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JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
HÅKAN HEDENMALM
An Hadamard maximum principle for the biplacian
on hyperbolic manifolds
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Journees Equations aux derivees partielles
Saint-Jean-de-Monts, 31 mai-4 juin 1999
GDR 1151 (CNRS)
An Hadamard maximum principle for the bilaplacian on
hyperbolic manifolds
Hakan Hedenmalm
Abstract
We prove the existence of a maximum principle for operators of the type AL^—IA, for
weights uj with logo/ subharmonic. It is associated with certain simply connected subdomains
that are produced by a Hele-Shaw flow emanating from a given point in the domain. For
constant weight, these are the circular disks in the domain. The principle is equivalent to the
following statement. THEOREM. Suppose uj is logarithmically subharmonic on the unit disk,
and that the weight times area measure is a reproducing measure (for the harmonic functions).
Then the Green function for the Dirichlet problem associated with An;"1
 A on the unit disk
is positive.
1. The classical maximum principle.
Let A be the Laplace operator in E" (n = 1,2,3,...). We say that a real-valued function u is
harmonic on an open subset of W1
 if Aiz = 0 there, and subharmonic if An ^ 0. A domain in R71
is an open and connected set. The classical maximum principle for subharmonic functions can be
given the following formulation. Let D be a bounded domain in R", and u a function continuous
on the closure of D. We then have the implication
0 < ^U\D and U\QD < 0 =» U\D < 0. (MP:A)
Moreover, unless U\D = 0, the conclusion can be sharpened to U\D < 0.
2. A biharmonic maximum principle.
It is natural to try to extend the classical maximum principle to higher order elliptic operators:
we focus on the simplest example, the bilaplacian A2
. In the same way that physically, the laplacian
corresponds to a membrane, the bilaplacian corresponds to a plate (there is also a connection with
creeping flow). In view of the nature of the boundary data for 'he Dirichlet problem, the maximum
principle we are looking for necessarily will involve two inequalities along the boundary of the
subdomain D, one for the functions, and another for the normal derivatives. We first need some
notation. A real-valued function u on a domain Q is biharmonic provided that A2
?! = 0 there,
and sub-biharmonic if A2
^ < 0 (one should think of A as a negative operator, which is the reason
why the inequality is switched as compared with the definition of subharmonic functions). In the
following we shall restrict our attention to the case of the plane R2
, which is identified with C, the
complex plane. Around 1900, it was known - more or less - that a variant of a maximum principle
can be formulated for circular disks. Let D be a circular disk and u a C1
 -smooth function on the
closure of D. The maximum principle reads
r^
^U\D .< 0, U\QD < 0, and — <, 0 =^ u\o < 0, (MP:A2
)
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whereby the normal derivative is calculated in the interior direction. Actually, unless u\o = 0, we
have U\D < 0. Let TD denote the Green function for the Dirichlet problem associated with A2
 on
D: for fixed C e D, the function rp(-,C) vanishes along with its normal derivative on 9D, and
A
2
!^', C) equals the unit point mass at C. The above maximum principle (MP:A2
) then expresses
•the following three basic facts:
0<rp(^,C), (z.QeOxD , (2.1)
0 < A.FD^, C), (^ C) e 9D x D, (2.2)
and
r\
^ _ A, Fp(z, C) < 0, (z, QeOD x D. (2.3)
These properties are easily verified by computation. In fact, with the normalizations used here,
the Green function F = FD for the unit disk D={z€C:|z|<l}i s expressed by
F(^ C) = \z- Cl2
 log ^—— + (1 - M2
)^ - |C|2
), (z, C) € ID x D. (2.4)
It should be mentioned that a local analysis of the behavior near the boundary shows that (2.2) is
an immediate consequence of (2.1), at least if we replace the sign "<" with "<". The connection
between (MP:A2
) and (2.1)-(2.3) is apparent from the symmetry of Fp together with Green's
formula:
u(z)= f r^,c)A^(c)dS(C)+^ I (Acr^.cAc) Qu
I -J^\~7-»/— — —\-»/'~——\^ / • f) / I •——(•>- ^^1 '•»/ "Q
ID 2 JQ[) \ an
9
JD 2 JQ[) \ Qn
A^FD^, C) n(C)) ^(C), z € D, (2.5) 9n(Q
where G?S is area measure, normalized by the factor Tr"1
, and da is one-dimensional Lebesgue
measure, normalized by the factor (27r)~1
.
Notation: We have normalized the laplacian A acting over the plane: it is the operator
1
 f y
 92
 \
^^-A^^W2
)- ^^'^
A locally summable functions / on some domain f2 in the plane is equated with a distribution by
the dual action
^,/)= /'^)/(z)dS(z),
J^i Jfl
for test functions y?: compactly supported (7°°-smooth functions on 0.
3. Hyperbolic geometry.
In his treatise on plaques elastiques encastrees ([3], pp. 515-641), Jacques Hadamard suggests
the possibility of a maximum principle of the type (MP:A2
) for more general subdomains D - in
fact, he conjectured that it would hold for all convex D. That was later disproved in a series of
papers of Duffin, Loewner, and Garabedian (see, for instance, [2]). With hindsight, it is possibly
to intuitively feel why the maximum principle (MP:A2
) fails in such generality; the circular disks
are natural domains for the bilaplacian. Let us keep the circles - to be more precise, we single
out the Gauss mean value property which they possess - but consider instead much more general
geometries.
Let f2 be a simply connected two-dimensional Riemannian manifold with metric ds. Then
topologically, 0 is either the sphere, the plane, or the open unit disk, by the Koebe theorem. We
III-2
shall be interested in hyperbolic metrics, in which case we have only the plane or the disk as
alternatives. We choose the disk, because it can be used to give local coordinates of the sphere
and the plane. Then f2 equals the unit disk ©, and we have a global coordinate chart. The metric
ds can be expressed as
ds(z)2
 = a{z) dx2
 -h f3(z} dy2
 + 7(2:) dxdy, z = x + iy € D,
where Q, /3, and 7 are assumed smooth, and the associated 2x 2 matrix is strictly positively
definite. The basic "circular" neighborhoods of points need not look circular in this first choice
of coordinates (they are instead elliptic), but this is easily remedied. We find a quasi-conformal
mapping from ID to itself which straightens out the ellipses to circles, and in the new coordinates,
the metric takes the form
ds(z)2
 = Q(z)2
 \dz\2
 = Q(z)2
 (dx2
 + dy2
)^ z = x + iy e ©,
for some smooth strictly positive function g on D. The Laplace-Beltrami operator A on the
Riemannian manifold is very simple to express in these coordinates:
A=———A.
Q(Z)2
The distributional derivative A log Q turns out to be independent of conformal coordinate changes,
and, in fact, p, = -Alog^ represents a fundamental geometric quantity: the local distribution of
Gaussian curvature. Let us assume that the Riemannian manifold is hyperbolic, meaning that the
Gaussian curvature is negative everywhere. Then u, is a negative locally bounded Borel measure
on D, and the /^-measure of a subset tells us how curved the space is on the set. It is natural to
consider first smooth metrics, and then consider more rugged metrics as a limit of smooth ones.
The natural generalization of the planar maximum principle is
A^lp < 0, U\QD < 0, and —on
< 0 =^ U\D < 0, (MP:A2
)
OjD
where D is a smooth precompact subdomain of D, and the function u is assumed smooth on the
closure of D. Of course we cannot expect the above to hold for all subdomains D, but rather, we
should look for the appropriate generalization of the circular disks. The first attempt probably
would be to consider the metric disks, but the above maximum principle definitely fails for them
(a counterexample can be worked out on the basis of Miroslav English work [1]). Instead we focus
on the mean value property. Bernard Epstein [6] has shown that a domain D in the complex plane
with ZQ G D and possessing the mean value property for some fixed r, 0 < r < -hoc,
/ h(z)dW=r2
h(zo)^
J D
with h ranging over all bounded harmonic functions on D, necessarily is a circular disk centered at
ZQ with radius r. A precompact subdomain D of the hyperbolic manifold D is said to be a mean
value disk - "centered" at ZQ € D with "radius" r,0<r<-(-oo - provided that
f h{z)Q(z)2
d^(z)=r2
h(zo)^
J D
holds for all bounded harmonic functions h on D. The mean value disks turn out to be uniquely
determined by the parameters ZQ and r, just like the circles, and they are simply connected (this
need not be true without the assuption of hyperbolicity). They are the result of a physical process,
a Hele-Shaw flow on the hyperbolic manifold. The Hele-Shaw flow models how the free boundary
evolves between an incompressible viscous Newtonian fluid and vacuum, which occupy the space
between two parallel, narrowly separated infinitely extended surfaces (parallel to the manifold), as
fluid is injected at a constant rate at the source point ZQ. The time parameter corresponds to r2
,
and initially, the space between the surfaces is empty.
III-3
The maximum principle (MP:AL<; 1A) probably holds true holds for mean value disks D. The
proof of this statement, however, remains to be found. Nevertheless, for mean va2ue disks D, we
have found the weaker principle [5]
f\
A^lp ^ 0, U\QD = 0, and -
tz
- <, 0 ==^ u\o < 0, (MP':A2
)
on QD
assuming u is smooth on D. It is convenient to state the result for D equal to ID, the unit disk.
Then the metric o(z) \dz\ should have the reproducing property
I h(z) g(z)2
 dE(z) = h(0) (3.1)
JB
for all bounded harmonic functions h on ©. The Green function for the Dirichlet problem associated
with the weighted biharmonic operator A^r^A on the unit disk D is denoted by Fg . The main
result - equivalent to (MP^A2
) - is the following.
THEOREM 3.1 Suppose the metric o{z) \dz\ is hyperbolic on D, and has the above reproducing
property (3.1). Then 0 <, rjpxD￾The statement of the theorem is false if we keep the reproducing property but scrap the log￾arithmic subharmonicity. These examples also show that hyperbolic geometry lies on the "right"
side of flat geometry, in the sense that for elliptic Riemannian manifolds (with positive Gaussian
curvature) there is no general theorem of the above type.
4. Ideas of the proof.
It is important to note that since A^^AF^-.O) is a unit point mass at <^ it follows that
A,r,(z, c) = e(z)2
 (G(z, 0 + H,(z^ o), (^ C) e D x D,
where
G(^,0=log
-C|2
1-zC
is the Green function for the laplacian, and the function Hg(z,Q is harmonic in z. We shall call
the kernel function Hg the harmonic compensator, because it solves the balayage problem
f (G(z, C) + H,(z, 0) h{z) Q(z)2
 cK{z) =0, C C D,
JD
for all bounded harmonic functions h on ED. An approximation argument allows us to assume
substantial regularity of the metric function ^, such as real analyticity up to the boundary. If the
Green function Yo is positive, then a local analysis near the boundary shows that A^F^(^,^) is
positive on T x ED, and using the harmonicity of Hg in the first variable, it follows that Ho is positive
throughout Dx D. It is much less obvious that if the harmonic compensator is positive for a certain
family of weight functions of the same type as g, then we can go the other way around and obtain
the positivity of F^. This is done with the help of a variational technique due to Hadamard (see
[3, 4]), along domains D(r) given by Hele-Shaw flow on the hyperbolic manifold, with injection
point ZQ = 0 and time parameter r2
, 0 < r < 1, ending with D(l) = D.
The harmonic compensator is related to the reproducing kernel function Qg for the space
HP2
 (ID), o) obtained as the closure of the harmonic polynomials with respect to the norm of L^D, o),
=f/'|/(^)|2
e(z)2
dS(^)'
\JD /
More precisely,
AcJ^O^^O'Q^C).
III-4
If the harmonic compensator is positive, then a local study of the behavior near T x T reveals
that Qe\j2\6^ <, 0, where J(T) = {(z, z) : z € T} denotes the diagonal. We are led to search
for some kind of reverse implication. We first study the reproducing kernel function Kg for the
space P^BD,^) which is obtained as the closure of the (holomorphic) polynomials with respect
to the norm of L2
^^). It is shown that Q^ = 2 Re Kg - 1, so that the information obtained
for Kg can be readily converted to information about QQ. This identity reflects the fact that
under the reproducing condition on g, the analytic polynomials and the antiananlytic polynomials
vanishing at the origin are perpendicular to each other in the Hilbert space HP2
 (ID), g). We obtain
a representation formula for Kg^
^M^^;^0
, (.,C)6©xI3),
whereby Lg is the reproducing kernel for some Hilbert space of analytic functions on D, which
may be called the deficiency space for P^D, g). Using this representation, we find that that Qg is
negative on T2
 \ 5(T), and in fact that
0^0^-(^+^) ^^, (.,C)eTxT\J(T), (4.1)
and just as previously this allows us to go backwards, to obtain the positivity of the harmonic
compensator, by means of a variational technique along the weighted Hele-Shaw flow. We use the
Hadamard variational method for the laplacian to write the Green function G as a negative integral
of a product of two Poisson kernels for the flow domains D(r) over r, 0 < r < 1. Noting that
Hg(' , C) is the orthogonal harmonic projection (with respect to the weight) of the function —€?(•, C),
we find that it suffices to show that the harmonic projection of a positive harmonic function on a
flow region D(r), with 0 < r < 1, extended to vanish on © \ D(r), is positive throughout D. This
is precisely what the estimate (4.1) permits us to do.
References
[1] M. Englis, A weighted biharmonic Green function, Glasgow Math. J., to appear.
[2] P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J.
Math. 1 (1951), 485-524.
[3] J. Hadamard, (Euvres de Jacques Hadamard, Vols. 1-4, Editions du Centre National de la
Recherche Scientifique, Paris, 1968.
[4] H. Hedenmalm, A computation of the Green function for the weighted biharmonic operators
AI^I-^A, with a > -1, Duke Math. J. 75 (1994), 51-78.
[5] H. Hedenmalm, S. Jakobsson, S. Shimorin, An Hadamard maximum principle for biharmonic
operators, submitted.
[6] H. S. Shapiro, The Schwarz function and its generalization to higher dimensions, University
of Arkansas Lecture Notes in the Mathematical Sciences, 9, Wiley-Interscience, John Wiley
& Sons, Inc., New York, 1992.
DEPARTMENT OF MATHEMATICS, LUND UNIVERSITY, P. 0. Box 118, S-221 00 LUND, SWEDEN
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