ANNALES DE L’I. H. P., SECTION C
KLAUS ECKER
GERHARD HUISKEN
Interior curvature estimates for hypersurfaces
of prescribed mean curvature
Annales de l’I. H. P., section C , tome 6, n
o
4 (1989), p. 251-260
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Interior
curvature
estimates
for
hypersurfaces
of
prescribed
mean
curvature
Klaus
ECKER
Department
of
Mathematics,
University
of
Melbourne,
Parkville
Vic.
3052,
Australia
Gerhard
HUISKEN
Department
of
Mathematics,
R.
S.
Phys.
S.,
Australian
National
University,
G.P.O.
Box
4,
Canberra,
A.C.T.
2601,
Australia
Ann.
Inst.
Henri
Poincaré,
VoL
6,
n°
4,
1989,
p.
251-260.
Analyse
non
linéaire
ABSTRACT. -
We
prove
that
a
smooth
solution
of
the
prescfibed
mean
curvature
equation
div
( v -1
Du)
=
H,
v
=
( 1
+
Du
I 2) 1/2
satisfies
an
interior
curvature
estimate
of
the
form
Key
words :
Mean
curvature,
second
fundamental
form,
capillary
surfaces.
RESUME. -
On
démontre
qu’une
solution
de
l’équation
Du)
=
H,
v=(l +
( Du
2) 1/2
satisfait
l’estimation
interieure
pour
la
courbure
Interior
gradient
estimates
for
solutions
of
the
prescribed
mean
curva-
ture
equation
Classification
A.M.S. :
53
A
10.
Annales
de
l’Institut
Henri
Poincaré -
Analyse
non
linéaire -
0294-1449
Vol.
6/89/04/251 / I O/t
3,00/ ©
Gauthier-Villars
252
K.
ECKER
AND
G.
HUISKEN
were
derived
by
Finn
[6]
and
Bombieri,
De
Giorgi
and
Miranda
[1]
in
the
case
H=0,
and
by
Ladyzhenskaya
and
Ural’tseva
[11],
Heinz
[8],
Trudin-
ger
([16],
[17])
and
Korevaar
[10]
in
the
general
case,
assuming
that
~H ~u~0.
The
exponential
form
of
the
estimate
where
xoERn,
and
c i,
c2
depend
on
n,
R
sup
H
I
and
BR
(x0)
R 2
sup
aH
cannot
be
improved
as
was
shown
by
Finn
[6]
in
the
case
BR (x0)
ox
p
Y
H -_- o.
From
here
one
can
obtain
interior
second
derivative
estimates
by
employing
standard
linear
elliptic
theory.
However,
the
estimates
thus
obtained
are
nowhere
near
optimal
in
terms
of
their
dependence
on
the
gradient.
As
far
as
we
know
Heinz
[9]
was
the
first
to
obtain
interior
curvature
estimates
for
minimal
hypersurfaces
(not
necessarily
graphs)
in
two
dimen-
sions,
which
were
later
generalized
to
the
case
n _ 5
by
Schoen,
Simon
and
Yau
[14].
Interior
curvature
estimates
in
all
dimensions
for
solutions
of
( 1)
were
recently
established
by
Caffarelli,
Nirenberg
and
Spruck
[2].
They
proved
an
estimate
of
the
form
where
A j
I
denotes
the
norm
of
the
second
fundamental
form
of
M = graph u,
assuming
aH >_ 0
>_ o.
Their
estimate
holds
in
fact
for
a
general
class
of
nonlinear
elliptic
equations.
However,
for
solutions
of
(1)
the
dependence
on
the
gradient
in
the
above
estimate
can
be
significantly
improved
by
exploiting
the
strong
geometric
information
contained
in
the
Codazzi
equations.
We
prove
the
curvature
estimate
where
~~
{ xo) _ ~
x
E
~n/~
x -
xo ~ 2
+ ~
u
(x) -
u
(xo) ~ 2
_
R 2 ~
and
the
constant
depends
on
n,
H
and
the
first
two
covariant
derivatives
of
H,
see
Theorem
2.1.
This
estimate
generalizes
an
estimate
for
minimal
graphs
obtained
in
[5],
which
led
to
a
new
Bernstein
type
result
for
entire
minimal
graphs.
Another
important
case
we
have
in
mind
are
capillary
surfacs
Annales
de
l’Institut
Henri
Poincaré -
Analyse
non
linéaire
253
INTERIOR
CURVATURE
ESTIMATES
(Corollary
2.2).
i.
e.
hvDersurfaces
satisfving
In
this
case
estimate
(3)
for
solutions
in
BR
(xo),
R
1
reduces
to
which
seems
to
be
natural
in
view
of
Concus’
and
Finn’s
interior
height
estimate
in
[3]
and
the
interior
gradient
estimate
obtained
in
[4].
1.
PRELIMINARIES
Let
M
be
a
hypersurface
in
D~n + 1
represented
as
a
graph
over
i~n
with
position
vector
x
and
upward
unit
normal
v.
We
define
the
height
of
M
by
and
the
gradient
function
by
where
denotes
the
(n+
1 )
st
coordinate
vector
in
~" + 1.
Note
that
x (x) _
(x,
u
(x)),
v
(x)
=
1
+
Du
(x)
and
v
(x)
=
(x) .
( -
Du
(x), ~ )
for
The
second
fundamental
form
of
M
is
given
by
where
V
denotes
covariant
differentiation
in
M
is
an
orthonormal
frame
for
M.
It
is
well-known
that
the
gradient
function
then
satisfies
the
equation
where A,
A
and
H
denote
Laplace-Beltrami
operator,
norm
of
the
second
fundamental
form
mean
curvature
of
M
respectively.
The
following
lemma
gives
a
generalization
of
inequality
(1.34)
in
[14]:
1.1.
LEMMA
VoL
6.
n’
4-1989.
254
K.
ECKER
AND
G.
HUISKEN
Proof -
From
( 1.20),
(1.29)-(1.31)
in
[14]
we
infer
the
relations
and
where
the
totally
symmetric
tensor
VA
is
given
by
~,
Since
for
fixed
i we
have
the
result
follows
in
view
of
Young’s
inequality.
1.2.
Remark. -
It
is
worth
noting
that
for
n = 2
the
inequality
holds.
The
next
lemma
was
proved
in
the
case
H = 0
in
[5].
1.3.
LEMMA. -
For
p,
q
>_
2
we
have
the
inequality
Proof -
Combining
(6)
and
(7)
we
infer
Inequality
(9)
then
follows
from
Young’s
inequality.
Annales
de
l’Institut
Henri
Poincaré -
Analyse
non
linéaire