,y) =
sup
QD
U
N
luj? I ,
„
e
-a < y < a .
-
%
oo
(4.22)
N+l
Iroofs
2
2
n y . sinh
+
(
2
2"
\ cosh n a
sinh “))
/.cosh
H(N,y) = ^;
ro|H
Then
The problem reduces to the maximization of
oo
z
N+l
(0,y)
(an cosh n y + cn sinh n y)
with respect to
Z
ao
o
p
(an
cosh
p
p
n a + cn
sinh n a) - 2
N+l
Then (4.22) follows by the maximization lemma.
36
Theorem 10.
If u(x,y) is of the form (4.4) on -a < y < a, and
Hull
(a)
(4.23)
-
m = e“c<a
1
^
(o)
Q
”y)* 0 '
Hull
I
m
1
,
’
+ Hull *
•"•el
CL
*
- 2 ,
then, for -a < y < a »
(4.24)
lul
<
L + H ,
where
L
s
!([<*],y) ,
H = H([o<],y) ,
iJiL
(4.25)
Proof;
L + H <
m
Moreover,
[cX-l]
[(f 51
G
-2nlyI
)2+ f1_e-2(i-iyi)J
It follows immediately by lemma 7 and lemma 8 that
luly
for every N.
*
l(N,y) + H(N,y)
Then (4.25) follows by simplification of (4.24)
M|H
are defined by (4.19) and (4.22).
37
5. Harmonic continuation, data given on a whole circle.
Suppose there exists at least one function u(r,©) having the
properties
(a)
u(r,©) is harmonic for r < 1 ,
(b)
llu - gll
- m =s a* , where
the L0 data function
a,
c.
g(©) is given,
(5.1)
(c)
Problem 1.
iiuii
x
*
i.
Find a function approximating every u satisfying (5.1).
let g(©) have the expansion
A
^00
g(&) = —+ />
y*2" —
(5.2)
(A^ cos n © + B sin n ©) a11 ,
n
n
’
and set
N
A
(5.3)
^
+
■y g
gN(r,©)
y
/■■■ i'"
(A
cos n © + B
H
sin n ©) r11,
r < oo
We let
be
our
approximation function and derive an a-priori
error bound.
Lemma 8. If u satisfies (5.1)> then
lu
” gKTr ^
L + H
»
where L and H are defined by (2.10) of theorem 2.
Proof:
We write
CD
(u
u
- S[«]) = (u - «[<*])[«] + (
00
= (u - g)[o<-] + U^-J
* 8[<X]>[«]
38
By (5.1 b), lemma 1, and (2.10),
l(u - g)[cX]lr
- I«(l><Lr) = L .
By (5.1 c), lemma 2, and (2.10),
oo
lu
[«]'r
£
H(M,r)=H.
Therefore, the lemma is proved.
Recall from (2.17) that L + H is, except for a factor of at
most two, the best possible bound.
Therefore,
the best possible truncation order.
ibnan
a11 - A.
n
[ ] is, in a sense,
Notice that (5.1 b) implies
B
rii
n
a'
^ V2
a
o<
while (5*1 c) implies
laj , lbnl *
VT,
or
|an a11 - °|,
|bn a,n - Ol
-
T/2
an .
In other words, we are given two sets of data for the individual
coefficients a an and b a11 of u on the data circle r = a .
n
n
The
first set claims that a„n
a11 andn
b_ an are within V2" a°^of the
coefficients A^ an and B^ an of the data function g, while the
second set claims that they are within
lf2 an of being zero.
Therefore by setting the coefficients of our approximation function
equal zero for n greater than ©< , we have used for each coefficient
the data with the best claimed accuracy.
The data function g clearly cannot be arbitrary if there is to
exist a function satisfying (5.1).
We derive conditions on g
which can be checked after the Fourier coefficients of gjj have been
calculated.
Notice that
1
H% - gll
a
= (llgll ^ “ 11%*' a )
is quite easily computed at each step.
2
39
Lemma 9. If there exist u satisfying (5.1), then g^] must
satisfy these a-posteriori compatability conditions:
(a)
lls
M - S11 a
(5.4)
(b)
Proof:
ll«[o<]llx
2
~
Given u satisfying (5.1)>
(aL)
•
then
l,u
- » .
l,uco - g” II
*
U - Si^a
(aH)
2,11 '
£
a ,
(5.5)
(bL)
(bH)
Since
UJJ
£
llu® II
* 1 .
- gjj contains no orders higher than N, (5.5 aL) implies
(5.5 aL') llu,, - SNII
Since u®
x
1 >
i N
x
* m(i)
_
contains no orders lower than N + 1, (5.5 bH) implies
(5.5. bh>)
- 1 aW+1
llu® lla
.
Then, using (5*5 aH), (5.5 bH1), (5.5 aL')> and (5.5 bL) we have
"
gl1
a
=
i
** a
”
"
*,gN
-
m + 1 a^+^,
“
U
N ** a
Hgjj ~ a^ll ^ + Hujjll ]_
-i
m(J)
N
+ 1 .
+
a
40
Finally,
(5.4) follows from (5.6) for N = [<*] .
This lemma implies that if there exist u satisfying (5.1),
then itself must almost satisfy (5.1).
The preceding formulation has the disadvantage that the inner
and outer constraints, m and 1, are ad-hoc assumptions whose
compatability is checked only after all the coefficients of g^
are computed.
However, we see by the following lemma that the
computed g^- disclose, step by step, a great deal of information
about what constraints are compatible for the given g.
Lemma 10.
Let m^ and MJJ be defined by
(a)
Hgjj “ gH
=
(t>)
llgNll1 = 2MN.
»
a
(5.7)
Then there exists no function u harmonic on r
(a) IIu - gll
-
a
1 satisfying both
mN,
(5.8)
0>)
Hull
^
x
MN
.
Proof: Suppose u satisfying (5.8) exists.
11
"SH
a
-
1
■
m
H
+ M
a
<
and therefore
IIgN ~ gH
a
<
which contradicts (5.7 a); or
M* aN+1
* mN,
and therefore
'%M1
aN+1
+
Either
%
N
<£ 2M.
U
which contradicts (5.7 b).
% •
’
Then (5.6) becomes
41
Now, let E denote the set of pairs (m,N) for which there
exists a harmonic function u satisfying the constraints
llu - gll
a
*
m ,
Also, extend the definition (5.7) to include
(m_l>
M
_i) =
°) *
Then, we see hy lemma 10 that the rectangles
m - 2mN,
M
are in E,
- 2Mn,
N
^
-1 ,
while the rectangles
m - mjj,
M
-
are in the complement CE.
MJJ,
N
- -1 ,
Moreover, one easily shows that E is
a convex set; hence, E contains the convex hull of the first
collection of rectangles.
the diagram below.
M
This information is illustrated in
42
It will often be sufficient to terminate computations at
some order N less than [©<].
IIu - gll
iiuii
1
=
o
Si
=
M
let m and M be defined by
m ,
.
Then
llu - gHll
a
-
m + 2ns,
llu - gHll
a
*
M + 2%.
By theorem 2 we get a bound for lu - gjjlr whose leading term is
log r
log a
(m + 2mN)
log r
(M + 2MN)
log a
It is therefore sufficient, for good approximation of u, to continue
until m^ is as small as the expected m, but to stop before MJJ
grows too large.
43
6. Some auxiliary three circle bounds,
jn section 7 we shall
have occasion to consider functions u having the properties
(a)
Q
u(r,©) = -£ +
(b)
Null
(c)
ianl
K
O*
(6.1)
Z1
V2
-
a
m = a'
ibj
»
*
9
1 ,
fixed positive integer .
of the class
(a.
n
Let C
0 K
uc unis sx
In this section we will derive
of theorem
certain precise bounds for such functions.
Lemma 11.
Let u e C 0,K'
22
log_r
(6.2)
Then, for a <■ m
2
M
(Z
Z
log a
lulr ^ m
-
m
-
£-[<*+! ]
(?) f- * z
0
For 0
- 1 ,
n
0
a^ ,
K
lu
(6.3)
'r
2
(2 z <!> f
f.
0
Proof:
K
Z
K
For a < m - 1,
le
60) 0
GO
everywhere that
z
appears in the proof of (2.20) of
[<X+l]
[©<+l]
theorem
and (6.2) follows.
lemma 1 and (6.16) we have
For 0 - m - a^, i.e.,
K
lulr
^
L(E,r) < m
(g)2n)
(2
0
o< ^ K, by
44
Lemma 12
Let u e CQ
Let uS0(r,O) be its second order
directional derivative in a fixed direction.
W
Then, for a ^ m - 1 ,
Let a
b ^ 1,
M
'V* * “t* (22
n4( )2n
s
F
(6.4)
K
z
8
n
2 .n
b •
[oC+l]
For 0 ^
m
aK ,
-
4 ( )2
iu
x ° *f•
ss(b*
(6.5)
Moreover, we have, independently of K ,
(6.6)
lu3elb
- D(m) , 0
- m i
1 ,
log b
where
D(m) « mlog
a
(o<+l)5/2
c(b) ,
and
CO
0(b) =
]T
(l+n)2 bn.
0
We write
u
ss = ’W's*
One easily shows that lrslb
In
I
ss b
£
b2
*
+
-
Vss
+ u
1 ,lre3Ib
ee(9s>
+
Vss
I© I
-
aJ|H
Proof:
45
Therefore,
*uss*b
*urr*b + b*u
r*b
...
.
“
+
u
,2*
©©*b
b -
+
.b 2*-u©*b
Now
E
(an cos n© + bn sin n©) n bn~\
ur(b,©) = 2^
1
E
u
(h.e) =
]T
K
u9(b,©) = 21
(
n-2
cos n© + b^ sin n©) n(n-l)
(~aG sin n© + b^ cos n©) n bn ,
1
K
u99(b,©) = 2^
(“an
cos
+
*>n
sin
bn
1
Therefore, we obtain
u ss I,b
E
1
/ 2 , . 2\2 2 , n
n
b
K + bn >
- ^s?
Since (6.1 b) implies
M
Z1
<an
+ b 2) a2n
n
- 2“2 ’
we have by the maximization lemma
[<*] i
Z(a
2
n
+
b 2 ?
n )
M
n2b» *
Z
m 2
*4
<!)2n)
If E - o( , we get the same result with [c*] replaced by E, which
proves (6,5).
obtaining
E
Next, we apply (6.1 c) to the high order terms,
_E
/ 2 , 2\2 2 , n
<an + bn ) n b
£*+!]
<
-
2
n
[o<+l]
2 ,n
b ,
46
which completes the proof of (6.4).
oo
in (6.4).
Finally, we may replace K by
Then,
M
1
CD
+
+1
^ b^
D
(n + [o<+l])2 bn
^
<
0
CD
*°<
(2[«W
X.
2
2[«+i]
+
(I*155ST)
>n
0
£
b* (<*+l) 2
<
00
16
Lb
1
X
2
(i + n)2
] .
0
Therefore, the proof is completed.
Notice that, for m - a"^, the bound (6.6) is linear in m;
while, for m > a , the bound (6.5) is linear on intervals of
constant [o<], but has jumps where [<*] changes from one integer to
the next.
We put the bound in a more useful form for automatic
computation.
First, define the step function
1
S(j,K) = a^ (2
13 V
]T
4
n
2n^2
(|)
:
1
(6.7)
K
8
n2 b11,
0 - j - K .
d+i
By (6.4) this is a bound for lu
m=a
(6.8 a)
-
aLJ.
I, when [o<] = j, since
BS D
We now force our bound to be non-decreasing. Let
B(m,K) =
min S(j,K),
0 - j ^ [*]
aK<
m
- 1;
47
(6.8 b)
B(m,K) =
min S(j,K:),
0 * j < K L
0 -
(6.8 c)
m
-
SS
L
D
(I’ ) ].
1
'f
B(m,K) = 3(0,K),
Since a bound on lu
211 1
Z
b
m > 1 .
for a given m is a bound for any smaller m,
B(m,K), as defined in (6.8 a) and (6.8. b), is a bound for lu
Since the outer constraint Hull
Hull
I, .
SS D
- 1 automatically implies
£L
- 1 , we may extend our bound to include m > 1 by means of
81
(6.8 c).
Notice that we have increased the bounds (6.4) and (6.5)
by a factor of at most (1/a) in obtaining B(m,K).
Thus, since
D(m) of (6.6) is an upper bound for (6.4) and (6.5), — D(m) is
an upper bound for B(m,K).
We have therefore proved the following
lemma:
Lemma 13.
Let u e CQ
(6.9)
Then
lusslb
*
B(m,K) ,
where B(m,K) is defined by (6.8).
B(m,K) is a non-decreasing
function of m, defined for m - 0.
B(m,K) is initially a positive
multiple of m, and thereafter a step function.
for m >■ a.
B(m,K) is constant
Moreover,
(6.10)
B(m,K) <
J
D(m), 0
ci
^
m
^
1 ,
where D(m) is defined by (6.6).
B(m,K) will frequently be a much better bound for luss^'
than D(m).
48
7.
points.
Harmonic continuation, data given at a finite set of
Suppose there exists at least one function u(r,&)
having the properties
(a) u(r,0) is harmonic for r < 1
(7.1)
(b) |u(rj,0^)
e
o ’
where the data values g^ are given at interior points
=
(r^,©j), 0
(c)
Hull
1
1? •••
-
9
P
?
1 .
let P be a simple closed polygon with vertices (r.,9„),
6 the
3 3
length of its longest side, and a and b the radii of the inscribed
and circumscribed circles with centers at the origin, 0 ^ a ^ b < 1,
Problem 2.
Find a function approximating every u satisfying (7.1).
First, we outline an explicit method which reduces Problem 2
to Problem 1 by means of solving a Dirichlet problem ( a wellposed problem) on the polygon.
Let u „ be the second directional derivative of u with respect
SS
to arclength along P„
we can bound
lul-n*
ss r
By means of the outer constraint (7.1 c)
the uniform norm of u
on P.
ss
let g be the linear interpolation of the g. onto the whole
3
polygon P. Then we have the explicit interpolation bound
(7.2)
|u - glp - lu - glp + lu - ulp
-
eQ +
*uss*P '
Now, approximate u on P° by solving the Dirichlet problem
for the Laplace difference equation,modified as in [8] to deal
with the irregular boundary, using boundary data f on P,
let
gAx denote this solution for a grid size Ax, using bilinear
interpolation between the grid points.
By means of the outer
constraint we can bound the fourth order partials of u on P°;
hence, the global truncation error on P° is O(Ax^).
Now, our
modified difference scheme with bilinear interpolation still
obeys the maximum principle; therefore,
49
(7.3) lu - gAxl
Q
-
lu - glp + 0(Ax2) ;
£
lu
or,by (7.2),
llu
- «Ax" a
(7.4) lu — g^xI Q
- SAx'a
-
=e^+0(5 ) + 0(Ax2) .
Finally, apply the methods of section 5 for problem 1, with
gAx(a>®)
as
data.
Next, we develop a method of linear inequalities which has two
great advantages over the preceding method. First, the work
requirement is usually greatly reduced. Second, the interpolation
error is usually greatly reduced, thereby yielding far
better
accuracy from the given data. The method, in brief, is the
following: linearize the outer constraint (7.1 c), then find a
harmonic function of finite order which closely fits the data and
satisfies the constraints.
We will use an approximation of order K, K to be chosen later.
Assume that u satisfies (7.1). Then, at the data points Ug.
satisfies
(7.5)
luK(r^,©j)
gj1 s Miye.j)
g-jl + lu® (r^,©..! - eQ + H(K,b),
where we have used the maximum principle and lemma 2 to bound
lu® (r.. ,©^) I .
We write (7.5) in terms of the Fourier coefficients,
a , bp,
f
, bg, of u-g-, and obtain 'the following
K
$ 000
2p linear inequalities:
g-j - eQ - H(K,b)
(7.6 a)
*
a
JL
— ™ + / ( a (r1?- cos n©.) + b v(r1} sin n©.) ) Y2* 4__
-
l
n' 2
gj + eQ + H(K,b) ,
3
n
0
j = 1, ... ,p.
y )
50
The given quadratic quter constraint,
K
1
9
0
implies the linear termwise constrainrs*
|a
n*
>
lbnl
^
V2 ,
or the 4K + 2 linear inequalities,
an s vr,
II
^
K ,
3
i2
0
-
K .
I
n = 1
Ml
i V2 ,
bn
Now, there are well-known numerical methods for the solution
of linear inequalities,,
See [ l],
[4], and [9] for references.
Moreover, if there exist no solutions to the inequalities, these
methods will tell us so, which gives an a-posteriori check on the
hypotheses (7.1).
B-^,
Therefore, find any solution, AQ,A-l,
... , Bg, to the linear inequalities (7.6).
... , A^,
Then let these
be the Fourier coefficients for an approximation function
v^;
that is, define
E
(7.7)
vE-(r,©)
to
VF
(A
cos n© + B sin n©) r11 .
v
n
n
z
4
*
We begin the derivation of an a-priori error1 bound.
Write
the error function in the form
(7.8)
u - vK = Eg + Ug. ,
where
E
K
= U
K ~
V
E *
By (7.6 a) ,
(7.9) lE^r^,©..) I
^ luK(r..,©.j) - g.. I + lvK(r..,©..) - g.. I
^ 2e0 + 2H(K,b).
51
Extend this to a bound for lEglp, the uniform norm of Eg on P
by means of the linear interpolation Eg?
(7.10)
lEjj-lp
£
lEKlp +
|EE
« 2e0 + 2H(K,b) +
For a fixed direction,
- Eg Ip
l(EK)sslp.
(Eg)gg is also a harmonic function; hence,
by the maximum principle,
(7.11)
E
^BK^ss*b
“
^ K^SS*P
Define m by
(7.12)
2m = IEgIp
Thus,
a
“
a
”
*BK*P
=
2m
*
Let a'
= (a
- A) and b'
= (b - B ) denote the Fourier
x
n
n
n'
n
' n
n
coefficients of Eg.
Then,
K
a’.
(a) EF(r,0) = —— + \ (a1
i/2
~~
cos n& + b'
n
sin n©) rn
n
(7.13)
(b) llEgll
(c) la*nl
In other words,
-
a
2m ,
lb*nl
^
2 VT .
Eg- is in the class CQ g of section- 6.
Therefore,
by lemma 4,
(7.14)
1
(%)Sslb
”
2
*
Then, by (7.11) and (7.14), the inequality (7.10) becomes implicit
in m:
(7.15)
m
-
eQ + H(K,b) +
B(m,K)
52
In (7.15)
cQ is the pointwise data error, H(K,b) the truncation
*2
error, and
B(m,K) the interpolation error.
interrelated by the
These errors are
implicitness of (4.15); yet it will be
convenient for the purpose of analysis to isolate their individual
effects.
By (7.15),
2
(7.16) m
* 3 max (eQ, H(K,b), ^-B(m,K)) .
2
Suppose ^ B(m,K) is the maximum above; then m is a root of the
equation
(7.17)
for some number
/3
t
+/Q
- 0 .
=
| 62 B(t
K)
If this is so, then we see by the
diagram below that there exists a largest root, I^, to the equation
(7.18)
t =
| 52 B(t K),
and that all roots of the equation (7.17) must fall below 1^..
If
there exists no root of (7.18) other than zero, this is merely a
-2
sign that it is impossible for ^ B(m,K) to dominate in (7.16), in
which case we define Ig = 0 .
53
We have therefore proved the following lemma:
Lemma 13.
If Eg is the difference of two harmonic functions Ug
and Vg whose Fourier coefficients satisfy (7.6), and if 2m is
the uniform norm of Eg on P, then
(7.19 a)
-
m
-
mg* = J max (e0»H(K,b), Ig) ,
where Ig is the largest root of
(7.19 b)
t = | 62 B(t K) ,
t 2 0 .
y.
Corresponding to mg
defined by (7.19), we define
o<* by
m** = a<** .
Then,
a
* =
m
3
K* ^
H(K,b) .> bK+1 ^ aK+1
.
Thus, it is always true that
(7.20)
cK
E+l .
Recall from 7.8 that
XL -
-n c®
Ug - Eg
+ Ug
.
By (7.13) and (7.20) we may use the bound (6.2), rather than (6.3),
for ^ Eg.
By (7.1 c) we may use the bound H(K,r) of lemma 2 for
u®. After simplification we obtain the following a-priori error
bound:
54
Lemma 14.
If u satisfies (7.1) and the Fourier coefficients of
Vg satisfy (7.6), then
(7.21)
lu -
Vr
log r
*
log a
m
K
[**]
r
8
f
Z
0
( )2n)2
Moreover, on P and its interior P°,
(7.22)
lu - vKlpo
- 2m* + H(K,b).
Considering (7.21), we see that, as far as the sums in the
bracket are concerned, it makes little difference where the
division point K between the second and third sums falls.
*
Therefore, what really matters is the smallness of nig .
Now eQ is fixed, H(K,b) is decreasing exponentially like
b
K
, and I„. is bounded but increasing slowly, since B(m,K) is
bounded for all K but increasing slowly.
Thus, the minimum nig
will occur either at that integer K at which H(K,b) first ceases to
dominate both e
and I™., or at the integer just before this.
0
n.
^
Choose K to be that integer at which the minimum mg occurs.
Let us compare the explicit method and the method of linear
inequalities with respect to work requirements and interpolation
error.
For the first method, we found in (7.4) that
(7.23 a)
lu - gAxI
=
eQ + 0(M2) + 0(62) ,
where 0(Ax ) is the global truncation error for the Laplace difference
o
2
equation on P° and 0(6^) is the data interpolation error on P.
method requires the solution of
(7.23 b)
linear equations.
0(A£2)
The
55
For the second method, the truncation error H(K,b) is
0(bK).
The interpolation error I£ is a root of equation (7.1 b);
hence, by the upper bound (6.10) for B(m,K), Ig satisfies an
inequality of the form
lQ
1
(7.24)
IK
2
-
g *>-
2
0(6 )
l0g
Hog IKI
IK
a
.
1
We absorb Ilog I^-l
2
into the exponent and obtain
l.9.K.±
T
1
K
^
~
et
I
K
"
C
e
x log &
6.2 ±
e
K
< «f
T
_
e
6
1+e-log b/log a
We therefore have
(7.25 a)
K
-1
lu - vF
n + 0(b ) + ofe
ill pUi eo
+
e
"
log t/log a
)
The method requires the solution of
(7.25 b)
4K + 0(6~1)
linear inequalities, for fairly evenly spaced data points.
In the limit, where b = a and all the data points lie on a
circle, we would use linear interpolation along the circumference to
obtain a bound exactly analogous to (6.6) for IUQQ1^.
(7.24) would become
1
(7.25)
2
2
IK
- 0(6 )
Hog IKI
IE
5 e*p(-
IK,
or,
(7.26)
•
Therefore
56
Let us suppose, for the purpose of illustration, that
2
log b/log a equals
+
e
an<
*
e
0
is zero.
Then let us
compare the two methods from two different viewpoints.
One
viewpoint, corresponding to many experimental situations, assumes
that the number of data points may not be increased without great
cost.
Our accuracy, therefore, is limited by the interpolation
error.
In the present example, the interpolation error is 0(64)
for the second method, as opposed to 0(5 ) for the first method, a
very great increase in accuracy for the given data.
Another
viewpoint, corresponding to many computational situations, assumes
that the major cost is involved in the computation, and that the
number of data points may be increased arbitrarily.
The accuracy,
therefore, is limited only by the number of calculations which we
are willing to perform .
Let
and 1^(7) denote, respectively, the number of linear
equations required in the first method, and the number of linear
inequalities required in the second method, to guarantee an error
on P° no larger than
Y[
.
Prom 7.23 and 7.25 we obtain
Nx(^) = 0(^_1) ,
“2(7) =
•
Therefore, we have the comparison
1
4
N2 = 00T, ) .
The linear inequalities method of problem 2 is quite similar
to the linear programming approach used by Douglas and Gallie [7]
for harmonic continuation on the half plane, by Douglas [5] for
analytic continuation, and by Cannon [2] for backward solution of
the heat equation.
Douglas [6] has also applied the linear
programming approach to improper integral equations.
57
8,
Analytic continuation.
The problems and methods for
harmonic continuation extend immediately to analytic continuation,
since the real and imaginary parts of an analytic function are
harmonic.
Notice that if
f = u^ + i Ug
is an analytic function with the Taylor series
00
an + i b^
0
fU) = -2-
Y~ '(an- i n
b ) zn
/
1
+
vr
then the Pourier series of u^ and that of its harmonic conjugate
Ug satisfy the
relation
<3D
Un(r,©) =
+
/
(a
n
1/2
sin n©) rn,
cos n© + b
n
(8.1)
CD
u9(r,©) =
y
+
V
(-b
2'
^
cos n© + a
sin n©) rn
Suppose there exists at least one function f(z) having the
properties
(8.2)
(a) f(z) = u-^(z) + i u2(z)
is analytic for Izl < 1,
(b) 11^ - g1ll
a
-
m = a
,
Hug - ggll
a
“
m > where the L2 data functions
g1(©) and g2(©) are given,
(c) llu1ll1
*
it^ii -
1 ,
1
58
Problem 1.
Find a function, approximating every f(z) satisfying
(8.2).
Let g-j^ and g2 have the expansions
at>
n
(An1 cos n© + Bti1 sin n©) a
+
,
1
(8.3)
OD
21
1
g2(a>©)
^“Bn2 cos n© + A^p sin n©) a31 .
By (8.1), all the coefficients except the constant terms may be
obtained from just one of the data functions.
Define the conjugate
harmonic functions
N
A
g
l
*
=
+
(Anl
2Z
V2
cos
+ B
sin
nl
**n,
r
^
® >
(8.4 a)
B
r
g2yu( »©) =' - +
N
("Bni
21
< *-
cos
n
® +
A
sin nQ
ni
)
~\
Then let the analytic function
(8*4 b)
gN(z) = gx
N(z)
+ i g£
H(z)
=
N
A
ol ~
1
B
O2
+
V2
Z
1
be our approximation function, for N = [o<] .
“
1
Bnl^
1,11
»
r < OD .
59
For checking purposes also calculate the corresponding
coefficients of g2« We have the following a-posteriori
compatibility condition between g^ and g2:
[I
1
(8.5)
£
[I f
«V - «n>2 + - V Vf
1
+
[I Z1
((A
n2 -
L
=
Ws±
- u-JI
a
a
n>2
-
'Bn2
+ llg2 - u2H a
"
^ 2m.
To (u^ - g, ) we may apply directly the a-priori bounds and
rl
*
compatibility checks of lemmas 8-10. For (u2 - g2 JJ) the only
difference is that instead of having
*,U2,N “
S
2,J^ a
"
m
’
we have only
(8.6)
c
-
^u2,N ~
S
I <bo -
B
2,W^ a
o2)2
+
2
Z
1
Ilu2 - gll ^ + lli^ - g-JI
<<an “
\ ~
A
nl)2
2m2 .
+
<bn ~
B
nl)2^2n»
60
Now, suppose that we are given only data values {g-. .,g0
.)
at a finite number of points (r^,©..), as in problem 2 for harmonic
functions.
If we merely treat an analytic function as a pair
(u^jUg) of harmonic functions whose Fourier coefficients are
related by the form (8.1), then the statement of problem 2 and
the results of section 7 extend in obvious fashion to analytic
functions.
See Douglas [5] for an alternative approach to analytic
continuation.
Instead of a Taylor's series representation, a
Riemann sum approximation to the Cauchy integral is used.
61
9.
Backward solution of the heat equation on a rectangle.
Let us begin with a brief discussion which shows that the problems
of backward solution of the heat equation on a rectangle with a
prescribed bound may be reduced to the problem of backward
continuation of periodic solutions with a prescribed bound.
0<t< a j
a rectangle R = j"(x,t): 0 < x ^ rc,
On
we consider
solutions u(x,t) of the heat equation which are continuous on
the closure of R.
Let B, T,
and
denote the base, top, and
two sides of R.
The backward boundary value problem, for example, is to find
u when given its values on the top and two sides.
That is,
suppose u satisfies
(a)
u
(b)
u
(c)
u
u
on R ,
=
Gr
on
=
G
on
xx =
t
i
T ,
s.,
1a±1 - 1,
where Gr and GL are known approximately.
Moreover, suppose
it
(d)
Hull
y
2
= (% J
i = 1,2.
u (x,y) dx'J
1
2
- 1, 0 * y ^ a.
o
If Gr-^(t) = G^Ct) s 0, then u(x,t) may be extended
antisymmetrically across x = 0 and x = w by means of its sine
expansion to a solution on all t > 0, periodic in x with period
2n.
If not, let H be any continuous function on B which agrees
with Gr1 and
at the corners and which satisfies IHI
- 1.
Then, let u' be the solution to the forward boundary“value
problem
(9.2)
(a)
u
= u
(b)
u'
=
G±
(c) u'
=
H
=
G-'
'xx
on
't
on
on
R
S± ,
B.
Let G-* be defined by
(d) u'
on
*
T
62
Notice that
lu* I
- 1
on
B.
implies
Hull
1? 0 - y - a.
O
tl
£
(c)
(a)
llu" II *
«y
11
on
R ,
= Gr
£
(!)
u"XX
= u!t
1
3
’ satisfies
(a)
(9.3)
-
ii
Then u"
«y
- G'
on
T ,
on
S. ,
x *
2, 0 - y ~ a »
Therefore, u = u' + u", where the periodic extension of u" is a
solution of the heat equation on t >0, has hounded
norm, and
is the backward continuation of G-". Moreover, since u' is the
solution to a well posed problem, small errors in G-^ lead to only
small errors in u', and therefore add only small errors to G".
We could consider the backward Neumann problem instead, in
which u , rather than u, is given on the sides. Again, by
subtracting the solution to a well posed forward Neumann problem,
we can leave u" = u - u* which has u" = 0 on the sides, and
which can therefore be extended symmetrically across x = 0 and
x = n to a periodic solution on t > 0 .
We have shown that it is sufficient to consider the
following problem for periodic solutions, which corresponds to
problem 1 of section 5. Suppose there exists at least one function
u(x,t) having the properties
(a)
u(x,t) is a solution of the heat equation for
t ^ 0, periodic in x with period 2ra ,
2
(9«4) (b)
(c)
a
llu - gll cl
- m = e
, where the L0c. function
g(x) is given,
Hull 0 - 1 o
63
Problem 1.
Find a function approximating every u satisfying (9.4).
Let
A
g(x) = —: +
y~
(9.5)
V7
(A
®
2
n a
cos n x + Bn sin n x)e"
,
i
and set
«
(9*6)
gw(x,t) = — +
n
-/2
\
(A
cos n x + Bn sin n x)e
_ 2+
n
^
‘
•
This, for H = [c<], is our approximation function
We can then easily obtain the a-priori error bound,
|u
-
<
1 +
H
’
as in lemma 8 (here L and H are defined in theorem 5), and exact
analogies to the a-posteriori compatibility conditions of lemmas
9 and 10.
Again, if the data for u across the top of the rectangle is
replaced by data at a finite number of interior points, we may
easily shew that it is sufficientto consider the following problem
for periodic solutions, which corresponds to problem 2 of section 7.
Let approximate values g^ for u be given at a finite number of
data points (x.,t.) in a single period. These points, with their
J <3
periodic images, either lie on the line t = a, with maximum
separation 5, or are the vertices of a simple polygonal path P
which separates y = G from y = a.
Let 6 "be the maximum segment
length of P, and let P itself lie between y = b and y = a, 0 - b - a.
Then the methods of section 7 carry through
in obvious analogy.
Cannon [2] has also treated backward solutions of the heat
equation.
The methods of problem 2 here are similar in many
respects to his.
64
10.
The Cauchy problem for Laplace's equation on a
rectangle.
We develop numerical methods only for the case,
corresponding to theorem 10 and corresponding to all the
numerical methods of previous sections, that the data curve he
embedded in the solution domain.
First, let us show that the
Cauchy problem on a rectangle with a prescribed bound can be
reduced to the Cauchy problem for periodic solutions with a
prescribed bound.
-a < y < a
On a rectangle R = |(x,y): 0
x < n,
we consider harmonic functions u(x,y) which are
continuous on the closure of R.
let B^,
Eg, ^1
^2 ^eno^e
the base, horizontal center line, top, and two sides of R.
convenience we also assume that u
For
is continuous in a neighborhood
of the end points of C.
Consider the Cauchy problem in which u and u^. are given on
the center line and u is given on the sides.
(10.1)
(a)
u
+ u
= 0
xx T yy
0>)
u = 5 on C ,
(c)
u
(d)
u = ^
y
011
R
That is, suppose
*
as F on C ,
on S^,
I G-± I
where G, F, and G^ are known approximately.
,,
12
- 1,
Moreover, suppose
(e)
1,
o
If G]_(y) = G2(y)
s
antisymmetrically across
then u(x,y) may be extended
x = 0
and x as n. to a periodic
harmonic function of the form (4.4)on-a<y<.a.
let
If not,
and EL, be any continuous functions on B^ and B^ which
agree with G1 and G^ at the corners and which satisfy |H^I - 1 ,
iHgl - 1.
Then, let u' be the solution to the Dirichlet problem
65
(10.2)
+ u"
=
(a)
u’
(b)
u' = G. on S.
l l
(c)
u* = H. on B.
XX
yy
I
I
Let O' and P* be defined by
(d) u' = G5 on C ,
(e) u'
y
= P1 on C
Then, u" = u - u' satisfies
(a)
' '
u"
xx
+ u"
0 on R ,
yy
(b) u" = G" = G - G* on C ,
(10.3)
(c) u"
= P" =
P - F‘ on C ,
U
(d) u" = 0 on
(e)
Hull
-
,
2
a
,
y - a
Therefore, u = u1 + u", where u* is the solution of a well-posed
Dirichlet problem and where the periodic extension of u" is of
the form (4.4) on -a < y < a .
Moreover, u', G', and even P’
(when its IlD^P* II
considered) depend continuously on G-^ and Gg.
norm is
Since u', G', and
P' depend linearly on G-^ and G2, it is sufficient to show they
are bounded when G-^ and G2 are.
(10.4)
Now, if
IG^I ^ e on Si,
then by the maximum principle
(10.5)
lu* I
^
s on R .
But, if u' is harmonic on a disc of radius 6, and lu* I
then Iu*
-
e there,
I
^ at tl*e center. This is shown by differeny
tiating under the integral in the Poisson integral formual. Thus
66
(10.6)
lF'(x)l
*
i|
,
where 6 is the distance from (x,0) to the boundary of R.
Now,
consider the periodic extension of F"(x) antisymmetrically across
x = 0 and x = w.
Since u'
is continuous on C even at the end
points, this is a piecewise continuous function with zero mean
value.
Hence, we may apply the bound (4.8 b) .
Let
F'(t) dt
It follows from (10.6) that
II(x)I < 6(2^ + K2 Ilog 16) »
where the constants
ratio |r.
and K2 depend only on the geometrical
The L2 norm of
+ K2 I log 16 is a finite number K,
hence,
Hill <
K e .
Finally, by (4.8 b) we have
IlD"1!" ||
(10.7)
£
Hill < K e ,
and from (10.5) we have
:
IIG-* II
^
e .
We therefore may conclude that small errors in
and G2 add only
small errors to G" and F".
We have shown that it is sufficient to consider the following
problem for periodic harmonic functions, which corresponds to
problem 1 of section 5.
u(x,y)
Suppose there exists at least one function
having the properties
67
(a) u(x,y) is of the form (4.4) on - a < y < a ,
* m = e”018, ,
(b) llu - ell
0
(10.9) - i
(c) 11D 1(u - f) II
(d) Hull
Q
-
m ,
\ + Hull \
-
2 ,
<3.
**"Cl
where g(x) and f(x) are given periodic functions with zero
mean value, g(x) in
and f(x) in 1^.
Problem 1. Find a function approximating every u satisfying (10.9).
oo
cos n x +
gU) =
sin n x ,
l
(10-.10)
GD
n C
f (x)
cos n x + n D sin n x .
n
n
and set
IT
gjj(x,y) =
21
(An cos n x +
sin n x) cosh n y
(Cn cos n x +
sin nx) sinh n y .
1
(10.11)
H
*
z
1
This, for N = [o<], is our approximation function.
We can then easily obtain the a-priori error bound
lu
"
s
Mly
<
1 + H ’
as in lemma 8 (here L and H are defined in theorem 10), and
close analogies to the a-posteriori compatibility conditions in
lemmas 9 and 10.
68
The problem for periodic harmonic functions corresponding
to problem 2 of section 7 is the following,.
We are given
approximate values g. and f. for u and u at a finite number of
j
j
y
data points is a single period. These points, with their
periodic images, either lie on the x axis, with maximum separation
6, or are the vertices of two simple polygonal paths P^ and Pg
which enclose the x axis between,
let 6 be the maximum segment
length of P-^ and Pg, and let P^ and
by the lines y = -b and y = +b,
themselves be enclosed
0 - b < a.
Then the methods of
section 7 carry through in obvious analogy.
Lavrentiev [12] also treats the approximate solution of the
Cauchy problem by Fourier expansion of the data.
However, he
derives lg rather than uniform error bounds and considers only the
case in which u is identically zero on the base and two sides of
the rectangle.
Moreover, his procedure involves a projection or
filtering process for the high order coefficients, with no
provision for truncation at a finite order.
Pucci [14] has considered a Cauchy problem in which the data
is given only at a finite set of points, as in problem 2 here.
See also Pucci [15] and Payne [13] for further references on the
Cauchy problem for Laplace’s equation.
References
[1] Agmon, S.,
The relaxation method for linear inequalities.
Canadian Journal of Mathematics, vol. 6(1954),
pp. 382-392.
[2] Cannon, J.R.,
Backward continuation in time by numerical
means of the solution of the heat equation in a
rectangle. Masters thesis, Rice Institute, I960.
[3]
Courant, R., and Hilbert, D.,
Methods of mathematical
Physics, vol. II, Interscience, Hew York, 1962,
pp. 226-231, 795-797.
[4]
Dantzig, G.B.,
Maximization of a linear function of
variables sub.iect to linear inequalities, Activity
analysis of production and allocation, edited by
T.Co Koopmans, John Wiley and Sons, New York, 1951,
pp. 345-547.
[5]
Douglas, J.,
A numerical method for analytic continuation,
Boundary problems in differential equations, edited
by R.E. Danger, University of Wisconsin Press,
Madison, I960, pp. 179-189.
[6] Douglas, J.,
Mathematical programming and integral
equations,
Symposium on the numerical treatment of
ordinary differential equations, integral and
integro-differential equations,
Birkhauser, Basel,
I960, pp. 269-274.
[7] Douglas, J., and T.M. Gallie,
An approximate solution of
an improper boundary value problem, Duke Mathematical
Journal, vol. 26(1959), pp. 339-348.
[8] Forsythe and Wasow,
Finite-difference methods for
partial differential equations, John Wiley and
Sons, New York, I960, pp. 283-288.
[9]
Gass, S.I.,
Linear programming:methods and applications,
New York, McGraw-Hill, 1958.
[10] Hobson, EoWo, The theory of functions of a real
variable and the theory of Fourier series,
2nd edition, Cambridge University Press,
London, 1926, pp. 551-553.
[11] John, F.j
Continuous dependence on data for solutions
of partial differential equations with a
prescribed bound, Communications on Pure and
Applied Mathematics, vol0 13(1956), pp. 551-585,
[12] Lavrentiev, M.M.,
equation,
^n the Cauchy problem for the Laplace
(in Russian), Izvestiya Akademii Nauk
SSSR. Seriya Matematiceskaya, vol. 20(1956),
pp0 819-842.
[13] Payne, LoE.,
Bounds in the Cauchy problem for the
Laplace equation, Archives of Rational Mechanics
and Analysis, vol. 5(1960), pp. 35-45,
[14]
Pucci, C.,
Sui problemi di Cauchy non. ben posti, Atti
Accad. Naz. Lincei. Rend. Cl. Sci. Pis. Mat. Nat.
(8), vol. 18(1955), PP. 473-477.
[15] Pucci, Co,
Discussione del problema di Cauchy per le
eouazioni di tipo ellitico, Annali di Matematica
Pura ed Applicata, series 4, vol. 66(1958),
pp0 131-154.
[16] Titchmarsh, E0C0,
The theory of functions, 2nd edition,
Oxford University Press, London, 1939, p. 172.