ON THE SOLUTIONS OF QUASI-LINEAR ELLIPTIC
PARTIAL DIFFERENTIAL EQUATIONS*
BY
CHARLES B. MORREY, JR.
In this paper, we are concerned with the existence and differentiability
properties of the solutions of "q-qasi-linear" elliptic partial differential equations in two variables, Le., equations of the form
A (x,
y,
z, p, q)r+ 2B(x,
>0
AC-B2>0,A
(
y,
Z,
p, q)s + C(x,
p=-,
oz
ox
q=-,
oZ
oy
y,
r=-,
z,
p, q)t = D(x,
02Z
ox2
s=--,
02Z
oxoy.
y,
z,
t=-
p,
q)
02Z)
oy2
.
These equations are special cases of the general elliptic equation
cf>(x,
y,
Z,
p, q, r, s, t)
= 0,
cf>rcf>t-
cj>82
> 0,
cf>r
> O.
The literature concerning these equations being very extensive, we shall
not attempt to give a complete list of references. The starting point for many
more modern researches has been the work of S. Bernstein, t who was the
.:fi.rstto prove the analyticity of the solutions of the general equation with 4>
analytic and who was able to obtain a priori bounds for the second and higher
derivatives of z in the quasi-linear type in terms of the bounds of I z I , I pi, I q.1
and the derivatives of the coefficients. He was also able to prove the existence
of the solution of the quasi-linear equation in some very general cases. He
assumed that all the data were analytic. However, his papers are very compli, cated and certain details require modification. On account of the results of
J. Horn, L. Lichtenstein, and many others,t the restriction of analyticity has
been removed. Some very interesting modern work has been done by Leray
and Schauder§ in a 'paper in which they develop a general theory of nonlinear functional equations and apply their results to quasi-linear equations,
* Presented to the Society, Apri111, 1936; received by the editors February 9, 1937.
t Particularly the papers: Sur la nature analytique des solutions des equations aux derivees partielles d·u second ordre, Mathematische Annalen, vol. 59 (1904), pp. 20-76 and Sur la generalization
du probleme de Dirichlet, Mathematische Annalen, vol. 69 (1910), pp. 82-136.
t For an account of some of these results, see the article by L. Lichtenstein on the theory of
elliptic partial differential equations in the Encyklopadie der Mathematischen Wissenschaften, vol.
II 32, pp. 1280-1334.
§ Jean Leray and Jules Schauder, Topologie et eq'uations fonctioneUes, Annales Scientifiques de
l'Ecole Normale Superieure, vol. 51 (1934), pp. 45-78.
126
ELLIPTIC DIFFERENTIAL
EQUATIONS
127
Schauder* has also obtained good a priori bounds for the solutions (and their
derivatives) of linear elliptic equations in any number of variables.
In the present paper, an elliptic pair of linear partial differential equations
of the form
is studied. We assume merely that the coefficients are uniformly bounded
and measurable. In such a general case, of course, the functions u and v do
not possess continuous derivatives but are absolutely continuous in the sense
of Tonelli with their derivatives summable with their squares (over interior
closed sets). However, certain uniqueness, existence, and compactness theorems are demonstrated and the functions u and v are seen also to satisfy
Holder conditions. These results are immediately used to show that if z(x, y)
is a function which minimizes
f fR!CX, y, z, p, q)dxdy,
among all functions (for which the integral may be defined) which take on
the same boundary values as z(x, y) and if z(x, y) satisfies a Lipschitz condition on R, its first partial derivatives satisfy Holder conditions. E. Hopft has
already shown that if p and q satisfy HOlder conditions, then the second
derivatives satisfy Holder conditions. In proving this fact, Hopf shows that
if the coefficientsin (1) satisfy HOlder conditions, the first partial derivatives
of u and v satisfy Holder conditions. The results of this paper concerning the
system (1) together with Hopf's result yield very simple proofs of the existence of a solution of the quasi-linear equation in certain cases, a few of which
are presented in §6.
The developments of this paper are entirely straightforward, being, for
the most part, generalizations of known elementary results analogous to the
step from Riemann to Lebesgue integration. The main tools by means of
which the Holder conditions of u and v in (1) are demonstrated are Theorems
1 and2 of §2 which state roughly that: if ~= Hx, y), 'I] = 'I] (x, y) is a 1-1 differentia13'letransformation of a Jordan region R into another Jordan region ~
in which the ratio of the maximum to the minimum magnification is uniformly bounded, then the functions ~ and 'I] and the functions x(~, '1]) and
y(~, '1]) of the inverse satisfy Holder conditions. Since these Holder conditions
* J. Schauder, Uber lineare eUiPt-ische Dijferentialgleichungen zweiter Ordnung, Mathematische
Zeitschrift, val. 38 (1933-34), pp. 257-282.
t E. Rapf, Zum analytischen Charakter der Losungen regultirer zweidimensionaler Variat-ionsprobleme, Mathematische Zeitschrift, val. 30, pp. 404-413.
128
C. B. MORREY
[January
are so important in the modern theory of elliptic equations, these two theorems may prove to be an important tool in this field.
We use the following notation: A function <j>(x, y) is said to be of class
C(n) if it is continuous together with its partial derivatives
of the first n
orders. If E is a point set, E denotes its closure and E* its boundary points.
If E and F are point sets, the symbol E·F denotes their product, E+F denotes their sum (E and F need not be mutually exclusive), and E c F means
that E is a subset of F. The symbol C(P, r) denotes the open circular disc
with center at P and radius r.
1. Preliminary definitions and lemmas. Most of the definitions and lemmas of this section are either found in' the literature or are easily deducible
from known results. We include the material of this section for completeness.
DEFINITION 1. A function u(x, y) is said to be strictly absolutely continuous
in the sense of Tonellit (A .C.T.) on a closed rectangle (a, c; b, d) if it is continuous there and
(i) for almost all X, a;£X;£b, u(X, y) is absolutely continuous in y on
the interval (c, d), and for almost all Y, c;£ Y;£d, u(x, Y) is absolutely continuous in x on the interval (a, b) and
(ii) V/v)d [u(X, y)] and Va(x)b [u(x, Y)] are summable functions of X and
Y, respectively, Vc(v)d [u(X, y)], for instance, denoting the variation on (c, d)
of u(X, y) considered as a function of y alone. It is clear that these variations
are lower semicontinuous in the large lettered variables.
DEFINITION 2. A function u(x, y) is said to be A.C.T. on a region R (or R)
if it is continuous there and strictly A.C.T. on each closed interior rectangle.
Remark. Evanst has shown that every continuous "potential function of
its generalized derivatives" § is A .C.T. and conversely, so that his theorems
concerning the former functions are applicable to the latter. Thus
1.§ If u(x, y) is A.C.T. on R, then aujax=ux and aujay=uy exist
almost everywhere in R and are summable over every closed subregion of R.
LEMMA
t) andy=y(s, t) isa 1-1 continuous transformation
of class C' of a region R of the (x, y)-plane into a region ~ of the (s, t)-Plane,
and if u(x, y) is A.C.T. on R, then thefunction u[x(s, t), yes, t)] is A.C.T. on ~
and
LEMMA
b
t
L. Tonelli, Sulla quadratura delle superficie, Atti della Reale Accademia dei Lincei, (6), vol. 3
(1926),pp.
t
2.§ Ifx=x(s,
357-362,445-450,
633-638, 714-719.
G. C. Evans, Complements of potential theory, II, American Journal' of Mathematics, vo!. 55
(1933), pp. 29-49.
§
G. C. Evans, Fundamental points of potential theory, Rice Institute Pamphlets, val. 7, No. 4
(1920),pp.252-329.
j
ELLIPTIC DIFFERENTIAL
1938]
129
EQUATIONS
almost everywhere.
DEFINITION 3. Let D be a region. By the expression almost all rectangles
of D we mean the totality of rectangles a ~ x ~ b, c ~ y ~ d in D for which a
and b do not belong to some set of measure zero of values of x, and c and d
do not belong to some set of measure zero of values of
both sets of measure zero may be vacuous.
y.
Naturally
either or
DEFINITION 4. Let 1>(D) be a function defined on almost all rectangles D
of a region R such that whenever D =Dl +D2, where Dl and D2 are admissible
rectangles having only an edge in common, we have that 1>(D) =rj:>(D1)
+1>(D2). We say that 1>(D) is absolutely continuous on R if for every e>O,
there exists a 0> 0 such that for every sequence of non-overlapping admissi[meas (Dn)] < 0, we have
I 1>(Dn) I <e.
ble rectangles {Dn} with
L
LEMMA
L
3. t Let f(x, y) be summable on R, let D denote a rectangle (a, c; b, d)
on which f(x, y) is summable
<t>(D)
if;(D)
f
=f
=
(this being almost all rectangles of R) and define
d
[j(b, y) - f(a, y) ]dy
=
ab
(j(x, d) - j(x, c) ]dx
= - -f D* fdx.
ID/dY,
Then a necessary and sufficient condition that f(x, y) be A.C.T.
f(x, y) be continuous and rj:>(D)and l{;(D) be absolutely continuous
region I1for which XC R. When this is true,
<t>(D)
=
II
D
aj dxdy,
ax
if;(D)
=
If
aj
D ay
on R is that
on each sub-
dxdy
for each rectangle in R.
DEFINITION 5. We say that a function 1>(x, y) is of class Lp on a region
R if
11>
I
p
is summable over R.
DEFINITION 6. We say that a function u(x, y) is of class Da on R or R if
it is¥.1.C.T. there and luxl a and lulll a are summable over every closed sub-
region of R.
LEMMA
4. Let {1>n(X,y) }, n = 1, 2, ...
, and 1>(x, y) be of class Lp on a
rectangle (a, c; b, d) with
t A summable functionf(x, y) for which cf>(D) and tj;(D) are absolutely continuous on each subregion ~ of R for which t:c R is said to be a "potential function of its generalized derivatives" and
this lemma is essentially the theorem of Evans mentioned above (ComPlements of potential theory,
lococit.).
130
C. B. MORREY
[January
> 1,
p
where G is indapendent of n. Let
=
<l>n(X, y)
f f
ax
c Y fjJn(~,
1J)d~d1J,
=
<I>(x, y)
f f
ax
c Y fjJ(g,
rJ)dgd1J
and suppose that the sequence {<I>n(X, y)} converges uniformly to <I>(x, y). Suppose also that {An(x, y)}, n=l, 2, ... , and A(x, y) are of class Lq on (a, c;
b, d), q=pj(p-l),
and that
lim
n~oo
Then
(i)
f" a
(ii)
lim
n--:,.co
f
f bf
c
dl
r:/J
a
C
f
a
"fd, c
/pdxdy
An -
=
A /qdxdy
o.
lim
n~ooinf f- a "f c d) r:/JnIpdxdy,
~
=
dAnr:/Jndxdy
f "f
a
c
dAfjJdxdy.
Proof. The first conclusion is well known. t
To prove (ii), choose €>O. For n>N1, we see from the Holder inequality
that
Ifa"
fd(An
-
A)r:/JndXdyl
~
[f"Ida
Jl/q[fbfd
c'An-A/qdXdy
a
c!fjJn/PdXdY
Now, let {Bk(x, y) } be a sequence of step functionsj
b
lim
.1:-"0
Then, for k>K (independent
<4'
€
such that
d
f f/
a
JI/P
c
Bk - A Iqdxdy
.-:. O .
of n),
t For instance, this result may be obtained by the method of proof used in Theorem 7, §1 of
the author's paper, A class of representations 'of manifolds, I, American Journal of Mathematics, vol. 55
(1933), p. 693.
t To form these, let Gk denote the grating formed by all lines of the form x=2-ki, y=2-kj. We
then define Bk to be a properly chosen constant on the part of each square of Gk which contains a
point of the rectangle.
ELLIPTIC DIFFERENTIAL
1938]
~
I fab
f
d(Bk -
[fbfda
c
A)~dxdy
~
I
131
EQUATIONS
B k - A I qdxdy Jl/q[fbfd.
a
c'
A I qdxdy Jl/q[fbfd
a
c
~n I pdxdy JIIP
<
4'
JIIP
<
4.
B ko is
con-
E
\
[fbfda
c
I Bk -
I ~ I pdxdy
E
Now, let ko > K. Then, for n > N2,
the Ri being the rectangular subregions of R over each of which
stant and Bko.i being the value of Bko on Ri.
Finally, if we let N be the larger of NI and N2, we see that
[fab
fd(An~n
- A~)dxdy
~ I Jab fdCAn
-
+ I fa J
b
I
I
A)~ndxdy
+ I Jab fdCA
dBko(~n - ~)dxdy
j
+ I fa
b
-
f
Bko)~ndxdy
d(Bko -
I
A)~dxdy
I
< E.
This proves the lemma.
LEMMA
points
5. Let cp(x, y) be of class Lp on R, p > 1, and let Rh be that subset of
(xo, Yo) of R such that all points
I x-xol
(x, y) with
<h,
I y-Yo!
<h are
in R. Let
h> 0,
be defined
(i)
i'n Rh. Then
ff
Rh I
;~ [f
CH)
<Ph
!Pdxdy
IRhl
is continuous
cph(X, y)
~
ff
~h Ipdxdy
RI
-
<P
and of class Lp on Rh and
IPdxdy,
f f) ~Ipdxdy J
=
lim
71->0
ff
Rho
I
~h
-
~
I Pdxdy
=
0,
ho
> o.
132
C. B. MORREY
Proof. This lemma is well known.
[January
t
DEFINITION 7. Let u(x, y) be defined on R or R. If u is of class D2 on R
or R with lux' 2 and I Ull!2 summable over the whole of R, we define
D(u)
Otherwise we define D(u)
6. Let {un(x,
= f fR (ui + u1l2)dxdy.
= + 00.
and {vn(x, y)} each be of class D2 on R with
D(un) and D(vn);[: G independent
of n, and suppose that {un(x, y)} and
{vn(x, y) } converge uniformly on R to functions u(x, y) and vex, y), respectively.
Let the sequences {an(x, y)}, {bn}, {cn}, {dn}, {en}, Un}, {gn}, {hn}, {kn},
and {In} be measurable and uniformly bounded and suppose the sequences converge almost everywhere on R to a, b, c, d, e, f, g, h, k, and 1 respectively. Then
LEMMA
y)}
(i)
u(x, y) and vex, y) are of class
(ii)
D(u) ~ lirn inf D(un), D(v) ;[: lirn inf D(vn) ,
n~oo
(iii)
on R,
D2
n~oo
f fR [(aux+bull+cvx+dvll+e)2+(jux+guy+hvx+kvy+l)2]dxdy
;£lirn inf
n~Ct:J
If
R
[(anunx+bnunll+cnvnx+dnvny+en)2
.
C/nUnx+gnUny+ hnvnx+ knVny+ln)2Jdxdy.
+
Proof. Conclusions (i) and (ii) are well known. t
To prove (iii), let M be the uniform bounds for an, etc., and let
1>
=
au",
+ buy + cv"' + dvy + e,
Then, for each h > 0, we see that cf>n(h), 1f;n (h), cf>(h) and 1f;(h) are uniformly
bounded on Rh, the proof for cf>n(h), for instance, employing the Holder inequality as follows:
t See Lemma
1, §1 of the author's paper, loco cit.
ELLIPTIC DIFFERENTIAL
1938]
each * denoting a term similar to the first term. By Lemma 4,
converge at each point to cP and If respectively. Hence,
If
If
~ lim inf If
+ t/;~)dxdy =
Rh (et>h2
133
EQUATIONS
lim
n-->eo
n-->eo
Rh (et>:h
R (et>:
cPnh
and
Y;nh
+ t/;:h)dxdy
+ t/;=)dxdy,
using Lemma 5. The statement (iii) follows from Lemma 5 by letting h-'70.
LEMMA 7. Let f and
be of class D2 on R, let D be a Jordan subregion of R
such that D* is a rectifiable curve interior to R on which ep is of bounded variation. Then a (j, ep) / a (x, y) is summable on D and
ep
f D* fdcjy
=
If
D
a(j, cjy)
a(x,
y)
dxdy,
the line integral being the ordinary Stieltjes integral.
Proof. That the Jacobian is summable follows from the Schwarz inequality, since f1 +fJ +<1>1+<I>J is summable over D.
have their usual significance as average functions. If I hi,
Let fh and
I k I < a, ih and ep k are defined on [) and moreover
is of class C' if h > 0 and
epI, is of class C' if k >0. Hence, by Green's theorem
<l>k
h
Letting k tend to zero, and using Lemmas 4 and 5 and well known theorems
on the Stieltjes integral, we see that
The result follows by letting h tend to zero.
2. Fundamental theorems on transformations. We state first
DEFINITION 1. We say that u(x, y) satisfies a condition A [~; M(a, d)]
on R if it is of class
D2
on Rand
134
C. B. MORREY
If
C(P,r)
+ ull2)dxdy
(Ux2
P
=
~ M(a,
[January
(x, y)eR,
o
(!.-)A,
a
d)
<r<
a,
> 0,
A
where a >0, d >0, and a+d is the distance of (x, y) from R*, M(a, d) depending on a and d and not on (x, y).
DEFINITION 2. We say that u(x, y) satisfies a condition B [}.L; N(a, d)] on
Rif
I U(X1, Y1) -
U(X2, Y2) I ~ N(a,
r
=
[(X2 -
X1)2
d) (:
+ (Y2
0
}',
-
a condition
satisfies a condition B [A/2; N(a, d)], where
N(a,
d)
=
a,
Y1)2]1/2,
provided that every point on the segment joining
distance;;:::;al2 +d from R*.
LEMMA 1. Let u(x, y) satisfy
<r <
(Xl, Y1)
to
(X2, Y2)
is at a
A [A; M(a, d)] on R. Then it also
8·3-1/2·A-1• [M(aI2,
d)]1/2.
Proof. First assume that u(x, y) is of class C' on R. Let PI:
and
P2: (X2, Y2) be two points of R which are such that every point on the segment
joining them is at a distance > al2 +d from R*. Next, choose axes so that PI
is the origin, and P2 is the point (2-1/2·r, 2-1/2·r). Then each square of the
form
o ::::;x ~ 2-1/2. rt,
is in a circle of radius
Let a = 2-1/2 r', then
f
o
atf
faa-at f
0
at(u}
rtl2
2-1/2·r·(1
+ Uy2 )dxdy
a CUll? + u1I2)dxdy
a-at
- u(O, 0)
=
x ~
-
- t) ::::;Y ~ 2-1/2·r
whose center is at a distance ;;:::
al2
:5 M(aI2,
d) (rtl a) "A,
~
d)(rtla)A,
M(aI2,
Now, for each (x, y) with O~x~a,
u(a,a)
or
o ~ y ~ 2-1/2·rt
(1 - t) ~ x ~ 2-1/2r,
2-1/2·r·
1ux(xt, yt)dt
+ y fo lUy(xt,
+
t(x -
+d
O~t<l,
O~t<1.
we have
O~y~a,
(x - a) fo1uil'[a
(Xl, Y1)
yt)dt
a), a
+
t(y -
a)Jdt
from R*.
ELLIPTIC DIFFERENTIAL
1938]
-
(y -
i
a)
+ t(x
luv[a
-
a), a
+ t(y
Integrating both sides of this equation with respect to
u(a, a) -
-
u(O, 0)
=
fo a
~2
i {i
a
1XU:I;(xt,
135
EQUATIONS
yt)dt}
dxdy
x
-
a)]dt.
and y, we obtain
+*
~2iafoa{fo\x-a)U:I;[a+t(X-a),a+t(y-a)]dt}dXdY-*
=1f
a2
-
0
1t31
{f
0
at
f
0
~U~(~, 7J)d~d7J
at
+*
the last being obtained by suitable changes of variable; the * denotes the
term in Y or 1] which is similar to the term preceding it. Using Schwarz's inequali ty on the interior terms, we obtain
1
u(O, 0) I:'S:
u(a, a) -
+* + [
0 1a-2t-3
{[f
fa a-at fa a-at(~ -
~ 4.3-1/2[M(a/2,
=
f
0 atf
at~2d~dY'f
Jl/2[f
a)2d~d7) Jl/2 [f
d)]1!2.(r/a)X!2
8X-13-1!2[M(a/2,
0
~
0
atf
0
atU~2d~d'I'J
Jl/2
a fa a_atu~2d~d'I'JJ1/2
a-at
+ * '} dt
ItX/2-1dt
d)]1!2·(r/a)l'!2.
If u(x, y) is merely of class Dz, Uh(X,
general result by letting h tend to zero.
y)
is of class C' and we obtain the
= 1] (x,
be a 1-1 continuous transformation of a closed region R into a closed region ~, which is of class C' in
R with ~.,7Jv-~v7J.,~O. Let (xo, Yo) be a point of R, x=x(O'), Y=Y(eT) (eT arc
length) be a regular curve such~that x(lTo) =xo, y(uo) =yo, x'(uo) =cos e,
y'(lTo) "':'sin e. If (~o, TJo)is the point of ~ corresponding to (xo,yo) and if ds is
the differential of arc length of the curve in ~ corresponding to the above, we
define the magnification of T at (xo, yo) in the direction e by I ds/ de I·
DEFINITION 3. Let T: ~=Hx,
y),
1]
y)
Remarks. Clearly this magnification depends only on (xo, yo) and e and
not on the curve chosen. It is given by
I dO'
ds =
/2
Eo cosz ()
+ 2Fo sin e cos e + Go sin2 8,
Eo
= E(xo,
Yo),
etc.
136
C. B. MORREY
[January
The square of its maximum and minimum (with respect to e) at Po are given
by
+ Go +
~ {Eo + Go -
~ {Eo
respectively,
tion is
+ Go)2
[(Eo + Go)2-
[(Eo
-
4(EoGo
F02) ]1/2} ,
-
4(EoGo - F02)
]1/2}
,
so that the ratio of the maximum to the minimum magnificaEo
Mo
= 2(EoGo
+ Go
_
F02)1/2
at Po. If this ratio is uniformly bounded in R, it is clear that the inverse transformation has the same property. In the foregoing remarks, E, P, and G have
their usual differential-geometric significance:
E
THEOREM
= ~} + 'rJ:?,
1. Let R and ~ be two Jordan regions, a, b, and c be three distinct
points of R*, and Cl, {3, and'Y be three distinct points of L*. Let {T}
of 1-1 continuous transformations
of the form
T:~=Hx,y),
'rJ
= 7](x,
be afamily
y)
which carry R into ~, which carry a, b, and c into a, {3, and'Y respectively, and
which satisfy the following hypotheses:
(1) each T is of class c' within R with ~x1]y-~y1]x¥=O,and
(2) the ratio of the maximum to the minimum magnification of each transformation at each point (x, y) is ~K, which is independent of x, y, and T. Then
there exist functions M(a), N(a), P(a), and m(a) which depend only on K, the
regions Rand ~, and the distribution of the points a, b, c, a, {3, and 'Y; and there
exists a number}.. >0 which depends only on K such that
0) M(a), N(a), m(a) >0 for a>O, lim,,~oM(a) =lima-+oN(a) =lima~oP(a)
=lima~om(a) =0,
(ii) all points of R or ~ which are at a distance ~ p > 0 from R* or ~* correspond to points of the other region at a distance ~m(p) from its boundary, and
(iii) the functions Hx, y), 1](x, y), x(t 1]), and y(~, 1]) all satisfy conditions of
the form A [2}..; M(a, d)] and B [}..; N(a, d)] with M(a, d) = M(a), N(a, d)
= N (a), and the equicontinuity condition
I
rjJ(al, (31) -
cp(a2,
Ca, (3)
(32)
=
I ~
P(a) ,
(x, y) or C~, 'rJ),
[(a2
cp
,
-
al)2
~, 7],
+
({32 -
x, or y.
(31)2]1/2
=
a,
ELLIPTIC DIFFERENTIAL
1938]
137
EQUATIONS
Proof. Since the ratio of maximum to minimum magnification is < K, it
follows that
J JR (~}
+
1];c2+ 1]y2)dxdy
~
2Km('2),
f f}; (X~2
+ XfJ2 + y~2 + y,nd~d1]
~
2Km(R).
~y2
+
From this,t follows the existence of the functions Pea) and mea) satisfying
the desired conditions.
Now, let Po belong to R, for example, being at a distance a from R*.
is carried into a Jordan subregion of ~
The circle (x-xo)2+(Y-Yo)2~a2
which is surely a subset of the circle (~-~o)2+(17-7}o)2~ [P(a)]2, (~o, 7}o)being the correspondent of (xo, yo). Let
A(r)
=
ff
C(Po,r)
lex,
o
Then
;?;;
y)dxdy
r ~
a,
= fr0 f
J(x,
+ p cos 8, Yo + p sin e)dpd8,
0 21rpl(xo
y)
= I ~;c1]y-
o
~y1];cI·
< r < a;
here, s denotes arc length on the circle (x-xo)2+(y-Yo)2=r2,
curve in ~ into which this circle is carried. Thus
and er is the
A'iA ~ 2/(Kr),
and hence
o
2
r ~ a.
Sin'te
we find that
Jf
C(Po,r)
(~;c2
+ ~y2 + 'YJ} + 'YJJ)dxdy
~
2Kll'[P(a)
]2(rla)2!K.
t See the author's paper, An analytic characterization of surfaces of finite Lebesgue area, I, American Journal of Mathematics, val. 57 (1935), Theorem 1, §2, p. 699.
1
,
r
138
C. B. MORREY
[January
Hence, we see that (Hi) is satisfied (remembering Lemma 1) if we choose
A
=
l/K,
M(a)
=
2K1I"[P(a)]2,
N(a)
=
8·2I/2·3-I/211"1/2K3/2·P(a/2).
DEFINITION 4. We say that a Jordan region R and three distinct points
a, b, and c of R* satisfy a condition D(L, do) if (1) the distances ab, ac, and be
are all ~ do >0, (2) R* is rectifiable, and (3) if PI and P2 are any two points
,...--...
--
,...--...
of R*, the ratio P1P2-T P1P2 ~L, where PIP2 is an arc of R* joining PI and
which contains at most one of the points a, b, or c in its interior.
P2
a, {3, and 1", and the
Theorem 1 and suppose
THEOREM 2. Let the regions R and ~, the points a, b, c,
family {T} of transformations satisfy the hypotheses of
(R; a, b, c) and (~; a, (3, 1") satisfy a condition D(L, do), do>O. Then the conclusions of Theorem 1 hold and, in addition, there exists a number M depending
only on K, L, do and the areas m("'2:) and meR), and a number p,>0 depending
only on K and L such that
f
ff
J
C(Po.r)
C(Po,r)-Z
·R
(~",2
+ ~y2 +
+ 'l]y2)dxdy
:::;
Mrf.l,
(X1;2
+ x,? + Yl;2 + y,nd~d'l]
::;;
MrJl.,
'1]",2
for any point Po in the Plane.
Proof. We need to prove only the last statement. Let Po be a point in
the plane and let O<r<do/2. Then the set C(Po, r) -R is vacuous or consists
of a finite or denumerable number of Jordan regions rn, the boundary of each
of which consists of (1) a finite or denumerable set of arcs of C*(Po, r)· R,
(2) a finite or denumerable number of arcs of R*, and (3) points of R* which
are limit points
of all of~ these arcs. All~ the points of (2) and (3) are on one
"-....,,,.-....,
of the arcs abc, bca, or cab of R*, sayabc. Clearly rn* and rn7 have at most
one point in common if n~n'. Let En.r be the set (1) above for each rn, let
Er='LEr,n,
let Un be the region of "'2:corresponding to rn, let u=Lun,
let
Cr.n =O"n*,let Cr=L:Cr.n, let rr.n be the totality of arcs corresponding to Er.n,
and let rr=L:rr,n.
Clearly Cr.n-Cr.n, is at most one point if n~n'.
Let
I(C) =L:1(Cr,n), let l(rr,n) be the sum of the lengths of all the arcs of rr,n
and let l(rr) ='Ll(rr.n).
It is clear that
[l(er)]2
~
47rm(lT).
Consider an rn, and the closed set R* -r n*. Proceeding along the arc (abc),
there is a first point P n and a last point Qn of this set. Then, there is an arc
of Er.n joining P n to Qn. Hence if ITn and Kn are the corresponding points
ELLIPTIC DIFFERENTIAL
1938]
139
EQUATIONS
--.
of ~*, they are on the arc a{3'Y and there is an arc of rr,n joining them. Now
--.
(arc of a{3'Y) ~L
easy to see that
l(lIn,
times the length of this arc of rr,n. Hence it is
Kn)
1
L+l
l(rr) ;:::--
I(C).
Any of the above sets may be null and for a set of measure zero of values of r,
l(Cr) and l(rr) may be infinite.
We may now proceed as in Theorem 1. Let
A (r)
=
ff
C(Po,r)·R
o ~ r ::; do/2,
J(x,
=
y)dxdy
0
r[p
= I L1]y
y)
J(x,
f f
-
Ep
J(xo
~y1]:I;
+ p cos 0, Yo + p sin 0) dB]
I,
(the integral being zero if the field of integration
A 'er)
is null). Then
f
= r ~ J(xo + r cos 0, Yo + r sin O)dO > 2.
K
6
(2K7rr)-I[fEr(~82
+ 1]82)1/2dsJ=
+ 1)-2 [l(Cr)
~ (2KlI"r)-I(L
]2
>
dp,
f
Er
(~82
+
1]82 )ds
(2KlI"r)-I[l(rr)]2
2K-I(L
+ 1)-2r-I·A(r).
Also A (do/2) ~m(~). Hence as before
A(r)
ff
C(Po,r)
(~:I;2
+
~y2
+
~ mCJ:)·(2r/do)2IK<L+I)\
1]:1;2
+ 1]y2)dxdy
<
2Km("j;)(2r/do)2IKCL+I)2.
Since A (r) ~ m(~) for all values of r, the theorem follows.
Let a, bl, b2, c, d, and e be measurable functions defined on a
bounded region R with I aI, I bll, I b21, I cl, I dl, I el ~M> 1 on R. Then there
exist sequences {an}, {bI•n}, {b2•n}, {cn}, {dn}, and {en} which are analytic
on R and uniformly bounded and which converge almost everywhere on R to
a, bI, b2, C, d, and e respectively. If bI = b2, we may choose bI,n = b2,n for each n.
If bl = b2 = band ac - b2 > m> 0 on R, we may choose the sequences so that there
exist numbers M and fit >0 such that
LEMMA 2.
Further, if ac -
b2
= 1, the sequences may be chosen so that ancn-
bn2
= 1.
Proof. Let D be a region containing R in its interior and define a = c = 1,
140
C. B. MORREY
[January
in D-R.
For h sufficiently small, ah, blh, b2h, Ch, dh, and eh
are defined and continuous in a region containing R and all are numerically
b1=b2=d=e=0
~M.
Now suppose b1=b2=b, ac-b2~m>0.
We know that ac-b2 is the product of the maximum by the minimum (for 0 ~ fJ ~ 27r, (x, y) fixed) of
j( x,
Clearly \f(x,
y; fJ)
y; e)
I ::'£
= a cos2 e + 2b sin 8 cos 8 + c sin 8
= H(a + c) + (a - c) cos 28 + 2b sin
2
28].
2M so that the minimum above ~ m/2M.
ahCh- bh2
m2/4M2
~
Thus
>0
for each h >0 and all (x, y) in Dh' The remainder of the proof is obvious.
3. Let a, b, and c be analytic in a region G which contains the Jordan
region R in its interior and suppose I ai, I b I, I c I ~ M, ac.-b2 = 1, a >0; let ~
LEMMA
be another Jordan region. Then there exists a unique 1-1 analytic map
~= (x, y), 'Y]= 'Y](x, y) of R on ~ which carries three given distinct points p, q,
and r on R* into three given distinct points 7r, K, P (arranged in the same order)
on ~*, and which satisfies
The J acobian does not vanish and the ratio of the maximum
magnification of the transformation is ;;£ M at each point.
to the minimum
Proof. Let D be a Jordan region contained in G and containing R whose
boundary is a regular, analytic, simple, closed curve. It is knownt that there
exists a solution X(x, y) of the equation
a
-
ax
a
(aXx
+ bXll) + -ay (bXx + cXu)
=
0
which takes on the values X=x on D*, which is analytic on D, and whose
first derivatives do not vanish simultaneously. Clearly there exists an analytic
conjugate function Vex, y) which satisfies the same equation and the relations
'"
Yx
= -
(bXa;
+ eXy),
Yy
=
aXx
+ bXlI•
The equations X=X(x, y), Y=Y(x, y) yield a 1-1 analytic map of (D and
hence) R onto a region ~ which carries p, q, and r into three points 7r', K',
and p' arranged on ~* in the same order, and for which XxYu-X!lYa;~O.
If
~=~(X, Y), 1]=H(X, Y) is the conformal map of ~ on l: which carries
7r', K', andp' into 7r, K, andp (respectively), it is easily seen that
t See Lichtenstein,
loco cit.
ELLIPTIC DIFFERENTIAL
1938]
~=
~(x, y)
=
E[X(x,
y), Y(x, y)],
7/
=
141
EQUATIONS
7/(x, y)
=
H[X(x,
y), Y(x, y)]
is a mapping of the desired type. That this mapping is unique follows from
the fact that if ~=~'(x, y), 1]=1]'(x, y) is another such map, then (e, 1]') are
related to (~, 1]) by a conformal transformation with 7r, K, and p fixed; thus
,stJ-t
=s, 1]=1].
LEMMA
4. Let ~ = ~(x, y),
gion R in which the functions
1]
= 1] (x, y) be a transformation
~ and
7/x
1]
=-
are of class D2. Suppose
~y,
7/y
almost everywhere in R. Then the above map
jugate harmonic functions.
= ~"
is conformal,
i.e., ~ and
Proof. Let D be a rectangle on the boundary of which
continuous, such rectangles being almost all rectangles
Lemma 7, §1, it follows that
L(SD)
=
I ID (EC - P2)1/2dxdy
=I
ID (~x'Y]y- ~yYJ
x)dxdy
defined on a realso that
= ~I
=
ID (~,,2
1]
are con-
is absolutely
in R. Then, from
1] (x,
y)
+ ~y2 + 'Y],,2+ 'Y]y2)dxdy
IDo ~d'Y],
SD being the surface ~=Hx, y), 7/='Y](x, y), (x, y)£D, L(SD) meaning its
Lebesgue area. Since the Geocze area G(S) of any surface with this boundary
curve must be at least as great as L(SD), and since L(S) ~G(S) for every
t
surface, we see that SD is a surface of minimum area bounded by its boundary
curve. Hence ~(x, y) and 1] (x, y) must be harmonic, since otherwise they
could be replaced by the harmonic functions having the same boundary valu~s to form a surface of smaller area bounded by the boundary of SD.
3. Let Rand
~ be Jordan regions, let p, q, and r be distinct points
on R* and let 7r, K, and p be distinct points arranged in the same order on ~*.
Let a, b, and c be bounded, measurable functions defined on R:
THEOREM
lal,
Then there exists a
into ~
Ibl,
1-1 continuous
IcJ<M,
transformation
ac -
b2
= 1.
~ = Hx, y),
(i) which carries p, q, and r into 7r, K, and p respectively,
(ii) which is such that Hx, y), 1](x, y), x(~, 1]), and y(~,
7J
= 7/(x,
y) of R
are of class D2
on Rand ~, x = x(~, 'I}), y = y(~, 1]) being the inverse transformation,
(iii) in which the conclusions of Theorem 1 aPPly to the functions Hx, y),
t See the
author's paper, loco cit., pp. 696, 698.
1])
C. B. MORREY
142
[January
y), X(~, '1]), and y(~, 1J) and in which those oJ Theorem 2 also apply if
(R; p, q, r) and (~; 11", K, p) satisfy a condition D(L, do);and
(iv) in which the Junctions Hx, y) and 1J(x, y) satisfy
1J(X,
7]x
=-
(btx
+ C~y),
7]y
=
+ b~y
a~x
almost everywhere on R.
Proof. Let {an} -+a, {bn} -+b, {cn} -+c almost everywhere on R, the an, bn,
and
Cn
being analytic on R and satisfying
I an I, I bn I, I en I
< M,
Let Tn: ~=~n(X, y), 'Y}='Y}n(X, y) be the unique analytic transformations of R
into ~ which carry p, q, and r into 11", K, and p respectively and which satisfy
Let X=Xn(~, 'Y}), y =Yn(~, 1J) denote the inverses. These transformations satisfy
the conditions of Theorem 1 and hence the conclusions of Theorem 1 and
also of Theorem 2 if (R; p, q, r) and (~; 11", K, p) satisfy a condition D(L, do);
it is easily seen that the ratio of maximum to minimum magnification of T n
is :Si\n+O"n2-1)1/2, An= (an+cn)/2, and is therefore :S2M. Hence a subsequence {nk} of the integers {n} may be chosen so that {~llk}, {'Y}nk}, {xn,,},
and {Yllk} all converge uniformly on R and ~ to functions Hx, y), 1J(x, y),
x(~, '1]), and Y(~, 1J) respectively and x=x(~, 'Y}), y=y(~, '1]) is the inverse of
T: ~=Hx, y), 'Y}=1J(x, y). Clearly T is a 1-1 continuous transformation of R
into ~ in which p, q, and r correspond to 11", K, and p respectively.
Also, since the ratio of maximum to minimum magnification ::;;2M, we
have
ff2
R (~nx
If
"k
2
(Xn~
+
2
~nY
+ 'lJnx + fJlly)dxdy
::;;
- m(~) ,
4M
+
2
Xn1/
2
+ Yn~2' + Yn1/)d~d7]
~
4Mm(R),
2
2
so that it follows from Lemma 6, §1 that Hx, y), 'Y}(x, Y), x(~, 1J), and y(~, 1J)
~"are all of class D2 on Rand ~. Using the same lemma, we see that (iii) also
holds and that
o
<
IfR
[(7] x
~ lim
n....•inf
oo
+ b~;J; + C~y)2 + (7]y -
ff
R [(7]n;J;
a~:c -
+ bn~nx + Cn~ny)2 +
so that (iv) is demonstrated.
b~y)2]dxdy
(7]ny -
an~n'"
-
bn~ny)2]dxdy
=
0,
ELLIPTIC DIFFERENTIAL
1938]
143
EQUATIONS
4. Let (R; p, q, r), (Z; 1r, K, p), a, b, and c satisfy the hypotheses
y), '1] = '1] (x, y) be the transformation
derived in
of Theorem 3 and let T: ~=Hx,
that theorem. Then T enjoys the following further properties:
(i) sets of measure zero and hence measurable sets of Rand
Z correspond,
THEOREM
and
(~x'1]y- ~y'1]x) and
of measure
(x~Y1/- X1/Ye) are defined
(ii) if cf>(x, y) and tf;(~,
cf>[x(~,
. (~x'1]y- ~y'1]x)are summable
(iii) if cf>(x, y) and tf;(~,
<P
possibly
on a set
Z, respectively;
zero in Rand
A~ Z, the functions
~ 0 except
and
'1])
are summable
'1]),
on
y(~,
on measurable
(xeY1/-xllYe)
'1])].
T(D) and T-l(f).)
subsets
and tf;[Hx,
respectively,
DC Rand
y),
'1](x,
y)
J
and
:3 respectively,
then
[x(~, '1]), y(~, 7])] and tf; [Hx, y), 7](x, y)] are of class D2 on :3 and R respec'1])
are of class
D2
on Rand
tivelyand
cpe = cpxxe
lh =
almost
t/;xx;
+ cPyYe,
+ 1/;yYe,
+ cPYY1/'
= t/;xXf/ + t/;YYlI,
cPII= cP"x1/
1/;11
+ cP1/'YJx,cPy = cPE~y+ cPlI'Y]y,
1/;x = tf;E~Q;
+
1/;y = tf;E~lI+ tf;'1'Y]y,
cPx = rpE~x
t/;f/'YJ",
everywhere;
(iv) T is uniquely
determined
by
(i), (ii), and (iv) of Theorem
3.
Proof. Let 0 be an open set in R, let 0 =:L::tRi where the Ri are closed
non-overlapping rectangles on each of which 7](x, y) is absolutely continuous,
and let Si be the surface Si: ~=~(x, y), 7] ='1'J(x, y), (x, y)r.Ri. Clearly L(Si)
is merely the measure of the closed or open region -i or ~i into which Ri or Ri
is carried, '2;; being a rectifiable curve. From Lemma 7, §1, it follows that
Thli's ~x7]y- ~y'1]:v
~ 0, and if Q';; T( 0), then
men)
= I Ia (~:v'Y]y-
~y'YJx)dxdy.
It follows very easily that a measurable set E in R is carried into one in ~ and
144
C. B. MORREY
[January
The same proof establishes the fact that a measurable set
into a measurable set D in Rand that
m(D)
= I III (X~Y1j
-
L1
in ~ is carried
x'1y;)dEd'l].
Hence the rest of (i) follows easily and we have proved (ii) for the case
</>
=tf; = 1.
Suppose </>,for instance, is bounded. Let {<;Dn(X, y)} be a sequence of
step functions which are uniformly bounded and converge to <I> almost
everywhere. It is clear that the functions <;Dn[x(~, 'f}), y(~, 'f})]. (x~Y'1-X'1Y;)
and <;D[x(~, 'f}), y(~, 'f})]·(x;Y'l-X'1Y~)
are all summable and dominated by a
summable function, and, for each n, it is clear from the above that
If
D rPn(X,
=
y)dxdy
I
r
et>n[X(~, '1]), y(~,
JT(D)
'1]) ] (XJ;Y'1 -
x1jy~)d~d'l].
The result (ii) for <;D
follows by a passage to the limit, since <;Dn[x(~, 'f}), y(~, 'f}) ]
converges almost everywhere on ~ to <;D
[x(~, 'f}), y(~, 'f})] and the latter function is certainly measurable. If <;D
is merely summable, let CPl =<;Dwhere <I>~ 0,
c,Vl=O
elsewhere,c,V2=-<;Dwhere 1><0, and 1>2=0 elsewhere, let 4>l,N=<;Dl if
<1>1
~ N, <;Dl,N = N elsewhere, cP2,N =<1>2 if 1>2
N, 1>2,N = 0 elsewhere. Then it follows that the functions
s
form monotone non-decreasing sequences of non-negative summable functions converging almost everywhere to 1>1
. (x~y'1 - X1jYE) and cP2' (xJ;Y'I- X1JY~) respectively, their integrals remaining bounded. Hence (ii) follows.
To prove (iii), let cP(x, y) be of class D2 on R, let G be a subregion of R
for which G c R, and let be the corresponding subregion of ~(r c ~ clearly).
Now, let.d be a rectangle of along which x(~, ?J) and y(~, 'f}) are absolutely
continuous. Then, using Lemma 7, §1, we see that
r
f
-f
,C,"
et>[x(~,
Ll" cp[x(~,
r
'1]), ya,
'1]) ]d'l]
= f D" <jJ(x,
1]), y(~,
7]) ]d~
= -
f
y)d'l]
D' cp(x,
=
y)d~
If
D (et>x7]u -cpy'l]
= -
If
x)dxdy,
D (c,t>x~u -
rPy~x)dxdy.
Clearly these relations follow for any rectangle ~ of G so that cP is A .C.T.
as a function of ~ and 1] by Lemma 3, §1. Using the same lemma, it follows
that
If
n
a~
drp d~d'l]
=
If
0 iJcjJ.
a~ (~x'l]y
-
~y1]x)dxdy
=
If
0(CPx1]y -
cPy'Y]x)dxdy,
ELLIPTIC
1938]
If
n
o<jJ
Or] d~dr]
=
If
0
DIFFERENTIAL
O<jJ'(~xr]y
Or]
-
145
EQUATIONS
=
~yr]x)dxdy
If (-
<jJx~y
0
+
<jJy~x)dxdy.
for every open set n in ~ with n c~. Hence, almost everywhere in R and ~,
1>xr]y
-
1>yr]x
1>x~y
+
1>y~x
-
Since
~x1]y -
= 1>;' (~xYJy
= 1>",' (~x7]y
-
~yYJx),
-
~y7]x).
except on a set of measure zero, it follows that
~y1]x~O
almost everywhere on R and ~. Setting 1>=x and
relations
+
x~~x
X",7]x
= 1,
X~~y
+
X",7]y
= 0,
y~~x
+
y
in turn, we obtain the
= 0,
y",7]x
y~~y
+
y",7]y
=1
almost everywhere, from which it follows that
almost everywhere on Rand l;; the relations for
follows that
(~xYJy
-
~yYJx)·
(x~y",
-
of
are proved similarly. It
= 1
X7JY~)
almost everywhere on Rand l;.
To show that 1>is of class D2 in (~, 1]) we see that
(1)~2
+ 1>",2),
(~x7]y
-
~yl],J
=
+
a1>x2
2b1>x1>v
+
;£
c1>v2
2M(1)}
+
1>y2) ,
using the relations above and the relations 1]x = - (b~x+c~y),
1]y = a~x+b~y,
ac - b2 = 1. The fact for of as a function of (x, y) can be obtained from the:
above and the fact that
ay;
x2
+ 2 by;
xY; y
+ ey;
1
y2
~
-=
('" + '"
2M
x2
y2)
•
To show (iv), let ~=e(x, y), 1]=1]'(x, y) be another map of R into lJ
satisfying (i), (ii), and (iv) of Theorem 3. Then i;' and 1]/ are of class D2 in ~
and and all the rules for differentiation apply. We see that 1]: = ~{ and
1]{ = -~: almost everywhere on l; so the map e=e(~, 'TJ), 'TJ'='TJ'(~,
'TJ) is
conformal by Lemma 4; Hence e=~, 1]'=1].
'TJ
3. A special elliptic system of partial differential equations
of the first
order. We prove nrst
THEOREM
that
1. Let D(x, y) and E(x, y) be oj class L2 on l;a = C(O, 0; a) such
146
C. B. MORREY
ff
C(P,p)
[January
1\
~ MpA,
E2dxdy
'~"
>
O.
Then the function
U(
x, y
) -
If (~~"
---------x)D(~, r])
(~ -
X)2
+
+ (r]
y)E(~,
y)2
(7] -
-
d
TJ)
~dr]
is defined and continuous over the whole space, and satisfies a uniform H iJUer
condition of the form
I
U(Xl,
Yl) -
U(X21
Y2)
I :::; NrA/2,
N
=
12(211'M)1/2[(2
r
=
1\)-1
-
[(X2
-
X1)2
+ 1\-1(1\ + 1)],
+ (Y2
-
Y1)2]1/2;
and
I
< 2(1 +
U(x, y)'
1\-1) (Z7l"M)
Proof. Define Dex, y) =0 for x2+y2>a2
U1(x,
y)
f f
+
(~-+
= If
=
00 _--.:...::(~
-00
(~_X)2
ex)
-ex)
~a (~
Let
hex, y; r)
= r1/2
x) 2
-
f2'11'
0
I D(x
__
1/2.
(x, Y)£~a'
(Za)A/2,
and let
x)_._
(r] D~
_ y)2 d~dr]
x) (7J·D-
y)
2
d~d'l'}.
+ r cos e, y + r sin eJ)·cos el
de.
Then
h2(x,
y; r)
= r [fa
the Holder inequality
smce
"By
27f
If
Oex,y:h)
D2d~d7J
=
2'"1
D cos e I deT
h2(x, Yi r)
211'
f f
0h
is summable on any interval O~r<h,
0 2'11'rD2drd8
~
211'
f
0 hr
> ~\2(X'
Thus
fa" h2(x, y; r)dr ~
27l"MhA•
[f
y;
0 2'11'D2 C052
r)dr.
Ode]
dr
ELLIPTIC DIFFERENTIAL
1938]
If we let H(x,
y; r)
= J:h(x,
we find by the HOlder inequality that
y; p)dp,
H(x, y; r) ~
If N is so large that
for r ~ N. Hence
If
•
Pf
0 211'
__
(~
-
~a
147
EQUATIONS
(21l"M)1/2rCHX}/2.
is in the circle C(x,
+
(_~_-_x)_D
X)2
(7] __
y)2 d~d7] I
= p-1/2H(x,
y; p) -
::; (1 + 'A-I)
(21l"M)
~
y; N),
f
E
then H(x,
Pr-1/2h(x,
y; e)
Cl/2H(x,
y; r) ~ (21l"MN
aX)1/2
y; r)dr
+ 21 JP
E
r-S/2H(x,
y; r)dr
I/2pX/2
for every e>0, and every p >0. Hence UI(x, y) is defined.
Now, let PI and P2 be any two points and let P be the midpoint of the
segment PIP2. Let p =PIP2 /2 and let a circle IJ(kp) of radius kp, k> 1, be described about Pas center. Let a be the inclination of the segment PIP2• Then
where
Thus
+ f -P [fro (k-I)p r-3/2h[x(s),
P
::; 2(1
+ 'A-1)(21l"M)1/2(k +
+ Jp_P -32 fro(k-l)p
.~ 6(27rM)1/2(1
(k
=
r-5/2H[x(s),
+ A-I +
2, x(s)
=
yes); r]drJds
1)'-/2pX/2
yes); r]drds
[2 ~ A]-1)pA/2
x
+ s cos a,
yes)
=
y
+ s sin a).
[January
C. B. MORREY
148
The result follows by using the similar result for U LEMMA
If
C(P,p)
1. Let D(x, y) and E(x, y) be of class
'~a
Then there exist sequences
on
~a
satisjying
If
;;2 M pt-,
D2dxdy
{Dn(x,
C(P,p)
y)}
'~a
Lz
EZdxdy
~'
U1•
on
~a
with
M
pA,
}.
and
{En(x,
y)}
lim
If
~a
[(Dn -
Proof. Define D(x, y) =E(x,
+ (En
D)Z
0,
of functions
the above condition with the sa;me M and
n-->oo
>
p
>
o.
I
-
/1..,
and such that
=
E)2]dxdy
analytic
O.
=0 for x2+y2;;:::a2, and let Dh and Eh be .
the usual average functions. Then Dh and Eh are continuous on l;a and
~~ I I~a[(Dh
y)
-
+ (Eh
D)2
-
= 0,
E)Z]dxdy
by Lemma 5, §1. Moreover for each h >0
the same being true for E. The lemma follows easily from this.
LEMMA 2. Suppose
,the condition
that Dn, En, D, and E are all oj class Lz on ~a:, satisfy
of Lemma 1, M and /I.. being independent of n, and the condition
If
lim
n~c.o
~lZ
[(Dn -
D)2
+ (En
-
E)Z]dxdy
=
O.
Then the junctions
n
U (
converge uniformly
x, y )
= -27f1
If (~~a
(~-
x)Dn
X)2
+
+ (7]('I] --
y)En
y)2
on any bounded plane region to the function
U(x,
y)
= -1
21T"
------d~dY).
If __(~+
z
11
(~-
x)D
X)2
('I]
('Y/ --
y)E
y)2
.;;
dl-d
7]
1938]
ELLIPTIC DIFFERENTIAL
149
EQUATIONS
Proof. Since the Un(x, y) are equicontinuous on any bounded set (in fact
the whole plane) it is sufficient to prove the convergence at each point
Po: (xo, Yo).
Choose € >0, and choose
! _1
271"
If
C(Po,Po)
Po
so small that
+
_(~_-_X_o)D_n_+_(_Y)
(~XO)2
(t) _-_y_o)_E_n
- YO)2 d~dt)
f
+ I~ f
271"
Then there exists an No such that for n
11271"
If
:3a-~a
~ _1_
271"po
S
(~-
-------------d~dY)
Xo)(Dn
D)
(~ - - XO)2
'C(Po,PO)
If
:3a
[(Dn
(2po)-1'71"-1/2·a·
D)2
-
xo)D
(E -
C(Po,Po)
I
-
a
yo)(En
YO)2
E)
-
[(Dn - D)2
+ (En
-
E)2]d~d'l]J/2<
This proves the lemma.
DEFINITION 1. Let D and E· be functions of class
and v are functions of class D2 on ~a, we define
K(u,
v)
J(u)
I
E)2]1/2dEdt)
f~
[f
yo)E
(t) -
> No,
+ (t)(t) --
+ (En
+
L2
on
~a'
+.
Then
if
u
= f f ~a..[vz(uz + D) + vy(uy + E)]dxdy,
=
ff
:3a
[(uz
+ D)2 + (uy + E)2]dxdy.
If the functions D and E have subscripts, we shall denote the integrals formed
by using the new functions by putting the same subscripts on J and K.
Remarks. It is clear that J(u+n
=J(u) +2K(u,
J(u)
~ 2D(u)
+ 2f f
~a
D(u)
<
+ 2f
f
:3a
2J(u)
S)
+D(S). Also
(D2
+ E2)dxdy,
(D2
+ E2)dxdy.
Accordingly, if H is the harmonic function which takes on the same boundary
values as u, we see that
150
C. B. MORREY
J(u)
< J(H)
D(u)
s 4D(H)
+2
~ 2D(H)
+6
ff
za
[January
ff
(D2
Za
(D2
+ E2)dxdy;
+ E2)dxdy.
2. Let D and E satisfy the conditions of Lemma 1, and let u*
be a continuous function defined on ~a*for which D(H*) is fini'te, H* being the
harmonic junction which takes on the given boundary values. Then there exists a
unique function u oj class D2 on ~a which takes on these boundary values and
minimizes J(u) among all such functions. The function u(x, y) is given by
THEOREM
U
u(x, y)
= H a(X,
ix,
= 1
y)
271"
y)
+ U a(X,
If (~~a
(~_
xX)2
)D
J
y),
+
+ (r)(r] -_
y)E
y)2
d~dr],
Ha (x, y) being the harmonic function which takes on the boundary values u* - Ua.
Proof. Let {Dn} and {En} be sequences of functions, analytic on ~a and
satisfying the conditions of Lemma 2 with respect to D and E. Let U(x, y)
be any function of class Dz on ~a taking on the given boundary values and
let un(x, y) be the unique solution, for each n, of
which takes on the given boundary values; each Un is the minimizing function
for In(u). Then J n(U)~J(U),
JnCU) > In(Un). Thus J(U) 2:lim supn~ooJn(U",).
On the other hand, the functions un(x, y) are given by
un(x,
y)
= Ha,n(x,
Ua,n(x,
y)
=
1
271"
y)
+
Ua,n(X, y),
If (~~
a
,~
+
x)Dn'n,'
(r] -
y)En
'"
d~dr),
where Ua,,,,(x,y) tends uniformly to Ua(x, y) so that Ha.n(x, y) tends uniformly
to Ha(x, y) and hence un(x, y) tends uniformly to the above u(x, y). Since
D(un) is uniformly bounded, u(x, y) is of class D2 and J(u) ~lim infn~oJn(un)
by Lemma 6, §1. Hence u(x, y) minimizes feu).
Let r be of class D2 on fa and zero on ~a*' Then
f(u
Since u(x,
y)
+ }.~)= J(u) + 2}'K(u, n + )..2D(r).
gives J a minimum, the middle term must vanish. Thus
J(u
unless r=O.
I
+ ~")
= J(u) + D(s) > feu)
!I
I
I
1
ELLIPTIC DIFFERENTIAL
1938]
151
EQUATIONS
Remarks. If we let Uo,a(X, y) be the minimizing function which is zero
on 2;a*, we see that it is of class D2• If Dn and En are as in Lemma 2, it is
clear that UO,a,n converges uniformly to UO.a and D(uo,a,n) is uniformly
bounded.
of Lemma 1 and let u be
the minimizing function for J(u) which takes on given continuous boundary values for which J(u) is finite. Then there exists a unique function vex, y) which
is of class D2 on 2;a, vanishes at the origin, and satisfies
THEOREM
3. Let D and E satisfy the conditions
vz
=-
(uy
+ E),
= Uz + D
Vy
almost everywhere. Any other function Vex, y) which is of class D2 and satisfies
these relations, differs from v by a constant. The function vex, y) is given by
vex,
y)
= Ka(x,
=-'1 If
V x
21l'
aC,y)
y)
;!;
a
+ Va(x,
y),
(~,.
x)E'n,' -
(17 -
y)D
'n
dd
~17,
where Ka(x, y) is the conjugate of the Ha(x, y) of Theorem 2.
Proof. Choose {Dn} and {En} as in Lemma 2, let un(x, y) be the minimizing function for In(U) with the given boundary values, and let vn(x, y) satisfy
Then
Vnex, y)
Va.n(x,
y)
= Ka.nex,
=
1
-211"
y)
+
Va,n(x,
If (~-
x)En
;O;a
-e~---x-)-2
y),
- C7] - y)Dn d~d17'
+-C-7]-_-y)-2-
It is seen by well known methods of differentiating
Va,n that
Va.n,y
the functions
=
Ua.n.a:
Ua,n
and
+ Dn
on ~a' Thus it is clear that Ka.n(x, y) is the conjugate of Ha,n(x, y) for each n.
Since we saw that Ha,n(x, y) converged uniformly to Ha(x, y) in Theorem 2,
and since Va,n(X, y) obviously tends uniformly to Va(x, y), it is clear that
Ka,n(x, y) tends uniformly to Ka(x, y) and vn(x, y) tends uniformly to the
above vex, y) on each closed subregion of ~a' Also D(vn) =J(un) and hence
D(vn) is uniformly bounded.
Now, let R be a closed subregion of ~a' Then vex, y) is of class D2 on R
with D(v) ~lim infn-.""D(vn) which are uniformly bounded. Hence, by Lemma
6, §1,
c. B.
152
o ;$
fIR
[(v:.
~ lim
n->eoinf
+
If
+ E)2 +
Uy
(Vy
MORREY
Ux
-
[January
D)2]dxdy
-
+ Uny + En)2 + (Vny
R [(Vnx
-
-
Unx
= o.
Dn)2]dxdy
Hence v(x, y) satisfies the desired relations almost everywhere. Now if V is of
class D2 and satisfies the same relations almost everywhere, V,;-Vx = Vy -Vy
=0 almost everywhere so that V -v is a constant.
4. If D and E satisfy the conditions of Lemma 1 and u(x, y) and
v(x, y) are of class D2 on 1;a and satisfy
THEOREM
Vx
=-
+ E),
(uy
Vy
=
Ux
+D
almost everywhere, then
y)
U(x,
= Ha(x,
+
y)
Ua(x,
y),
y)
VeX,
= KaCx,
where Ua and Va have their previous significance and
harmonic functions.
+ VaCx,
y)
and
Ha
y),
are conjugate
Ka
Proof. Let b < a so that v is absolutely continuous on "'2:, this being true
for all values of b < a excepting those in a certain set of measure zero (using
Lemma 2, §1). Then if r is of class D2 on hb, we have
If
.
Hence if
S
:;!;b
(r xVy
-
=
ryvx)dxdy
f
/dv
:;!;b
.
is also zero on 'h;;, we see that
Thus u is the minimizing function for J(u) on
is its "conjugate" as in Theorem 3. Hence
u(x, y)
= Ht
(x, y)
+ -1
27r
=
vex,
y)
Hb(x,
= K;
y)
(x, y)
27f
= Kb(x,
y)
with J(u) finite, and vex, y)
If --------d~d'l}
(~- ++
~b (~-
+ Ua(x,
+ -1
1;b
x)D
X)2
y)E
y)2
.
y),
If (~-
+ Va(x,
('I} .:(71 -
1:b
(~-
xX)2
)E
+
.
cC
7J
-
(17 -
y)D
y)2
d~d7J
y).
Clearly Hb and Kb are independent of b, and, since HI, and Kilare
harmonic functions, it is easily seen that Bb and Kb are.
conjugate
1938]
ELLIPTIC DIFFERENTIAL
153
EQUATIONS
4. A more general linear elliptic pair of partial differential equations. In
this section, we consider the pair of equations
(1)
Vx
= -
(bzux
+ CUy + e),
where we assume that a, bl,
Jordan region R with
(2)
lal,
Ibll,
Ibzl,
Icl,
[dl,
bz,
lel
Vy
C, d,
=
aU:rJ
+ bluy + d,
and e are measurable on a bounded
~M,
4ac-(bl+b2)2~m>O,
a>O.
1. Let R be a bounded Jordan region and H* a function continuous on R and harmonic on R for which D(H*) is finite, and suppose d = e = 0
on R. Then there exists a unique pair of functions u(x, y) and vex, y)
(i) which are of class Dz on Rand R respectively,
(ii) which satisfy (1) almost everywhere on R, and
(iii) for which u(x, y) =H*(x, y) on R* and v(xo, Yo) =0. These functions
also satisfy
(iv) conditions A [2~, M'(a, 5)] and B [A, N'(a, 15)] on R where ~ depends
only on M and m, and M'(a, 5), N'(a, 15) depend only on M, m, R, and the
maximum of I H*/ on R, and the maximum of I u\ depends only on the bound
for I H*\ on R*.
THEOREM
Proof. Approximate to a, bI, bz, C, d, and e by sequences
{ bz.n},
{cn},
of functions analytic on R for which
{an},
{bl,n},
and approximate to H* by harmonic functions Hn* for which D(Hn*) ~G and
such that the solution Un of
a
(3)
-
ax
a
(anunx
+ b1.nuny) + -
(b2,nUnx
ay
+ cnuny)
=
0
which coincides with Hn* on R* is analytic on Rt for each n; M, fii, and G are
independent of n. For each n, there exists a unique function vn(x, y), with
vn(xo, Yo) =0, which satisfies
Vnx
Vny
= - (bz,nunx + Cnuny),
= anunx + bl,nUny.
Define An(x, y), Bn(x, y), Cn(x, y) by
t This can be done by taking a sequence of regions Rn, each bounded by analytic curves, which
closes down on R and then assigning proper analytic boundary values on Rn*.That the solutions of (3)
exist under these conditions follows from the results stated in the article by L. Lichtenstein on the
theory of elliptic partial differential equations in the Encyklopadie der Mathematischen Wissenschaften, val. II 32, pp. 1280-1334.
154
C. B. MORREY
(5)
+ 2anbl.nUnxUny + (1 + bl.n)uny
An = ------------------,
anU~:/; + (bl,n + b2,n)UnxUny + Cn~~y
2
2
anb2.nUnx + (anCn + b1.nb2•n - 1)Un<lOUny+ bl.nCnUny
Bn=------------------------,
an.u2 x + (bl•n + b2.n)UnXUny + CnUny
(1 + b2.n)Unx + 2b2.nCnUnxUny + CnUny
Cn =
2
2
anUnx + (bl•n + b2,n)UnXUny + CnUnll
2
2
.
[January
2
2
anUnx
2
2
2
2
where these expressions are defined; otherwise, let An=Cn=l, Bn=O. It is
easily verified that An, Bn, Cn are measurable for each n and that
AnCn
(6)
l·(Ux
2
-
Bn2
+ Uy)
2
= 1, I An I, I Bn j, I Cn I < K(M,
2
< AnUx 2 + 2BnUxUy + CnUny::;
m) ,
L·(U:/;
2
+ Uy),
2.,
O<l5,L<oo,
where K, l, and L depend only on M and m (U is any function of class D2
on R).
Let p, q, and r be three distinct points on R* and 7r, K, and p be three distinct points arranged in the same order on ~*, the boundary of the unit
circle. Map R on ~ by functions ~=~n(x, y), "I]=7]n(X, y) where p, q, and r
correspond respectively to 7r, K, and p and where
(7)
Using the relations (4), (5), and (7) and Theorem 4, §2 we find that Un(~, 7])
and vn(~' 7]) are of class D2 on ~ and satisfy Vn~ = -UnT/, VnT/ =Un~ almost everywhere. They are therefore conjugate harmonic functions (in case Un:/; = Un1/ = 0,
it is clear that Vn:/;=Vnll=0
at almost all of these points; hence at almost all
corresponding poin ts, Vn~ = VnT/ = Un~ = UnT/ = 0; this takes care of points for
which An=Cn=l, Bn=O}. Since the transformations (7) are equicontinuou5
both ways, a subsequence {nk}, of the integers may be chosen so that the
corresponding transformations and their inverses converge uniformly to a
certain 1-1 continuous transformation of R into ~ and its inverse. Thus the
sequences {Unk(~' 7])} and hence {unk(x, y)} converge uniformly to certain
functions u(~, 7]) and u(x, y) respectively, u(x, y) coinciding with H* on R*
and u(~, 7]) being harmonic; and the functions Vnk(~' 7]) converge uniformly
on each closed subregion of ~ to v(~, 7]), the conjugate of u(~, 7]), and so
t It maybe shown that u=u,,(x, y), v=v,,(x, y) carriesR into a region on a finite sheeted Riemann
surface. The transformation (7) merely amounts to mapping this Riemann region conformally on the
unit circle i5 in a (~, 7)) plane. We shall use this transformation in the proof of Theorem 2 where this
interpretation is not valid.
ELLIPTIC DIFFERENTIAL
1938]
Vnk(x,
y)
155
EQUATIONS
converge uniformly on each closed sub region of R to a certain func-
tion vex, y).
It is easily verified that (see the proof of Theorem 4, §2)
(8)
If
D (AnU:JJ2
+ 2BnU
(1;Uy
+ CnUy2)dxdy
=
If
~n
(U~2
+ Und~d1J
for any function U of class D2 (in either (x, y) or (~, 'Y1», D and ~n being corresponding regions of R and ~. Since each Un is harmonic, it follows that
If
2
~ (Un~
2
+ Un'l)d~d1J
<
If ~
(Hn~*2
*2
+ Hn'f/)d~d7]
~
L·G.,
where L is the constant in (6) (using relations (6) and (8». Hence, by (6)
and (8), D(un) ~L·G/l (independent of n). Thus u(x, y) and vex, y) are of
class D2 on Rand R respectively. Furthermore, by Lemma 6, §1, it follows
easily that
I IR
[(v
(1;
+ b2U(1; + CUy)2 +
(Vy
-
aU:JJ -
b1uy)2]dxdy
= o.
Thus the existence and properties (i), (H), and (iii) are established.
To show (iv), define A, E, and C by (5) in terms of a, b1, b2, c, and u,
and perform the transformation (7). The relations (6) hold with M and m
replacing M and rn, and u(x, y), vex, y) are carried into conjugate harmonic
functions. By Theorem 4, §2, the functions ~(x, y) and 'Y1(x, y) satisfy conditions A [2A, M(a)] and E [A, N(a)] where A depends only on M and m and
M(a) and N(a) depend only on R, (p, q, r), (1r, K, p), M, and m. Thus it is
easy to see how to complete the proof of (iv).
Now, suppose U, V is another pair of functions obeying the conclusions
(i), (ii), and (iii). Then U and V, where U = U -u, V = V -v, satisfy these
conditions with H*=O. By defining A, E, and C by (5) in terms of U, a, b1, b2,
and c, and performing the transformation (7), we see that U(~, 'Y1) and V(~, 'Y1)
are conjugate harmonic functions for which U(~, 'Y1) =0 on ~* and V(~o, 'Y10)
=0. Thus U=V=O.
2. Let R be a Jordan region and p, q, and r be distinct points
on R*; we assume that (R; p, q, r) satisfies a D(A, do) condition. Let H* be continuous on R and harmonic on R with D(H*) finite. Suppose a, b1, b2, c, d, and e
satisfy (2) where d and e are not necessarily zero. Then the conclusions of Theorem 1 hold except that (iv) is replaced by
(iv)a u and v satisfy a condition E [A, N'(a, 0)] on R and I ul ~p on R,
t,
THEOREM
(ivh
J ID (u",2 + uy2)dxdy
~ N
156
C. B. MORREY
[January
for each closed subregion 15 of R; here A depends only on M, m, A, and do,
N'(a, 0) and P depend only on these, on (R; p, q, r; 7r, K, p), and on the maximum
of I H* I on R*, and N depends only on these and on the distance of 15from R*.
Proof. Approximate as in Theorem 1 to a, bl, b2, C, d, and e by sequences
{bl.n},
{b2,n},
{en},
{dn},
and {en} and to H*
of analytic functions {an},
by harmonic functions Hn* in such a way that if Un is the solution of
a
-
a
+ bl,nUny + dn) + -
(anunx
ax
(b2•nunx
ay
+
CnUny
+
=.0
en)
which coincides on R* with H n*, then Un is analytic on Rand D(H 11.*) ;f: C,
independent of n. Define An, En, and en by (5) and perform the transformation (7). We find as in Theorem 1 that Un and Vn are of class D2 in (~, 1']) on ~
and satisfy
(9)
almost everywhere, where
(10)
Also
AnCn 1· (U,,2
Bn2
=
+ Uy2)
1, 'An
~ AnUx2
I, I Bn I, I en' ~
K.(M,
+ 2BnUxUy + CnUy2
in),
+ UJ),
~ L·(Ux2
O<l~L<co,
(11)
\;,
ff
C(P,p)
.2:;
(Dn2
+ En2 )d~dTJ
~
N l(M,
in, A,
do) . p2\
A
>
0,
A
= A(M,
rn, A),
where U stands for any function of class D2, and D and ~n denote corresponding regions of R and ~ respectively, and K, Nl and A depend only on the
quantities indicated, and land L depend only on M and rn. These results
are obtained by straightforward computation, the use of Theorems 3 and 4
of §2, and the relations of Theorem 1.
Thus
(12)
where UO,n has the significance of §4 and Hn(~, 1']) is the function harmonic
on :3, and taking on the boundary values of Hn*(~, 1']). Now, as in Theorem 1,
we see that D(un) ~L· G/l and that a subsequence {unk} converges uniformly
on R to a function u(x, y) and {vn,J converges to a function vex, y) uniformly
ELLIPTIC DIFFERENTIAL
1938]
157
EQUATIONS
on each closed subregion of R. Thus u and v are of class D2 and, as in Theorem
1, are seen to satisfy (1) almost everywhere. It is clear that (i), (ii), and (iii)
are fulfilled. That the pair (u, v) is unique follows from Theorem 1, since if
u' and v' were another pair with the same boundary values, u' -u and v'-v
would satisfy (1) with d = e = 0 and obey the conclusions (i), Cii),and (iii) of
Theorem 1 with u'-u=O
on R* and v'(xo, yo)-v(xo, yo)=O. That lunl ~p
which depends only on M, rn, A, do, and the maximum of I H* I on R* follows
from Theorems 1 and 2 of §3.
Using Theorem 3, §2, and (12), we see that (iv) holds except that M
and rn intervene instead of M and m. Using (11), we see that
If
D (u:
+ u:)dxdy
~ lirn
_00inf
~ -
If
If
If
D (u~x
lim inf
1
n->oo
l
= -.1l lim
(Anunx
2
D
inf
n-4CQ
+ U~y)dxdy
J1.n
(Un~
2
+ 2BnUnxUny + Cnuny)dxdy
2
+ unfl)d~d1]~
2
N,
where N depends on the quantities indicated in the theorem except that M
and rn intervene instead of M and m; this is true since
f f~
(u~,o~
+ u~,o'1)d~drJ::: f f~
(D: + E:)d~drJ
and regions D at a distance ;::: from R* are carried into regions ~n at a distance ~m(a) from ~*, where mea) depends only on M, rn, and (R; p, q, r;
1r, K, p) and in such a region I Hn~1 and I Hnfl! ~2hn1r-l[m(o)]-lwhere
hn denotes the maximum of I H n* I on R*. To get rid of M and rn, merely define
A, B, C in terms of a, bl, b2, c, and u and perform (7); all the conclusions then
hold with M and m replacing M and rn.
(5
THEOREM 3. Let {an}, {bl,n}, {b2,n}, {cn}, {dn}, and {en} be sequences
of measurable functions which converge to a, bl, b2, c, d, and e respectively almost
everywhere and which satisfy
I
ani,
Fo..r each n, let Un and
I
Vn
bl.ni,
I b2•nl,
I enl,
dnl,
be of class D2 on Rand
almost everywhere with I un\ ~G on R*(M,
Then
I
I
enl ~ M,
R respectively and satisfy
m, and G being independent
of n).
c. B.
158
MORREY
[January
(i) lal, Ibll, Ib21, Icl, Idj, lel ~M, 4ac-(bl+b2)2~m>O,
a>O,
(ii) {Un} and {vn} are uniformly bounded and satisfy uniform Holder conditions on each closed subregion D of R, which depend only on M, m, R, G, and
the distance 0 (D) of D from R*,
(iii) J D(Un:/J+u~y+v;:/J+v;y)dx'dy < P [M, m, G, R, o(D)], and
(iv) if the subsequences {unk}, {vnk} are chosen to converge uniformly on
each closed subregion D of R to functions u and v, then u and v satisfy (ii) and
(iii) and
f
almost everywhere on R. The same conclusions
each Un is of class D2 on R with I Un I ~ G.
hold if we merely assume
that
Proof. (i) is obvious and (ii) and (iii) have been proved in Theorem 2.
To prove (iv), let D be a closed subregion of R. The conclusions (i), (ii),
and (iii) hold for D and {unk} and {vnk} converge uniformly on D to u and v
which are of class D2 on D. Then, by Lemma 6, §1,
o s;
I ID [(Vx + b2u:/J + CUy
< lim
If
inf
n->oo
D [(Vnx
+
+
e)2
b2,nUnx
+
(Vy
-
au"
-
<P(X1,
YI)
-
c/J(X2,
yz) I
<Ni",
d)2dxdy
-
+ Cf/,uny+ en)Z
+ (Vny
anunz
-
THEOREM 4. If a, bl, bz, c, d, and e satisfy
I
bIuy
A.
> 0,
r
=
-
bl.nuny
hypotheses
[(X2
-
Xl)2
dn)Z]dxdy=O.
-
(2) and satisfy
+
(Y2
-
YI)2]
1/2,
y) standing for a, blJ b2, c, d, or e) on R, then if u(x, y) and vex, y) constitute a solution of (1) with I ul ~G, then u", Uy, Vx, and Vy satisfy a uniform
H b'lder condition with the same exponent "A on any subregion D of R where the
constant N' depends on M, m, N, "A, R, and the distance of D from R.
C1>(x,
t
5. Applications to the calculus of variations. In this section, we shall discuss the differentiability properties of a function z(x, y) which minimizes
(1)
f fR!(X,
y, z, p, q)dxdy
among all functions having the same boundary values. We shall assume thatf
is .continuous together with its first and second partial derivatives for all
values of (x, y, z, p, q) and that the second derivatives satisfy a uniform
Holder condition on each bounded portion of (x, y, z, p, q) space, with
t See E. Hopf,
loco
cit.
ELLIPTIC DIFFERENTIAL
1938]
EQUATIONS
fpp
159
> 0,
everywhere.
t If
z(x, y) is a solution of (1) on a region R, which solution satisfies
a uniform Lipschitz condition, then
LEMMA 1.
for almost all rectangles D in R.
Proof. Let r(x, y) be any function which satisfies a Lipschitz condition
on R and which vanishes on R*. Then it follows in the usual way that
Now let D:a;£x;£b,
</>(x,
y)
=
c;£y~d,
be any closed rectangle in R, and define
f
y,
a xfz[~,
z(~,
y), p(~, y), q(~, y) ]d~.
Then, if r is any function satisfying a uniform Lipschitz condition on Rand
zero outside and on D*, we see that
I ID
(S</>:r:
+ Sxfp + Syfq)dxdy
=
I IDsifp
-
</»
+ Syfq)dxdy = 0.
Thus, it follows from a theorem of A. Haart that
for almost all rectangles
3: of D, and
hence that
Since R may be written as the sum of a denumerable number of rectangles D,
the lemma follows.
t This lemma is equivalent to Haar's equations for a minimizing function in a double integral
problem, first stated and proved by him in the case that p and q are continuous. See A. Haar, Uber
die Variation der Doppelintegrale, Journal fur die Reine und Angewandte Mathematik, vol.149 (1919),
pp.1-18.
t A. Haar, Uber itas Plateausche Problem, Mathematische Annalen, vol. 97 (1926-27), pp. 124158, particularly pp. 146-151.
C. B. MORREY
160
THEOREM
[January
1. Ij z(x, y) satisfies the hypotheses of Lemma
1, then it is con-
tinuous together with its first and second partial derivatives and the latter satisfy
uniform Holder conditions on each closed subregion oj R. Ij f(x, y, z, p, q) is
analytic, z(x, y) is also.
Proof. Let h be a small rational number and let (xo, Yo) be an interior
point of R. We define the region Rh as the set of all points (Xl, Yl) of R such
Y=Yl if h>O or xl+h~X~Xl,
Y=Yl if
that (1) the segment xl~x~xl+h,
h < 0 lies in Rand (2) (Xl, Yl) may be joined to (Xo, Yo) by a curve, all of whose
points satisfy (1). If I hi -;;;.ho, Rh is a non-vacuous Jordan region (ho rational).
Let D be a closed sub region of Ri for I hi <hI, Rh contains D. Also, if His
any rectangle out of a certain set of "almost all" rectangles of D, we have
J
H*
+ h,
{jp[x
(2)
y, z(x
- Jq[x
=f
+ h, y), p(x + h, y), q(x+h,
+ h, y, z(x + h, y), p(x + h,
+ h,
fHJz[x
y, z(x
+ h,
y), p(x
+ h,
y) ]dy
y), q(x
y), q(x
+ h,
+ h,
y)]dx}
y) ]dxdy,
fR' {jp[x, y, z(x, y), p(x, y), q(x, y) ]dy
- Jq[x, y, z(x, y), p(x,
(3)
=
f
fHJz[x,
y, z(x, y), p(x,
y), q(x, y) ]dx}
y), q(x, y) ]dxdy,
for all rational h with I hi <hI.
Let
Zh
= -h1
f
x x+h
z(~,
y)d(
Then
z(x
,.
Ph(X,
y)
=
=
y) -
z(x, y)
h
p(x
ph:r-
+ h,
+ h, y)
h
'
-
qh(X,
y)
h
= 1 f:r-+h
'"
p(x, y)
q(x
phy
=
+
. y)d~,
q(~,
h, y) h
q(x, y)
=
qhx,
almost everywhere Cl hi <hI). Clearly Ph satisfies a Lipschitz condition on D
for each h with 0 < I hi < hI, and hence is of class D2• Also I Phi is bounded
independently of h.
Subtracting (3) from (2) and dividing by h, we find that
ELLIPTIC DIFFERENTIAL
1938]
f
(4)
H' {[ah(X,
y)Phx
+ bh(x,
+ dh(x, y) ]dy
y)phx + Ch(X,
y)Phy
-
[bh(x,
161
EQUATIONS
+ eh(x,
Y)PhY
y) ]dx}
=0
on almost all rectangles H of D (for each rational h with I hI <hI)' Here we
may take
ah(x,
y)
=
i
+
bh(x,
y)
=~
eh(x,
y)
=
dh(x,
y)
=
f
+ th,
Ifpp {x
+
t[p(x
y, z(x,
h, y) -
Ifpqdt,
y)
p(x,
+ t [z(x + h,
y)],
q(x,
=~
Ch(X, y)
y)h -
z(x,
y)
J 1fpxdt + ------z(x + h,
y)h -
z(x,
y)
fqxdt
1
o
- h1 f
x x+hfz[~,
+
t[q(x
z(x,
+
y)],
p(x,
h, y) -
q(x,
y)
y) ]}dt,
Ifqqdt,
+ ------z(x + h,
o
y)
y) -
f
f
0
fqzdt,
1
0
fpzdt
1
y, z(~, y), p(~,
y), q(~, y) ]d~,
where the arguments which are not indicated in bh, Ch, dh, and eh are all the
same as that in ah· Clearly I ahl, I bhl, I chi, I dhl, and I ehl are uniformly
bounded for I hi <hI and it can easily be shown as in the proof of Lemma 2,
§2, that
for all h with I hI <hI. Also from (4), it follows that, for each such h, there
exists a function Vh(X, y) which satisfies a Lipschitz condition on R and which
satisfies
Vh(XO, Yo)
=
0,
Vhx
= -
(bhUhx
+ ChUhy + eh),
Vhy
= ahUhx + bhUhy + dh
almost everywhere. Also, if {hn} -+0 (all rational with I hn I < hI) {ahn}, {bhn},
{dhn},
and {ehn} tend to fpp [x, y, z(x, y), p(x, y), q(x, y)], fpq, fqq,
{Chn},
jpx+fpz'
p-fz,
and fqx+fqz' p respectively. Hence, from Theorem 3, §4 and
the fact that Phn tends to p almost everywhere, it follows that p(x, y) satisfies
a uniform Holder condition on each closed subregion Li of D. Similarly it
may be shown that q(x, y) also satisfies the same type of condition.
If we choose a subsequence so {Vhn} -+v, then we have that
.",
V:v
= -
(jpqPX
+ fqqqx +
[jqX
+ fqzP
D,
Vy
= fpppx + fpqpy +
almost everywhere. From Theorem 4, §4 it follows that
px
(Jpx
+ fpzP
and
py
- fz]
satisfy
162
C. B. MORREY
[January
uniform Holder conditions on each subregion of R; the same statement holds
for qx and qy. This proves the theorem. The last statement has been shown
to hold by E. Hopft if it is known that p(x, y) and q(x, y) satisfy Holder
conditions which fact we proved above.
6. Applications to quasi-linear elliptic equations. In this section, we shall
consider the equations of the form
(1)
a(x,
(2) a(x, y, z, p, q)zu
+ 2b(x,
y)zu
+ 2b(x,
y, z, p, q)zxll
+ c(x,
Y)ZXY
+ c(x,
y)zYlI
y, z, p, q)zllY
= d(x, y),
= Ad(x, y,
z,
p, q),
in which we assume that the functions a, b, c, and d are defined and continuous for all v~lues of their arguments with ac - b2 = 1, a> 0, and that these
functions satisfy a uniform Holder condition in each bounded portion of the
space in which they are defined. Then it is knownt that there exists a solution of (1) which is defined in the unit circle ~ and vanishes on ~* and that
its second derivatives satisfy a uniform Holder condition on each closed subregion of ~. A more precise statement is given in Lemma 1 below.
Let z(x, y) be the solution of (1) which vanishes on ~*, let k be
the maximum of I dl /(a+c) on ~, and let l be the maximum oj (a+c) on~. Then
LEMMA 1.
k
I zl
I
::: -,
2
pi,
/ q/
:::;
120ZSk
on ~, and p and q satisjy unijorm Holder conditions on each closed subregion X
oj ~ which depends only on k and l and the distance of ~ jrom ~*, and zxx, ZXll'
and z'1IY satisfy uniform H b'lderconditions on each closed subregion ~ of ~ which
depend only on the above and on the H iRder conditions satisfied by a, b, c, and d.
Proof. First, suppose' that a, b, c, and d are analytic on ~ so that z is
analytic on ~. It is known§ that if d1(x, y)~d(x, y) ~d2(X, y) on R and if Zl
and Z2 are the corresponding solutions of (1), then Zl ~ Z ~ Z2. Hence if we
choose
d1(x,
51
= k(l-
x2
y)
= -
+ c),
we see that k/2 ~z~ -k/2
see that
t Loc.
k(a
-
y2)/2,
Z2
d2(x,
on ~. Also, since
y)
Zl-Z~O
= - Zl,
= k(a + c),
and
Z-Z2~O
on ~, we
cit.
it is known (see Lichtenstein, loc, cit.) that the solution exists if a, b, c, and d are analytic
and the result follows from Theorem 4, §4 by approximations.
§ See for instance in S. Bernstein, loco cit., second paper.
t For
1938]
ELLIPTIC DIFFERENTIAL
aZ2
az
-=k>-?:.-k=ar
- ar -
on
163
EQUATIONS
aZ1
ar
~*. Since az/ae =0 on ~*,it follows that
I pi, I q I ::: k on
Now
p
~*.
and q satisfy the equations
apz
If we set u=
+ 2bpy + cqy = d,
-p, v=q,
(3)
and v=p, u=q, the equations become
a
Vx
= -
Uy,
Vz
=-
(2ba
(4)
2b
vy=-u{J;+-uy+-,
c
c
Ux
a
+ .:.-
Uy
-
d
c
a
.!..-),
respectively, each of which is a system of the type treated in §4. Thus, by
Theorem 3, §4, we see that p and q are bounded and satisfy uniform HOlder
conditions as desired in the lemma, and from Theorem 4, §4, we see that
Zzz, Zxy, and Zyy satisfy the desired conditions; these conditions hold whether
a, h, c, and d are analytic or not.
To see what the bounds are for p and q, let R be the unit circle and let
'ir, K, P and p, q, r be three equally spaced points on ~* and R* respectively.
Clearly (~; 1(', K, p) and (R; p, q, r) satisfy D(L, do) conditions where
L=47r·3-s/2 and do=31/2. In making the transformation (7) of §4, we see
that the ratio of the maximum to the minimum magnification at each point
is given by ,u+(,u2-1)1/2 where
p,=
<
A+C
2
2
Ca
+
2
a
2b
-u,?
+ -C
C
UiIJUy
+ uy2
C)2
-
1
since a+c?:.2. Thus the K of Theorem 2, §2 is l2_2, and hence
ff
C(Po.r).R
(X~2
+ XT/2 + y~2 + y,,nd~d'TJ
<
27r(l2 -
2) (3-1/2.
2r)2/[(l2_2)(L+1)].
Since, on R, D = Cd/ C)YfI' E = - (d/ c)y; in (3) and D = (d/ a)xfI' E
in (4) (using relations (10) of §4), it follows that
=-
(d/
a) x;
164
C. B. MORREY
If
If
C(P,p).R
(D2
+ E2)d~d71;:£
(D2
+ E2)d~d71;:£
max
_47r_(_l2
2_)._d2
31/2
c2 p2",
max
, ,_.p2\
47r([2
- 2) a2
d2
31/2
(x,y)on~
(x,y).~
C(P,p),R
(January
A
= ------,
1
(l2-2)(L+l)2
in cases (3) and (4) respectively. Let us call the first 'Yp2A and the second O'.p2A.
Referring to Theorem 1, §3 and using the notations of Theorems 2 and 3
and the remark, we see that
in cases (3) and (4) respectively so that
I UQ,lj,
I VQ,11
~
8(1
+ [2A]-I)
27r
(1)1/2
' I
I
UQ,II,
vo,11
~
8(1
+ [2A]-1)
27r
(a)1/2
in (3) and (4) respectively, so that, :finally
I u I, I v I
S; 8 ( 1
+ 2A1)(1)1/227r + k,
I u j,
j
v
I ~ 8(
1
+ 2A1)(a)1/227r + k,
in (3) and (4) respectively. Since I u!, I v I are merely I pI, I q I in one order or
the other, and since 21 dl /(a+c) is between I dl / c and I dl / a, we may substitute 4k2 for d2/ a2 and d2/ c2 so that we obtain
I pI, I ql
8[1 + ~(12
~
-
2)·(47r·3-S/2
+ 1)2]2-1/2·3-1f4·[.2k
~
120ZSk
since l~ 1.
1. A function z*(x, y) is said to satisfy a three point condition
with constant Ll on ~* if for each plane z = ax+by+c which passes through
three points of the curve z =z*(x, y), (x, y) on ~*, we have a2+b2 ~ .:l2.
DEFINITION
Let z*(x, y) satisfy a three point condition with constant Ll on ~*,
and let z'be the solution of (1) which takes on these boundary values, d(x, y) being
assumed to be identically zero on ~. Then
LEMMA 2.
"
(i)
(ii)
(x,
p2
min
y) on ];*
+ q2
z* ~ z(x, y) ~ (x,max
z*, (x, y)£~ in z(x, y), and
y).z·
~
A2.
Proof. This is well known. t
THEOREM
1. Let z*(x, y) satisfy a three point condition with constant
t See, for instance, J. Schauder,
elliPtische Di.ff~entialgleichungen,
.:l
on
Uber das Dirichletsche Problem im Grossen fur nichtlineare
Mathematische Zeitschrift, val. 37 (1933)1pp. 623-634.
1938]
ELLIPTIC DIFFERENTIAL
165
EQUATIONS
~* and let us assume that A=0 in equation
z(x, y) of (2) which coincides with z* on ~*.
(2). Then there exists a solution
Proof. Let M denote the maximum of I z* I on ~* and let l be the least
upper bound of a(x, y, z, p, q) +c(x, y, z, p, q) for all functions z(x, y) which
satisfy Lipschitz conditions and for which I z I ~ M, p2+q2 ~J~2.
Now, let zo(x, y) be the harmonic function taking on the given boundary
values and define functions Zn(X, y) for n 2:: 1 as the solutions of
which coincide with z* on ~*. At each stage
and an, bn, Cn satisfy Holder conditions which depend only on A, M, and l,
where an stands for a(x, y, Zn, Pn, qn), for instance. Since Pn and qn satisfy
the equations
and I pn I, I qn I ~A, it follows from §4 that pn and qn satisfy uniform Holder
conditions on each closed sub region J) of ~ which depend only on E, A, and
the distance of jj from ~* (and not on n). Thus, by Theorem 4, § 4, Zn,xx,
Zn,xy, and Zn,yy satisfy similar conditions, since an-l, bn-1, and Cn-l satisfy uniform Holder conditions on subregions of ~ which depend only on the above
quantities. Thus a subsequence {nk} may be chosen so that the sequence {Znk}
converges uniformly on 1; to a function z and so that the sequences {Znkox},
{ZnkoY}'
{Znk'XX}'
{Znk'XY}
, and {Znk'YY}
converge uniformly on each closed subregion of ~ to the corresponding derivatives of z. Clearly Z is the desired solution.
THEOREM 2. If I A I is sufficiently
small, the equation (2) possesses a solution
which vanishes on ~*.
Proof. Let l(z) and k(z) denote the least upper bounds of
a[x,
y, z(x,
y), p(x,
y), q(x,
d[x,
a[x,
y, z(x,
y), p(x,
y, z(x,
y)]
+ c[x,
y), p(x,
y), q(x, y)]
y, z(x,
y), q(x,
+ c[x,
y), p(x,
y), q(x,
y)],
y), q(x,
y)]
y)]
y, z(x,
y), p(x,
respectively. For all z with z2+p2+q2~a2,
l(z) ;i:l, k(z) ~k, Now if z(x, y) is a
function for which Z2+p2+q2 ~a2 and for which p(x, y), q(x, y) satisfy a uniform Holder condition on each closed subregion of :Z, the solution Z of
a(x,
y, z, p, q)Zxx
+
2b(x,
y, z, p,
q)ZXY
+
c(x,
y,
Z,
p, q)Zyy = t..d(x, y, z, p, q)
166
C. B. MORREY
which vanishes on ~* is such that Zzz, ZZy, Zyy satisfy Holder conditions on
each closed subregion [j of ~ which depend only on the Holder conditions
satisfied by a, b, c, 'Ad and z, p, and q, and Zz and Zy satisfy Holder conditionswhich depend only on a, k, l, and 'A. Also, by Lemma 1
(22
+ p2 + Q2)1/2
<
120.31/2.l3·k·1
AI
which is ~ a if 'A is small enough. The successive approximations
carried through as in Theorem 1.
may be
3. Let L(G) and K(G) denote the least upper bounds of l(z) and
k(z), respectively, for all z with Z2+p2+q2 ~G2. If, in addition to our hypotheses,
we assume that
THEOREM
. L3(G) . K(G)
hm----=O,
G
G-WJ
the equation (2) possesses a solution on ~ which vanishes on ~* for each value
of
'A.
Proof. For each.u>O, there exists a number
120·31/2·L3(G)·K(G)
Np.
such that
:::;;p,G
for all G > NI" Let M I' be the least upper bound of 120L3( G)K (G) .3112 for all
G ~ N p.' If we let Pp. be the larger of N p. and M-I. Mp., we see that
120l3(z)·
for all z for which Z2+p2+q2~Pl.
solution of
a(x, y, z, p, q)Zu
+ 2b(x,
k(z) . 31/2
~
p,Pp.
Thus, if z is such a function and Z is the
y, z, p, q)ZZY
+ c(x,
y, z, p, q)Zyy
=
Ad(x,
y, z, p, q)
which vanishes on ~*, then
(Z2
which is :::;;
P Jl if
1 and 2.
I 'A I ~
+ p2 + Q2)1/2<
I
AI·wpp.
M-I, The remainder of the proof proceeds as in Theorems
UNIVERSITY 0]' CALI]'ORNIA,
BERKELEY, CALIF.