In this paper we prove the C~-smoothness of Lipschitz-continuous graphs of C 2 with ... The graph of a Lipschitz-continuous function u: ft--+R will have Levi ...
Acta Math., 188 (2002), 87 128 (~) 2002 by Institut Mittag-Leffier. All rights reserved
Smoothness of Lipschitz-continuous graphs with nonvanishing Levi curvature by G. CITTI
E. LANCONELLI
Universit& di Bologna Bologna, Italy
Universit~ di Bologna Bologna, Italy
and
A. MONTANARI Universit~ di Bologna Bologna, Italy
1. I n t r o d u c t i o n In this paper we prove the C ~ - s m o o t h n e s s of Lipschitz-continuous graphs of C 2 with smooth and nonvanishing Levi curvature. Let ~t be an open subset of a 3. Given a C2-smooth function u: ~ - ~ R the Levi curvature of its graph at the point (~, u(~)), ~ E ~ , is the real number s
k(~,u) := ( l §
(1)
,
where f-.u := Uxx + Uyy + 2aUxt + 2buyt + ( a 2 + b2) utt,
(2)
and a = a ( V u ) , b = b ( V u ) depend on the gradient of u as a, b: R 3 --+ R,
a(p) - P2-PlP3
l+p~ ' b(p)-
-Pl-P2P3
l+p~
(3)
In (1), (2), ~ = ( x , y, t) denotes the point of R 3, ut is the first derivative of u with respect to t, and analogous notations are used for the other first- and second-order derivatives of u. The notion of Levi curvature for a real manifold was introduced by E . E . Levi in 1909 in order to characterize the holomorphy domains of C 2. Since then, it has played a crucial role in the geometric theory of several complex variables. In looking for the polynomial hull of a graph, Slodkowski and Tomassini implicitly introduced in 1991 the following definition of Levi curvature for Lipschitz-continuous graphs [16]. Investigation supported by University of Bologna Funds for selected research topics.
88
G. C I T T I , E. L A N C O N E L L I AND A. M O N T A N A R I
Definition 1.1. Let ~ be an open subset of R a and k a given function defined on ft •
The graph of a Lipschitz-continuous function u: ft--+R will have Levi curvature
k(~,u(~)) at any point ~Ef~ if there exist a sequence (u~) in C2(f~) and a sequence of positive numbers en-+0 satisfying the conditions: (i) There exists M > 0 such that IIunlIL~(a)+NVUnlIL~(fl) 0 such t h a t for every x, y, z
d(x, y) 0 dependent on M such that for every r
(42) P
c / ( , 2(Jkl+ ITkl)+ IV~ ,l 2) ITzl~,4- / T f T z r 6. J
We will make use of the following simple property:
J
108
G. CITTI, E. L A N C O N E L L I AND A. M O N T A N A R I
Remark 4.1.
From identity (22) and the definition of T it immediately follows that
for every function f, r
we have
f TXfr f (1+~/2 ~ 0 : j~TfXvr /TfX*. Analogously
Proof. We
differentiate equation (15) with respect to T, then we multiply by
and integrate:
JTfTzr
f T(X2z+Y2z+T2~)Tz~ 6 = [by Remark 4.1]
= I:+...+I9Let us consider a few terms separately:
1/(E~,Xlz+XTz)X~zr
Tl~+TXz)r
Ii + I4 = ---~
1 /Tyz([y,T]z+Tyz)r _ 1_2/(IT, Y]z+YTz)YTzr 6- -~
(2:) i / ((TXz)~+(XTz)2+(TYz)~+(VT:)~)r 2 1
=--~
1
((TXz)2+(XTz)2+(TYz)2+(YTz)2)r 6
1/((Xv)2+(yv)~)(Tz)2r
+~
Tzr 6
SMOOTHNESS OF GRAPHS WITH NONVANISHING LEVI CURVATURE
109
On the other hand, using identity (21) in/2 and/6, we get
I2+I5+Is = / YvTzwXvTzr
XvTzwYvTzr 6
+I /x((Tz)~)Xvwr
/ y((Tz)2)Yvwr
1 f T~((Tz)2)Tevwr + -~ Canceling the first two terms and integrating by parts the last three terms by means of the identities (20), we get
I2+I5+Is = --~ 21/(Tz)2Xvw(Yv_wXv)r
1/(Tz)2yvVoor
5
"."-'S
'''.+
, Si.zl...i..vl.+o-2l
2
Using the fact that v=arctanut in the terms I, 4 and 7, and using Proposition 2.1 in the last term, we arrive at
I.+I5+Is=--~
'I(Tz)'wT(k(l+a'+b2)312)r6 -3 f (Tz)'w(XvXr162162162
5
2 f (r.)'((x,)' +(yv)'+(T~.)'),6 = _12 f (Tz)h~162 32ff (Tz)2~&(l+a2+b2)l/2(aTa+bTb)r
-3 f (Tz)'~(XvXC+YvZr162162
_ 12f (Tz)'((xv)'+(rv)'+n)l~r
~
~
