A NNALES DE L ’I. H. P., SECTION C
J. I. E. U RBAS
Boundary regularity for solutions of the equation of prescribed Gauss curvature
Annales de l’I. H. P., section C, tome 8, n
o5 (1991), p. 499-522
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Boundary regularity for solutions of the equation
of prescribed Gauss curvature
J. I. E. URBAS
Centre for Mathematical Analysis, Australian National University,
GPO Box 4, Canberra 2601, Australia
Vol. 8, n° 5, 1991, p. 499-522. Analyse non linéaire
ABSTRACT. - We
study
theboundary regularity
of convex solutions of theequation
ofprescribed
Gauss curvature in a domain Q c ~n in the casethat the
gradient
of the solution is infinite on somerelatively
open,uniformly
convexportion
r of Under suitable conditions on the datawe show that near r x R the
graph
of u is a smoothhypersurface (as
asubmanifold of
1)
and that u~r
is smooth. Inparticular,
u is Holdercontinuous with
exponent 1 /2
near r.Key words : Gauss curvature, boundary regularity, free boundary, Legendre transform.
1. INTRODUCTION
In this paper we
study
theboundary regularity
of convex solutions ofthe
equation
ofprescribed
Gauss curvatureClassification A.M.S. : 35 J 60, 35 J 65.
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire - 0294-1449 Vol. 8/91/05/499/24/$4.40/© Gauthier-Villars
on a domain Q c in the case that
for some
relatively
open,uniformly
convexportion
r of aS2. Theboundary
condition
(1.2)
is to beinterpreted
asmeaning
for
all y
E r. This situation arises in a number ofproblems
connected withequation ( 1 . 1 ),
and infact, proving
that( 1. 2)
holds on a suitableportion
r of aS2 is a
key step
in theproof
of the interiorregularity
of solutions of( 1.1 )
in the papers[17], [18], [19].
Before
stating
ourresults,
we recall theproblems
in which the above situation arises. Let us suppose that Q is auniformly
convex domainin ~n and is a
positive
function. It iseasily
shown that thecondition
where ffin is the measure of the unit ball in
fR",
is necessary for the existence of a convex solution of(1.1).
In order to solve the Dirichletproblem
for( 1.1 )
witharbitrary boundary
data cp of class it is known from the papers[3], [15], [17], [19]
that two conditions are necessary.Namely,
werequire
to ensure the
validity
of an apriori
bound for the solution u, and in additionin order to obtain a
boundary gradient
estimate. As shown in[8],
thiscondition is not necessary if instead we
impose
some restrictions onI
1 ~a~~. If( 1. 5) holds,
but not( 1. 6),
then the Dirichletproblem
may not be solvable in the classical sense(see [15], [19]). However,
it is shown in[18]
that there is aunique
convex solution of( 1.1 )
which solves the Dirichletproblem
in a certainoptimal
sense: the solution u is the infimum of all convexsupersolutions
of( 1.1 ) lying
above cp on oralternatively,
u is the supremum of all convex subsolutions of
( 1 . 1 ) lying
below cp onThe Dirichlet
boundary
condition isgenerally
satisfied in the classicalsense on some
parts
of aS2 and not at otherparts.
Atpoints
at which it is not satisfied wehave |Du= oo, provided
K satisfies a mildgrowth
restric-tion near
aS2,
forexample
for some p > n. Infact,
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
is
sufficient, by
an examination of theproof
of Lemma 3.6 of[18] (see
also
[21],
Lemma2.1)
and Remark(v) following
theproof
of Lemma 2.2.In the extremal case
it is shown in
[17], [19]
that there is a convex solution of( 1. 1 )
which isunique
up to additiveconstants,
andfurthermore,
ifK e LP (Q)
for somep > n
(as
abovesuffices), then
on ao.At
present
theonly
theorems whichyield
any information about theregularity
of u near r are the Holder estimates of[19].
These are validmuch more
generally however,
and theirproof
makes no use of theboundary
condition(1.2).
It is reasonable therefore toexpect
that betterboundary regularity
can be obtainedby using ( 1. 2).
Our results are in fact valid for more
general
Gauss curvatureequations
of the form
where v denotes the downward unit normal to the
graph
of ugiven by
and K E
(03A9 R
X is apositive
function. Here Sn denotes the lowerhemisphere
of the n-dimensional unitsphere
Sn~Rn+1.
The situations described above forequation ( 1. 1 )
also arise forequation ( 1. 8)
if weimpose
a number ofhypotheses
on K(see [18],
Theorem1.1,
and[19],
Theorem
4.10).
Our main result is the
following.
Asusual, BR
denotes the open ball in (~n with radius R and centre 0.THEOREM 1. - Let SZ be a bounded domain in (~n and suppose that
for
some
x0~~03A9,
which we may take to be theorigin,
I-’ = aSZ~ BRO
is aconnected, C2 ~ 1, uniformly
convexportion of
aS2.Suppose
thatK E
C 1 ~ 1 (S2
x f~ x is apositive function
and u EC2 (SZ)
is a convex solutionof ( 1 . 8) satisfying ( 1 . 2)
and such thatand
where ~1, ~,2 are
positive
constants. Then there is a number p E(0, Ro), depending only
on n,Ro, h,
~,1 and ~2, such that thefollowing
hold:(i)
u EC°°1~2 (SZ n
and for any x, y E S2n BP
we have
Vol. 8, n° 5-1991.
where
C1 depends only
on n,Ro, r,
~,o, ~.l and ~2.(ii) graph
is ahypersurface for
any al,
and we havewhere
C2 depends only
on n,Ro, r,
~o, ~.2 and a.(iii )
u is of classfor
any a1,
and we havewhere
C3 depends only
on n,R0, r,
0, pi, 2, a and infu.q
(iv) If
in addition to the abovehypotheses
we haveand
for
someinteger k >__
2 and some a E(0,1 ),
thengraph (u hypersurface,
u isof
class and wehave ~
and
where
C4
andCs depend only
on n,Ro, r,
po, ~1, ~,2,K, k,
a and inf u.q
If
If the K and r areanalytic,
thengraph (u
and u|0393~B)
areanalytic.
hypotheses
of Theorem 1 aresatisfied
with r =aSZ,
then we obtainthe conclusions of Theorem 1
in {x ~03A9:
dist(x, ~03A9) 03C1}
for a suitablep > 0. We may then
apply
the interiorregularity
result[20],
Theorem1.1,
to extend the estimates to all of
SZ.
We state thisspecial
caseseparately.
THEOREM 2. -
Suppose
Q is aC2~1 uni.f’ormly
convex domain in(~n,
K E
(03A9
x R x is apositive function
and u is a convex solutionof (1 . 8)
such that(1 . 10)
and( 1. 11) hold,
andThen the assertions
(i)
to(iv) of
Theorem 1 hold with Q nBP replaced by
Q and r
by
aSZ. Thedependence of
the constantsC1,
...,Cs
onRo,
rshould now be
replaced by
Q.We shall prove Theorem 1 in the
remaining
sections of the paper. In Section 2 we shall derive somepreliminary
estimates forgraph
u near r.We also reformulate our
problem
as a freeboundary problem
for afunction w, obtained
by expressing graph
ulocally
as thegraph
of afunction w over a suitable subdomain of the
tangent hyperplane
tograph
u at
(0, u(0)).
In Section 3 we show how the information obtained in Section 2 can be used to deduceappropriate regularity
assertions for theLegendre
transform of w, from which the assertions of Theorem 1 follow.Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
As will become clear
later,
some of thehypotheses
of Theorems 1 and 2 may be weakenedslightly. First,
u may be assumed to be ageneralized
solution of
( 1. 8)
in the sense of Aleksandrov[2] (see
also[18])
rather thana classical solution.
However,
under the conditions of Theorem1,
u is aclassical solution on
Q n Bp
for some p E(0, Ro) depending only
on n,Ro, r,
0and 1 [see
Remark(ii ) following
theproof
of Lemma2.2]. Second,
the
boundary
condition(1.2)
may be weakened to a measure theoretic condition[see (2.20)]
which arisesquite naturally.
Infact,
it is this condition which is first shown to be satisfied in the papers[17], [18], [19];
the
stronger
condition( 1. 2)
is then deducedby arguments
similar to those used in theproof
of Lemma 2.2.Analogous boundary regularity questions
for solutions of the nonpara- metric least areaproblem
were studiedby
Simon[14].
His results wereimproved
and also extended to the mean curvatureequation
in situationscorresponding
to those mentioned above for the Gauss curvatureequation by
Lau and Lin[11] ]
and Lin[12], [13].
Ourapproach
is similar to[ 11 ], [12], [13]
in that we also reduce theproblem
to a freeboundary problem.
In our case,
however,
the freeboundary problem
is not apriori uniformly elliptic,
andconsiderably
morepreliminary
information isrequired
beforewe can
apply
standard results.It will be evident that the
techniques
of Section 2 are alsoapplicable
tocertain other
Monge-Ampère equations
of the formfor which the
boundary
condition( 1. 2)
arises in a similar fashion as for the Gauss curvatureequation, provided
weimpose
suitable conditions onthe function
f.
Inparticular, sufficiently
fastgrowth
off
withrespect
to Du is necessary.Furthermore,
in the case ofequation ( 1. 8)
we may also allow some unboundedness ordecay
to zero of K as xapproaches
r. Weshall make some remarks about these extensions later. In these
situations, however,
we obtain a freeboundary problem
which ingeneral
isnecessarily nonuniformly elliptic
ordegenerate,
and thetechniques
of Section 3 do notyield
any furtherregularity. Nevertheless,
even in these cases it isperhaps
reasonable toexpect
somewhat betterregularity
ofu Ir
than isprovided by
the Holder estimates of[19].
For
Monge-Ampere equations
of thegeneral
form( 1. 18)
theboundary
condition
( 1 . 2)
also arisesif f grows
fastenough
as xapproaches ao,
even without
imposing sufficiently rapid growth on f with respect
to Du.However,
we cannotexpect regularity
assertions foru Ir
such as those ofTheorem 1. For as
Cheng
and Yau[5]
haveshown, uniformly
convex domain in ~8n
(Q)
satisfies( 1.19) A dist (x, _, f ’ (x) _
B dist(x,
Vol. 8, n° 5-1991.
for some constants
A,
B >0,
a > 1 andP
n +1,
then the Dirichletproblem
has a
unique
convex solution u EC2 (SZ) (~ C° (SZ)
for anyboundary
data cp, and we
have |Du = o0
on aS2. Similar assertions are also valid forMonge-Ampere equations
of thegeneral
from(1.18),
wheresatisfies 0 and
for all
(x,
z, x R x[R",
whereA, B,
a,P,
y are constants such thatA, B > 0, cx> 1,
and(see [16],
Theorem2.19).
2. ESTIMATES FOR GRAPH u
The main results of this section are a Holder estimate for the normal vector field v to
graph
u and a strictconvexity
estimate forgraph
u, bothof these
being
valid at anypoint (xo, u (xo))
with Theseresults hold under weaker
regularity hypotheses
than stated in Theorem1;
namely,
we needonly
assume r to beuniformly
convex and of classand the bound
( 1.11 )
is not needed.We
begin
with somepreliminary
estimates for u. From[19], Corollary 3.11,
we have thefollowing preliminary regularity
result.LEMMA 2.1. - There is a number
a E (0,1 /2], depending only
on n, suchthat u E
C°~ °‘ (SZ (~
we havewhere C
depends only
on n,Ro,
r and ~.°.Remarks. - The estimate
(2.1)
does notdepend
on theboundary
condition
( 1. 2);
it usesonly
theregularity
of r and the lower bound of(1 . 10),
and isindependent
of anyboundary
condition. Theproof
of(2 .1 ) given
in[19] yields
theexponent cx = 1/2 n,
but this can be increasedslightly by
an iterationargument
described in[19].
Theoptimal exponent
is not known atpresent,
but for our purposes anypositive exponent
suffices.From Lemma 2.1 we see that
by replacing Ro by Ro/2
we may assumefrom the start that ue
(Q (1
It is also convenient toreplace
S2by
Q (~BRO.
This involves no loss ofgenerality
and has theadvantage
that Q is convex, which makes the
proof
of thefollowing
lemmatechnically simpler.
It is a refinement of theargument
used in[17], [18], [19]
to showthat Du|
= 00 on a suitableportion
of ~03A9.Annales de l’lnstitut Henri Poincare - Analyse non linéaire
LEMMA 2.2. - There are numbers El, C >
0, depending only
on n,Ro,
rand 1,
suchthat for
each x E~{
x E S2 : dist(x, ~03A9)
El1}
we haveProof. -
Choose coordinates so that thepositive
xn axispoints
in thedirection of the inner normal to aQ at 0. It suffices to prove
(2.2)
forx = (0, ~2) with ~~~1=~1(n, Ro, r,
because a similarargument,
withRo replaced by Ro/2
and~1/2
if necessary, willyield
the estimate at anypoint
of(~ { x E SZ : dist (x,
Since r is
uniformly
convex, there is anenclosing
ballBR (zo)
at 0 i. e.BR (zo) n
aS2== { 0 ~
with the radius R estimated from aboveby
a constantdepending only
on r. LetB2R (zl)
be anotherenclosing
ball at0,
and fors > 0 consider the set
where
x’ _ (xl,
..., It is clear that for ~>0sufficiently small,
sayE __
Eo = Eo(r),
we have dist(03BE0, ~03A9)
=E2
where03BE0
=(0, E2),
and further- more,letting E
=aB2R (z 1
+ço),
for some
positive
constants8, Al
andA2 depending only
on r andRo.
Here
(0)
denotes the open ball in[Rn-l
with radius R and centre 0.Let
Then
Lt
is aclosed,
convex,(n -1 )-dimensional hypersurface
inS2 and ~o
is enclosed
by Et.
LetGt
and G be thegeneralized
Gauss maps ofLt
andE
respectively.
Themeaning
of G isclear,
since E issmooth,
while foris defined to be the set of outward unit normals of
supporting hyperplanes
ofLt
atç,
and for any set E cLt,
.=
E2
+ E( ~
and letG
be the Gauss map of ~. Since eachEt
encloses~o, it
is evident that if T is any(n- I)-dimensional supporting plane
of at03BE0, T
can be translated in thenegative xn
direction to
yield
aparallel (n -1 )-dimensional supporting plane T~
ofEt
at some
point of Et
nLJ.
ThusVol. 8, n° 5-1991.
by
virtue of(2. 3)
and anelementary geometric
argument.To
proceed
further we need to define some set functions. LetSince Q is convex, M is a
complete,
convexhypersurface
inAs in
[18],
we define(2. 7) Xv ( y)
={ P E
asupporting hyperplane
For any set E c
Q
we defineA result
of Aleksandrov [1] (see
also[5])
asserts that ifA,
B aredisjoint
subsets of
Q,
thenXV (A) U XV (B)
has measure zero.Furthermore,
if E isa Borel set, then so are
x" (E),
x"(E)
andxv (E) (see [18]).
Continuing
now with theproof
of thelemma,
we see from(2. 5)
thatbecause each nonhorizontal n-dimensional
supporting plane
T of Myields
an
(n -1 )-dimensional supporting plane Tt
= T(~ ~
x : xn + 1=t ~
ofEt
foreach From
(2 . 8)
we haveHowever, by (2 . 4)
the second set on theright
hand side of(2.10)
iscontained in
xv (r)
and so isempty,
since on r. Thus from(2.9)
we obtainor
recalling
the definitionof v,
where
po = Du (xo)
and Xt. is defined as above with vreplaced by
u.Next, using
the fact that u is a convex solution of(1 . 8),
we haveAnnales de l’lnstitut Henri Poincaré - Analyse non linéaire
where h
(t)
=( 1
+t2) -
(n +2)/2 , Defining
for any
b > o, using (2 .12),
andestimating
the left hand side of(2 .13)
from
above,
we obtainfor suitable constants
C1
andC2.
It follows thatfor E small
enough,
sayRecalling
now thatPo = Du (ço)
and~o
=(0, ~2),
we see that we haveproved
since
E2
= dist(ço, lQ).
Theproof
of the lemma iscomplete.
Remarks. -
(i )
Theproof
of Lemma 2.2 does notrequire
u to be aclassical solution of
( 1. 8).
All werequire
is that u is ageneralized
solutionin the sense that
for any Borel set EcQ.
Furthermore,
if u is not differentiable at x, then in the statement of the lemma we needonly replace
Du(x) by
theslope
of any
supporting hyperplane
ofgraph
u at(x,
u(x)).
(ii)
If u is ageneralized
solution of( 1. 8)
and thehypotheses
ofTheorem 1 are
satisfied,
then u is in fact a classical solution onBp
for some
positive
p > 0. For if xo E Qn Bp,
then for p smallenough
wehave
Vol. 8, n° 5-1991.
where po is the
slope
of anysupporting hyperplane
T ofgraph
u at(xo, u (xo)).
We assert thatT n graph
u cannot contain a linesegment
with anendpoint
on SQ x R.Clearly
we cannot have such a linesegment
with anendpoint
on r xI~, by
virtue of theboundary
condition( 1. 2),
while if there were such a
segment
L with anendpoint
on(an - r) x IR,
thenby
the uniformconvexity
of r there would be apoint (xl,
u(xl))
E Lwith
dist (xl, bo
for somepositive
constant~o depending only
onRo
and r. But
then,
since T is also asupporting hyperplane
ofgraph
u atu (x 1 )),
we havesince u is convex. This contradicts
(2.18)
if p is smallenough,
so ourassertion is
proved.
TheC~ regularity
of u on Qn Bp
now follows in astandard way, see for
example [5], [20].
(iii)
In theproof
of Lemma 2 . 2 theboundary
condition( 1. 2)
was usedonly
to assert that the setx,* (0 ~ ~03A9)
in(2 . 10)
isempty.
All that isrequired
for theproof
is that this set have measure zero. Thus theboundary
condition( 1. 2)
could bereplaced by
the weaker measure theor-etic condition
(iv)
Refinements of Lemma 2. 2 arepossible.
Forexample
if we havewhere
K E Lq (Q)
for someq > (n + 1 )/2,
then inplace
of(2 .14)
we obtainwhich leads to the estimate
Similarly,
iffor some
constant ~3 > -1,
we obtain(v)
From theproof
of Lemma 2 . 2 and(iv)
it is evident that the interiorregularity
result of[18],
Theorem1.1,
is valid under theslightly
weakergrowth
restriction withq >_ (n + 1)/2
rather than q > n. One cansimilarly improve
the interiorregularity
assertions of[19],
Theorems 4. 8 and4 . 10,
and[21],
Theorems 1 and2,
in the case thatImpro-
vements
along
these lines arepossible
even if~03A9~ C1°1,
but we shall notpursue these
questions
here.Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
(vi)
Minor modifications of theproof
of Lemma2 . 2, specifically
in(2.13)
and(2. 14),
lead to similar results for moregeneral Monge-Ampere equations,
forexample, equations
of the form( 1.18)
wherefor all where
(S~)
and cx >n is a constant.Arguing
as above we obtainFurther modifications
along
the lines of(iv)
are alsopossible,
but we shallnot state these.
Since
r E C 1,1,
the distance function d definedby
is of class 1 near r
n
say for x Ed (x) 1 ~
forsome
03C31>0 depending only
on r(see [6],
Lemma14.16). By replacing
Si 1 from Lemma 2. 2by
a smaller constant if necessary, we may assume that03C31. We may extend p, the outer unit normal vectorfield to
T,
toBRo/2 n { x
E Q :d (x) 61 ~ by setting
The
tangential gradient
u, ..., of u relative to r is definedby
for all
points
of Q :d (x)
~1 ~ .
Using
Lemmas 2.1 and 2. 2 we now obtain thefollowing.
LEMMA 2 . 3. - There is a number
/3 > o, depending only
on n, such thaton
n ~
x E o :d(x)
El~,
where Cdepends only
on n,Ro, F,
po and ~1.Proof. -
From[19],
Lemma3.8,
and itsrefinement
to the caseu E
(SZ) (see [19], equation (3 . 37)),
we havefor all
n ~ x E SZ : d (x) E 1 ~,
where a > 0 is theexponent given by
Lemma 2.1 and C
depends only
on n,Ro, r,
po and pi. The estimate(2 . 28)
now followsby combining (2 . 29)
withLemma 2. 2.
Using
Lemmas 2. 1 to 2.3 we can obtain a Holder estimate for the normal vectorfield tograph
u and a strictconvexity
estimate forgraph
uat any
point (xo,
u(xo))
with xo First ofall,
we observe thatsince I Du =
00 on r and theonly supporting hyperplane
ofVol. 8, n° 5-1991.
at a
point (xo,
u(xo))
E r x (~ is a verticalhyperplane tangent
to r x R.Since M is a convex
hypersurface,
it follows that the unit normal vector-field to M is continuous at any
point (xo,
u(xo))
E M with xo E r. Thus the unit normal vectorfield tograph
ugiven by (1.9)
has a continuousextension
(which
we also denoteby v)
tograph u n (r
xR),
andLEMMA 2. 4. - There is a number y >
0, depending only
on n, such thatfor
allXo Ern BRo/2
and x~03A9~ BRo/2’
where Cdepends only
on n,Ro, I-’, oandl.
Proof. -
Itclearly
suffices to prove(2 . 31 )
forxo = 0
andfor other xo it can be obtained
by
a similarargument,
while the caseis
trivial.Choose coordinates so that the
positive
xn axispoints
in the direction of the inner normal to r at 0. Then v = ~, onr,
and inparticular, v (o) = - en
where ..., en + 1 denote the unit coordinate vectors in~n + 1.
then at x we have
for some
by
virtue of Lemmas 2 . 2 and 2 . 3. Infact,
we may take y =P/2 (1
+P), where P
is the constant of Lemma 2. 3.Next,
if x is anypoint
ofBRo/2 n ~ x E S2 : d (x) ~ 1 ~,
the sameargument
as above shows that
where x
is thepoint
of r nearest to x. It follows thatAnnales de l’lnstitut Henri Poincaré - Analyse non linéaire
since v = ~. on
r,
r is of class andThe lemma is
proved.
Next we come to the strict
convexity
estimate.LEMMA
2. 5. -
There are numberssuch that
if
T is theunique
tangenthyperplane
tograph
u atXo
=(xo,
u(xo))
where xo~0393~BR0/2,
thenfor
anypoint
X =
(x,
u(x))
E(graph u) n (xo)
xR)
we havewhere II is the
orthogonal projection
onto T andCo depends only
on n,Ro, r,
~oand 1 .
Proof. -
As usual it suffices to prove this for the other casesbeing
obtainedby
a similarargument.
We suppose for convenience thatu (0) = 0,
and let E2,Ci,
...,C10
denote variouspositive
constantsdepend- ing
on some or all of thequantities
n,Ro, r,
~,o and ~,1.By
Lemma 2. 4 there is a numberE2 > 0
such that(graph u) n (BE2 (0)
xIR)
can be written as thegraph
of a function wdefined on a suitable subdomain U of T obtained
by projecting (graph u) (~ (BE2
xR) orthogonally
onto T. Infact,
if we choose newcoordinates ya
= xa for a n, yn = - xn + 1and yn
+ 1= xn, thenby
Lemma 2. 4we have
for some R > 0
depending only
on n,Ro, r,
~o and ~1, wheregraph m c T
is the
orthogonal projection
ofgraph (u Ir)
ontoT, and y’ = (yl, ... ,
We may also assume that
In
addition, by
Lemma 2.1 we havewhere a is the
exponent given
in Lemma 2 . 1 and denotes the open ball in(~n -1
with radius R and centre 0.We also
represent
r x (~ near(0, 0)
as thegraph
of a function~ v ~ y E Il~n v ( y~ ~ R ~ -> (~. ~r
isindependent
of yn, and since r E 1 isuniformly
convex we haveVol. 8, n° 5-1991.
for
all y
EBR. Clearly
we also havefor all
(y’,
D.Let y =
( y’,
E D andlet y
be theunique point
on the lowerboundary
of D such that
y’
=y’;
thusUsing
Lemma 2. 1 we havewhere and a is the
exponent
of Lemma 2.1.If yn >__ 2 C2 ~ y’ ~°‘,
thenfrom
(2. 36), (2. 39)
and(2.41)
we obtainsince ~, ? 2,
whileif ;g
2C21 y~ ~°‘
we haveThus in any case we have
for all A similar estimate for all
y E U
followsby
theconvexity
ofw. The estimate
(2 . 32)
with 8 = 2 À followsimmediately
fromthis,
so thelemma is
proved.
Remark. - Holder
continuity
estimates for v similar to(2 . 31 )
can alsobe obtained if we allow some
decay
to zero orgrowth
toinfinity
of K asx
approaches
r. Forexample,
if we havewhere
K E Lq (Q)
for some q > n, and ~, > 0 and8 E [o, 1)
are constants, thenby [19], Corollary 3 .11,
we haveAnnales de l’Institut Henri Poincaré - Analyse non linéaire
for all x, y eQ
n
where Cdepends only
on n,Ro, r, ~,
and 8. This leads to thetangential gradient
estimateOn the other
hand,
as we havealready observed,
the secondinequality
of(2.43)
leads to the estimate(2. 23).
The combination of(2. 23)
and(2.45)
then leads to a Holder
continuity
estimate for v of the form(2 . 31 )
withy > 0 and C > 0 now
depending
in addition on eand q, provided
we haveStrict
convexity
estimates such as(2 . 31 )
may also beproved
under these somewhat moregeneral hypotheses.
Further refinements of these resultsare also
possible,
for solutions of(1 . 8)
as well as for solutions of moregeneral Monge-Ampere equations,
but we shall not pursue this here.However,
since theexponent
a obtained in Lemma 2.1 and its variousanalogues
for otherequations
is notgenerally sharp,
it is clear that theexponents
obtained in Lemmas2. 3,
2.4 and2. 5,
and conditions such as(2.46),
are notoptimal.
To conclude this section we make some further observations about the function w defined in the
proof
of Lemma 2.5. First ofall,
fromLemma 2 . 4 and the fact that u E
C2 (Q),
we have w EC2 (D) n C 1 (D)
andwhere
E = { y E aD : yn = ~ ( y’) }
is the lowerboundary
of D. Inparticular, since B)/
isindependent
of yn, we haveFurthermore,
it is clear that w is a convex solution of anequation
of theform
where
K
is obtained from Kby
the relationSince
(2. 34) holds,
there is no loss ofgenerality
inassuming
that w infact satisfies an
equation
of the formwith f~C1,1 (D
x [R x[R") satisfying
Vol. 8, n° 5-1991.
for almost all
( y,
z,p) E D
x R x whereand ~2 depend
on ~,1 and2 in ( 1. 10), ( 1. 11 )
and in addition on n andsup I Dw ~.
.D
We see therefore that w is a solution of a free
boundary problem,
andthe
regularity
assertions of Theorem 1 areequivalent
to suitableregularity
assertions for the free
boundary X
and for w near L. From Lemmas 2.4 and 2. 5 we see that for anyy0~03A3 and
anyy~D
we have the Holdergradient
estimateand the strict
convexity
estimatewhere
ye(0, 1)
andb >_ 2 depend only
on n andC,
codepend
in additionon
Ro, r,
~o and 1. Inparticular, setting yo = 0
in(2 . 55)
we obtainfor all
y e D. Using (2. 55)
and theconvexity
of w we also obtainfor all
y E D.
The free
boundary problem
obtained above is similar to those studiedby
Kinderlehrer andNirenberg [9],
and infact,
their results wouldimply higher regularity
of 03A3 and w near X if wealready
knew thatL E C1
andWE C2 (D U L).
Of course, we do not know suchregularity
at thisstage,
butnevertheless,
the method of[9]
can still be used.3. THE LEGENDRE TRANSFORM
To
proceed
further in theproof
of Theorem 1 weadopt
atechnique
used
by
Kinderlehrer andNirenberg [9]
andstudy
the freeboundary problem
for the function w obtained at the end of Section 2by studying
its
Legendre
transform. In our casehowever,
we may use the fullLegendre
transform rather than a
partial Legendre
transform as in[9].
Since is convex and w solves
(2 . 47), (2.51) with f positive,
themapping ~ :
y --~ Dw( y)
defines adiffeomorphism
of D ontoan open subset D* of
(~ + _
Infact,
sinceW
is of classC~
1and
uniformly
convex withrespect
toy’,
we may assume that~ E
C1 (D)
nC° (D)
is ahomeomorphism
ofD
ontoD*. Using
the esti-mate
(2. 57)
weeasily
see that for any y° E En BR/2
and any E E(0, R/4]
we have
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
and
where zo = ~
(yo)
= Dw(yo), ~
is as in(2. 57),
andC1 depends only
on n,Ro, r,
yoand 1. Using
the estimate(2 . 54)
we see that for any Yo E Ln BR/2
and any E E(0, R/4]
we haveor
alternatively,
for all6 > o,
such that a, we havewhere yo
= ~ -1 (zo),
y is as in(2 . 54),
and 60,C2, C3
arepositive
constantsdepending only
on n,Ro, r,
~,o and Infact,
the estimates(3 . 2), (3 . 4)
and
(3. 6)
can beimproved,
since w = Dw = onX, and B)/
is of classC~ and uniformly
convex withrespect
toy’.
The
Legendre
transform of w is the function w* onD* = Dw (D) given by
where
z = Dw (y).
It iseasily
verified thatand
where =
[D2 w] -1.
ThusD2
w* ispositive
definite inD*,
and we havew*
(D*) (~ C°~1 (D*),
w*(0)
= 0 and w* >_ 0 in D*by
theconvexity
of w and the fact that w
(0)
= 0. Since(2 . 47)
holdsand B)/
Eis uniformly
convex with
respect
toy’,
we see that we haveand
for all
(z’, 0)
E ~* = Dw(E)
c whereC4, A,
B are controlledpositive
constants.
Vol. 8, n° 5-1991.
Next, using
the relations(3 . 8)
and(3 .9)
it iseasily
verified that w*solves the
equation
where f*~C1,1 (D*
x R x isgiven by
for any
(z, t, q)
ED*
x R x Thusby (2 . 52)
and(2 . 53)
we havefor almost all
(z, t, q)
ED*
x R x for some controlledpositive
constants0, 1
and 2.
It can be shown that the estimates
(2.54)
and(2.55)
for wimply analogous
estimates for w* at eachpoint
of ~*. Moreprecisely,
from(2. 55)
we see thatand, using (2 . 54)
and(2. 55),
forany
sufficiently
small and any zeD* we have andwhere y, b are as in
(2.54), (2.55)
andCs, C6
are controlledpositive
constants. We shall not prove these estimates because
they
are not neededto
complete
theproof.
The
following
lemma nowessentially completes
theproof
of Theorem l.LEMMA 3 . 1-. - Let u E
C2 (BR ) n (BR )
be a convex solutionof
theequation
where
and suppose that
1 (BR
x I~ xsatisfies
for
almost all(x,
z, f~ x(~n,
where(3o
is apositive
constant.Suppose
also that
and
Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
for
all(x’, 0)
ETR
=BR n ~n -1
andall ~
E~n -1, where (31 and
7~ arepositive
constants. Then there is a number p E
(0, R), depending only
on n,R, (30, j31,
~,and (’ u II
~1 ~, such thatfor
any oc E(0, 1)
we havewhere C
depends
on the samequantities
as p and in addition onae(0,1).
Proof -
If wealready
knew that the estimate(3.22),
with C
depending
in addition on , would be a consequence ofBR
well known
global
second derivative Holder estimates forfully nonlinear, uniformly elliptic equations (see [6],
Theorem17.26, [10],
Theorem5.5.2).
So it suffices to prove
for suitable
positive
constants ~ E(0, R)
and Cdepending only
on n,R,
. ~
Let where A > 0 is fixed so
large
that 1 in . Choosenew coordinates for exn, and Then
graph u
can be
expressed
as thegraph
of a convex function vdefined
on a subset V of thehyperplane yn + 1= o,
so we haveThus the lower
boundary T
ofU, given by y" = u ( y’, 0), I y’ I R,
isuniformly
convex and of classC2°1, by
virtue of(3.20)
and(3.21).
Furthermore,
weclearly
havev IT = 0
andv E C2 (U) n 1 (IJ)
satisfies anequation
of the formwhere satisfies
for almost all
(y,
z,p)
e U x [R x wherePo depends only
on n,Po
andsup |Du |
.BR
Our aim now’ is to prove that
for suitable
positive
constants p and C. Once we havethis, (3 . 23)
followsimmediately
and the lemma will beproved.
A bound for
D2 v
on aun BR/2
would follow from the resultsproved in [4],- [6], [7], [15]
if we knew that v were of classC2
onU.
We do not know thisyet,
so we need to make our estimates on a suitableapproximat- ing sequence
of solutions. First we note thatby [20],
Lemma2. 2,
there isVol. 8, n° 5-1991.