Table Of Content(
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ALDO ANDREOTTI
I
,i .
I
Complexes of ,I
Partial
Differential
Operators
Yale Mathematical Monographs 6
I
I
Editorial Committee
Nathan Jacobson
Shizuo Kakutani
William S. Massey
George D. Mostow
Complexes of Partial Differential
Operators
by ALDO ANDREOTTI
//1
New Haven and London, Yale University Press, 1975
Contents
Published with assistance from the foundation Preface
established in memory ofAmasa Stone Mather
I. Elementary and Levi Convexity
ofthe Class of 1907,Yale College.
2. On the Hans Lewy Problem II
Copyright © 1975by Yale University.
Allrights reserved. This book may not be 3. Complexes of Differential Operators with Constant Coefficients 19
reproduced, inwhole or inpart, inany form
4. Boundary Value Problems for General Complexes of
(except by reviewers for the public press),
without written permission from the publishers. Differential Operators 27
Library of Congress catalog card number: 75-8440
International standard book number: 0-300-01887-8 5. Complexes of Differential Operators with Variable Coefficients 38
Set inTimes Roman type. Bibliography
Printed inthe United States of America by
Eastern Press, Inc.
Published in Great Britain. Europe. and Africa by
Yale University Press. Ltd.•London.
Distributed inLatin America by Kaiman &Polon,
Inc.•New York City; inAustralasia and Southeast
Asia by John Wiley &Sons Australasia Pty. Ltd.•
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Ltd.•Delhi; inJapan byJohn Weatherhill. Inc., Tokyo.
Preface
The lectures that comprise this monograph were greatly inspired by the work
of Malgrange on operators with constant coefficients. I have presented a
summary of some joint work with H. Grauert, C. D. Hill, S. Lojasiewicz,
B.Mackichan, and M. Nacinovich (in the third chapter). The aim is to carry
the classical results of the Cauchy-Riemann equations to systems of linear
partial differential equations.
I am not a professional "analyst". Therefore I beg indulgence from the
reader for any lack of elegance, but I have tried to be clear.
A.A.
Corvallis. Oregon
1975
.,
1. Elementary and Levi Convexity
1. Elementary Convexity
a) We consider R" as an affine space.
Letnbean open set inR"and let cp; n ---+ R beaCoo function. At a point
a Enwe consider the Taylor expansion of cp
cp(x) = q>(a) + ~ox, cp(a)(xi- ail +j~ox,x, cp(a)(xj- aj)(xj - aj) +0(IIx - a113).
The quadratic form
H(cp).(v)= ~O"'j cp(a)vjvj
is called the Hessian of sp at the point a En.
An affine change of coordinates on R" acts on H(cp).as a linear change
ofcoordinates
v ---+ Av, det A#-O
and therefore the number ofpositive and the number ofnegative eigenvalues
ofthe Hessian at a does not-depend on the affine choice ofcoordinates inR".
n En
A function cp on iscalled q-conuex ifat each point a the number of
strictly positive eigenvalues is ~ n - q.This notion isan affine notion.
Let a Enbe a point where (dcp). = O.Then any differentiable change of
coordinates near a acts on H(cp).as the linear change of coordinates
v ---+ J(a)v
where J(a) isthejacobian matrix of that change of variables.
Therefore at a critical point of cp (i.e. where do = 0) the signature of the
Hessian is independent of the choice of differentiable local coordina tes.
b) Let nbe an open set in R" with a smooth boundary. Bythis we mean
that there exists a Coo function q>;R" ---+ R with the following properties;
i) n = {xERnlcp(x)< O},
ii) at every point Xo E an = n - n, (do), #-O.
James K. Whittemore Lectures. May 1974.
2 Elementary and Levi Convexity
Elementary Convexity
This means that near any point XoEOn, ({Jncan be taken among a set of local (r affine linear one to one, OP = {r ERPILrt < 1}) such that
differentiable coordinates so that, locally, near Zoisdiffeomorphic to a half
space {({J ~ OJ. The boundary on of n is given by the equation {({J = O} r(O) = Xo
and the tangent space at Xoto on has the equation r(OP) - {xo}ERn - n.
La,; ({J(Xo) Vi = O. Similarly there exists a disk of dimension q = q(xo)
Let us consider the Hessian of ({J restricted to the tangent space T,,(an):
La,." ({J(x).viv)
affinely imbedded in R" such that
H«({J)I(v)T,,(OOI:: _
1 r(O) = Xo
La., ({J(x).v, - O. ,(OQ) - {xo} c n.
This is a quadratic form in n - 1 variables. One realizes readily that the The maximal values of p and q having these properties characterize the
number of negative or positive eigenvalues is independent of the choice of numbers p(xo) and q(xo) when the Hessian is nondegenerate.
affine coordinates in RO. c) An open set n c R" with asmooth boundary such that VXoEOn we have
Suppose that IjI is a choice of another function defining n as an open set
p(xo) ;;:, n - I - k
with a smooth boundary. Then near any point XoEan we must have
will be called k-convex.
({J = hljl
Exercise. Let n be k-convex and relatively compact. Show that there
where h is a C'" function with the property h(xo) > O. exists a Coo function ({J in n with the following properties
Now i) V c ER the sets
d. d({J = d. (h d({J + dh. IjI) = h d. dljl + 2 dh. dl/t + I/t d. dh. Be = {xEnl({J(x) < c}
Thus
are relatively compact in n
.
H«({J)IT,,(00) = h(o)H(I/t)ITx (00) ii) The function ({J is k-convex on n.
Proof (ex) Let g: n ------> R and let f be defined in an open set of R
as, on T,,(an),dljl = 0 and IjI(xo) = O. containing g(n).
Therefore: Assume f,g to be Coo and fconvex (i.e. f" > 0, f" > 0). If g(x) is q-convex,
If n is WI open set in RO with a smooth boundary at any point Xo E on the so is f(g(x». Indeed
n,
signature of theaHne,ssian of any defining function for restricted to the tangent ddf(g(x» = I", (g(x))ldgl2 + f",(g(x» ddg(x).
space at Xoto has a signature which is independent of the choice of affine
coordinates ill R'' and of the choice of the defining function. Let p(xo)(q(xo» be (/3) Using this remark it follows that for c > 0 sufficiently large ec<p- 1,
the number of strictly positive (strictly negative) eigenvalues of H«({J)IT (00) . where ({J is a defining function for n, is k-convex in a neighborhood of on in
"
We must have R". Thus, as the function-lit is convex on t < 0, the function on n e
~C/",
p(xo) + q(xo) ~ n - 1. is a(y)k-cAodndvienxg ftuoncetio~cn/", tohne nfunncetairononC'tLhxaft gwoeitsh toC'CXJ>w0henandwelaarpgpe,roacwhe geotn.a
=
As an exercise one can show that there exists a disk ofdimension p p(xo) function on n with the desired properties.
affinely imbedded in RO: Remark. One could take as definition of a k-convex subset of RO the
properties given in the previous exercise. One obtains a definition of k-
r:OP ------> R"
convexity that does not make use of the smoothness of the boundary nor of
4 Elementary and Levi Convexity Elementary Convexity 5
the fact that n is to be relatively compact in R" (definition of "k-conoexity (b) Any a-convex open set nin the general sense in R" has the
in the general sense"). property Hj(n, R) = 0 if i > O.
F.xample. Consider the set in RP x R? +1, n = p + q + I,
Y The proof of Hadamard's theorem can be obtained from the following
n = {(y, x) E RP x Rq + ll( J1+-I-~f - 1 4- ~Yf - i < O}. remarks. Let n in R" be aO-convex domain and letXoEan. We consider the
Gauss map: an --L..... S"- I which associates to Xothe unit exterior normal
Topologically n = DP +1 X S''. Taking into account the geometric defini- vector v(xo) to an at xo,translated to the origin. The O-convexity assumption
shows that the differentiable map g has always a nonvanishing jacobian.
tions of the invariants p(xo),q(xo) at each point XoEan, one verifies that
Ifn ~ 3,S" - Iissimply connected and gmust be one to one; ifn = 2,a direct
p(xo) ~ p V XoEOn. argument establishes the same fact.Then one can show that the tangent space
n
T•.(an) has all of n on one side. Hence is an intersection of half spaces
Therefore n isa q-convex set.
and thus it is convex. So also must be its interior n.(2)
For p = 2, q = 0, n isa disjoint union oftwo balls in R3.
From this theorem and an inspection of the example given above one is
For p = 1,q = I, n isa solid torus in R3.
led to conjecture that
d) We have the following
ifn is q-convex (or q-convex in the general sense) then one must have
Theorem of Hadamard (see [12J and [I3J): Ifn is a domain (I) in R" with a
smooth boundary relatively compact and a-convex then nis elementary convex H'(n, R) = 0 for i> q.
(i.e. if x, YEn then the segment {Ax + (I - J.)yIO :0::; J. < I} joining x and
e) The last conjecture can be proved by making use of standard Morse
y is all contained in n).In particular n is contractible to a point and therefore
theory. One has actually the following (See [3J).
Hj(Q, R) = 0 for all i > O. Theorem: Let nbe a q-convex open set in R" in the general sense,
then
If0isa-convex domain inthe general sense, then n isaunion ofan increasing
sequence ofa-convex domains relatively compact in Q: and
Hq(Q, Z) has no torsion.
n, c n2 c ...,n = u nj.
Proof It isnot restrictive to assume that wehave on n aq-convex, proper
Indced let cp: n ~ R be a a-convex function on n which is proper function cp: Q ---+ R having only nondegenerate critical points, because
(i.e. VcE R, {cp(x) ,,;; c} is compact). Select a sequence {c..}E R, where c.'is wecan "approximate" as wellas welike with first and second derivatives the
not a critical value of tp and set n', = {xE Qlcp(x) < c.}. We can assume given function exhibiting the q-convexity, with one having this property. At
c, < C2 < C3 < .,. lim c.' = +OC!so that n'l c n~ c Q'3 c ... and a critical point, the number ofnegative eigenvalues of H(cp)cannot exceed in
n = un'j. Each OJisrelatively compact with a smooth boundary. Thus nj number n - (n - q) = q.Thus the index i(p)ofeach critical point p is ,,;;q.
has a finite number of connected components. Let {K.} be an increasing Hence Qhas the homotopy type ofacellular complex with cellsall ofdimen-
sequence ofconnected compact subsets of'Dwith KI c K2 C ... .K, C Rj+1, sion :0::; q.Consequently Hj(Q, Z) = a ifi > q,and also for the same set of i's,
0= uKj• Hj(n, K) = a where K isany field. We have also by the universal coetTicent
Sleet c.' such that 0', ::> K, and set 01 = connected component of K, theorem
in 0'". Select c",> c, such that 0'" ::> K2 and set O2 = co~nected com-
0= Hq+1(0, K) = (Hq+1(0, Z) ® K) EB Tor (Hq(O,Z), K).
ponent ofK2 in0',.Continuing in this way we get the conclusion. Then we
get the following , Thus Hq(Q,Z) cannot have torsion.
Corollary: (a) Any a-convex domain in the general sense in R" is elementary NOTE: Hj(Q, R) = Horn (Hj(O, Z), R) = a if i > q, as expected by the
convex. conjecture above.
1. Domain means open and connected. 2, Another proof ofthe theorem willbegiven later.
6 Elementary and Levi Convexity Levi Convexity 7
I
2. Levi Convexity
I
2'(cp) (v) ==
a) Let n be an open set in C", cp: n - R a Coofunction. Using complex T:,(Ofl)
coordinates Zl' ... ,z, and theirs complex conjugates, the Taylor expansion
of cpat a point a Enhas the form
is hermitian form in (n - 1) variables whose signature is independent of the
cp(zl = c+p(at)L+ozL.zap,~(ac)p((za.)(z.- -a.Haz.,)6+- LaOpi)"cp(a)(z. - a.) fcuhnocitcioenofclpoocal hulomorphic coordinates near Zoand of the choice of the defining
+tLai cp(a)(z. - a.Hz,6 - ap) If p(zo) is the number of strictly positive eigenvalues of .2"(CP)T"Po) and
i
+ Lo,;6•cpp(aHz.- a.)(z,6 - a,6) + O(llz - aI13). q(xo) the number of strictly negative eigenvalues of 2'(cp)ITa'"(0'A,,) •'''.:ve have
The hermitian quadratic form p(xol + q(xo) ~ n - 1.
2'(cp).(V) = LOz;./1 cp(a) v,ifl' Moreover there exists an analytic disk of dimension p = p(zo);
iscalled the Levi form of cpat a. r :DP ----> C"
Any bihotomorphic change of coordinates near a acts on 2'(cp). with a (DP = {tECPILltil2 < I}, r holomorphic one to one) such that
linear change of variables
r(O) = Zo
v ----> J(a)v r(DP) - {zol c en - n.
where J(a) is the jacobian matrix at a of the change of variables. Thus the Similarly there exists an analytic disk of dimension q = q(zo)
number of positive and the number of negative eigenvalues of 2'(cp). does not
depend on the choice of local holomorphic coordinates.
A function cpon n is called q-conoex if at each point a En, 2'(cpl. has at
with
least n - q strictly positive eigenvalues.
Remark. If(dcp~ " O.then one can choose new coordinates z',holomorphic r(O) = Zo
near a, with Z'l = Loz.cp(a)(z. - a.) + tLOz.Z,6cp(a)(z. - a.)(z,6 - a,6)'Then r(Dq) - {zo} c n.
in these new coordinates the Taylor expansion of cphas the form
1fT isassumed to be biholomorphic, then the largest values of p and q with
cp(z') = cp(a) + 2Re z~ + 2'(cp).(z') + 0(1Iz'113) these properties give the geometric meaning of the invariants p(zo) and
q(zo) when the Levi form is nondegenerate.
(so that 2'(CP)a = H(cp).). c) An open set n c C" with a smooth boundary is called k-conoex if for
b) Let n be an open set in en with a smooth boundary and a defining every ZoEan we have
function cpoAt each point ZoEan one can consider the analytic tangent space
p(zol ~ n - 1 - k.
T~,(an):
More generally one can define an open set n c R" as k-conoex ifwe can
find a function ip:n --+ R with the following properties:
which is the maximal linear complex subspace contained in the real tangent i) for every c E R the sets
space. As in the case of elementary convexity we now have the following Be = {xEnlcp(x) < c]
statement.
are relatively compact in n,
At every boundarv point Xo E on andfor any choice of the defining function ii) cpis k-convex on n.
<p,the Levi form of cp restricted to the analytic tangent space to on at Xo:
8 Elementary and Levi Convexity Applications of Levi Convexity 9
a
One shows (with the same argument used for elementary convexity) that, Theorem: Let be a O-convex open set in en with a smooth boundary. Then
ifa is relatively compact and k-convex in the first sense, it is also k-convex ais an open set of holomorphy.
in this more general sense. n-k n Proof (ex) Let {x.} c nbe a divergent sequence. If {x.} is divergent in
Example. The set a = {z E enl:Elz;j2 - :E IZjl2 < I} is a k-convex en we can find a polynomial p with s1;lplp(x,)1 = 00. We can therefore
subset of en. I n-k+1 assume that {x.] is bounded in en and, passing to a subsequence, that
d) Let us denote by (!) the sheaf of germs of holomorphic functions on en. 1i!TIx, = ZoE aa.
One has the following Let a = {zEenlcp(z) < O},where cP is a defining function for a.We can
a
Theorem (H. Cartan and J.P. Serre; see [9] and [25]): If is a O-convex find a coordinate neighborhood U of Zosuch that, on U,
open set of"en (in the general sense),
rhell cp(z) = Ref + .2':(cp)zo(z) + 0(1IzIl3), where fis holomorphic.
Hi(a, (9) = 0 for all i > O. Thus {zEUIf(z) = O} (l n= I),as the Levi form is assumed to be positive
definite at zo, provided U is taken sufficiently small. Also replacing tp by
And more generally one has the following eC~- 1with c > 0 large, we may assume that !t'(cp)zo is positive definite in
a
Tlieorem: (see [2]), If" is a q-conuex open set in en (in the general sense), U (provided U is sufficiently small). Let Q: U ---+ R be a Coofunction with
1/)('/1 the properties
H i(r2, (n) = 0 [or all i > q. o < Q(x) V xE U, supp Q(x) c U, Q(zo) > O.
Retnark: In the previous statements the sheaf (9 can be replaced by any
Consider the function tp - £Q = CPl' This is O-convex in U, if£ issmall, and
analytic coherent sheaf.? defined on r2.
The proof of these theorems can be found in any standard reference book 11 = {zeenlcpl(z) < O}
(e.g. [9]) For the q-convex case see [2].
isstill O-convex with a smooth boundary. Set V = {zE UlcpI(z) < O}so that
11= n uV.
({3)We make use of the Mayer-Vietoris sequence: Let X be a paracompact
3. Applications of Levi Convexity
topological space and let Xi' X2 be open sets of X such that X = XI U X2.
a Let ~ be asheaf ofabelian groups on X. One has the following exact seq uence
a) Open sets of holomorphy. An open set c en is an open set of holo-
morpliy iffor any divergent subsequence {x,} c a(i.e. without accumulation 0-- HO(X, ff) ~ HO(X" ff) EElHO(X2, 3"')~ HO(XI U X2, .F)
points in a)one can find a holomorphic function on a with ---+ HI(X, ff) ---+ ...
syplf(x,)1 = 00. (here exand f3 are defined by the restriction maps).
If0 ---+ ff ---+ ygO ---+ ygl ------> yg2 ------> ... isa flabby resolution
The following theorem is due to E.E. Levi ([17]).
Theorem: Let abe all open set of holomorphy in en with a smooth boundary. of ~ we have
Thenfor any boundary point Zo E aa one has 0---+ r(XI, ygq) ---+ F(X, ygq) EErl(X2, ygq) ---+ r(XI (l X2, ygq) ------> 0
exact with obvious definition of the maps. Tnis gives an exact sequence of
complexes whose cohomology sequence is the Mayer- Vietoris sequence.
yt aa (y) Apply the Mayer-Vietoris sequence to 11= a u V and to the sheaf (I.
I.l· II/(' lcii [onn rcslricterl to the unul ic ianqent space to at Zo is positive
We get, as H1(11,(!) = 0 by the theorem of Cart an and Serre, a short exact
'('III/{i<'ii /I;Ii'
sequence
rile theorem of Carlan and Serre enables us to invert this theorem in the
Ioilowing form (solution of the Levi problem: see [llJ).
