50
REGULARITY AND SINGULARITY FOR ENERGY MINIMIZING MAPS
R. t1. Hardt *
1. INTRODUCTION
We will consider the occurrence of singularities in a class of
boundary-value mapping problems. Suppose M is an m dimensional
smooth compact Riemannian manifold with boundary and N is a smooth
comJ)act Riemannian manifold without boundary. Via an isometric
embedding. we view N as a Riemannian submanifold of IRk . We will
consider the following type of problem:
Given a smootll tp : aM
-+
I = tp.
N • find a least energy u : M -+ N w1t11 u aM
While various general energy functionals may be treated, we will mainly
discuss. for 1 < p < oo • the ordinary p-energy
Here, the most important case is p = 2 where critical points are
17armonic maps. In local coordinates x1,x 2 , • • • .xm on M. the
expression I'Vu !P should be interpreted as
l~o:.B ~i.j
(aui/exo:)go:.B (euj/ex 8)]P 12 and the volume element dM as
(det g)!1 dx where g = 9o:$ = [go:,B t 1 is the matrix representing the
metric of M in these coordinates. Since only the topology and geometry
of N will be relevant for our discussion of regularity and singularity, we
will, for simplicity of notations, assume that M Is an open subset of
IRm wlt/7 tl7e standard Euclidean metric.
*
Research pert1e11y supported by the Net1one1 Sc1ence Foundet1on
51
2. SINGULARITIES BY TOPOLOGICAL OBSTRUCTION.
Topological obstructions may be relevant for the existence o1·
regularity of least
maps. Perhaps
simplest example concerns
the case
maps
tile unit ball IBm in !Rm to the sphere 5i 11H = 81Bm.
Here an elementary topological condition is
(;.<) there exf...,ts a
map u : [im ~ ;sm-1
onl._1-1
u a!W' IJas degree 0 .
I
For p 2. rn , (~)
that there exists no function u :iBm
I
-v
s;m-l o;
flmte p-energ!J whose trace u a!W' has nonzero degree.
Jn fact, for p > m , a finite energy u would be essentially continuous by
Sobolev embedding while
p = m , one could suitably approximate u by
a continuous map to contradict ( * ). Thus for p ;: m , the least energy
boundary-value problem iT1ay be meaningless.
On the other hand, if 1 < p < m , t~1en there is such a finite p-energy
extension given, for example, by the r1omogeneous-degree-0 extension,
w(x) = tp()</lx I) . In fact, using spherical polar c:oOi'dinates,
In t11is case, the boundary-value problem is meaningful, but (*) implies
that a t:''Oiut!on vdl! necessari(t; be discontinuous on JBrn If the given tp
has nonzero degree (e.g. tp = identity).
ln general. if there exists some finite p-energy extension of tp ,
one easily obtains the existence of a solution of the least p-energy
boundary value problem by
methods using the weak compactness,
lower semi-continuity, and trace theory in the space L1·P (M.IRk) [KJF] of
functions of finite p-energy. Of relevance here is the fact that any
sequence weakly convergent in L1·D Is strongly convergent in L0·D = LD .
In particular, a weakly convergent sequence in
t~1en
L1·P (!'1
= L1·P
,!RI:::) n { u : u(x) t: N a.e, in M }
52
, and we may minimize in tr1e class
has limit in
the sense
L1·P traces)) .
3. SINGULARITIES WITHOUT TOPOLOGICAL OBSTRUCTION.
example
a smooth map <fl : sm~1 ..,
there is a definite
between the two numbers
ln
degree 0
In particular the energy minimizer must have a singularity even though
there is no topological restriction in the sense that there does exist some
continuous finite-energy extension of IJ) . The gap may be made
large
choice
(p . To explain the idea this
construction we
describe an analogous problem where 1 < p 2 = rn ,
N = $ 1 , and M is a region in the plane shaped
a barbell
a thin
handle of length L . The boundary data lfl is given by a unit vectorfield
on aM as shown.
0
L
As a map from aM to s 1 , <P
degree o . By considering a comparison
function that is approximately constant on the handle and homogeneous of
degree 0 on each end, we readily check that
53
On the other hand, suppose that v E nc 0([8m, ~m-l) . Then, fot' all
0 <A.< L • v 8M, has degree o where M>, is the subregion of M
indicated above. Then the restriction of v to the ver-tical slice S:;, must
almost cover ~ 1 because v oM"~~= 'fl SMA~~ almost covers § 1 .
This, along with Holder's inequality, gives a lower bound for the energy of
the slice:
I
I
By Fubini's U1eorem, JMI\7v1P dx L
I
cL
_, oo as L "* oo
4. PARTIAL REGULARITY THEORY.
Having seen tr1at singularities in solutions are often unavoidable, we
now discuss estimations on the their size. First we give a brief summary
of what is known.
In case p = 2, and there is some restriction on N (e.g. negatively
curved or lying in a coordinate neighborhood) there are several interesting
early works, e.g. [ES]. [Hal. [HKW]. For discussion of these and many other
works, we refer to the excellent surveys of S. Hildebrandt[Hi] and J. Jost
[J 1]. [J 2] . In case p > 2 and N lies in a coordinate neighborl'wod, the
work of N. Fusco and J. Hutchinson [FH] and of M. Giaquinta and G. Modica
[GM] implies, among other things. the partial C1·o: regularity of an energy
minimizer, for some 0 < 0< < 1 .
With no restriction on N and p = 2 , the fundamental work of R.
Schoen and K. Uhlenbeck [SUd (See also [GG].[SU 2J.[JM].[SU 3 ] ) showed
that tr1e interior singular set of an energy minimizer has Hausdorff
dimension at most m- 3 . The study of liquid crystals leads to
consideration of a more general energy functional with quadratic growth
for mappings from 3 dimensional spatial domains to ::0 2 . For these
minimizers. R Hardt. D. Kinderlehrer. and F. H. Lin [HKLd showed that the
singular set had 1 dimensional Hausdorff measure zero. For p > 2 and
more general energy functionals with p-power growth, S. Luckhaus
[L 1UL 2 ] established the C1·« regularity of minimizers away from a
singular set of dimension m- p . ln the independent
[HL 2 t p-energy
minimizers, for any 1 < p < oo, were shown to be C1·o: regular away from
an interior singular set of dimension at most m-[p]-1 . For more general
with p-power growth, arguments in
lead to the (not
necessarily optimal) dimension estimate m-p-e for some positive £ .
Next we will sketch some of the arguments used in [HKL 1], [HL 2l. and
[HKL 2]. The part ia I regularity proof goes in two steps:
Step L Partial Holder continuity
Step ll. Locally Holder continuity implies higher regularity
For Step ll, one may localize in M to reduce
the case that N is a
graph, {(y. f(y)) : y E 0 } for some open 0 c IR~"~ and smooth f : 0 -? IRI<-n .
Then we may write a minimizer u as u(x) = (u(x), f( u(x))) . hence.
V'u = (\i'u, V'f o\i'u) • and examine the functional minimized by u . For
p =2 ,
satisfies an elliptic system of diagonal type to which standard
regularity theory [G, VII, 3] applies. For p ;.: 2 , more argument is required
and the highest regularity that can be expected is C1·o: for some o < 0< <1
(although there may be a partial higher regularity result ). For p > 2 , the
corresponding problem with N replaced by IRn was first treated in the
work of K. UhlenDeck [Ul. To obtain Step ll in [HL 21 for all p witr1
1 < p < 2 • we combine arguments of E. DiBenedetto [Dl and P. Tolksdorff [T].
u
For Step I an important notion is that of the normalized ener{/...11
Note the appropriateness of the factor rp-m ; for a homogeneous degree 0
function u . 1Er.o (u) is independent of r . The use of normalized energy
in [HL 2). as previoiusly in [SUd and [HKL 1J. is motivated by
MORREY'S LEMMA.[M, 3.5.2] If c >0. 0 < 0< < I . and lEr,a (u) 5 c r po: for all
balls !Br(a) c IB = rs,(o) tl7en u I~ E co.o: .
I
While we wish to show such uniform power decay. we can at least get the
normalized energy arbitrarily small by taking a sufficiently small bail
centered about most points in the domain by the following:
55
function u E L1·P (M, N) , the ener_t1f:l density
B(a)"" lim supr.!.o!Er.a(u) equals 0 atallpoints aEM~Su lo.rsr.1.rneset Su
DENSITY LEMMA. For
having m- p dimensional Hausdorfl measure zerr..J.
This follows, as in [SU 1 , 2.71. from an elementary covering argument. In
case u is a p-energy minimizer, Holder continuity off of Su novv follows
by iteration and scaling from the:
REGULARITY LEMMA There exists a pos!live number e < 1 so that ;f
u E L1·D (!B,N) /sap-energy minimizer with IE 1•0 (u) < e , then
IEe.o(u) < ~IE,,o(u).
This may be proven by arguing by contradiction or "blowing-up". Here, as
in similar situations in geometric measure theory and elliptic systems, a
key problem is controlling the "blow-up sequence" whose convergence is
initially only known to be weak in a Sobolev norm. The extra ingredient is
a "caccioppoli-type" inequality which in this context follows from the:
COMPARISON LEMMA. There exi<>ts positive constants
0 < A< oo and any \j) : 8!B
funct /on
w : IB
~N
with
Jao/'V tan o/ IdS < e . there exists a
I = 'P
~ N w;th w 818
c. c. so that for any
and
This was first proven for p = 2 in [SU 1 • 4.3] and generalized for p "'2 in
[HL2l and lL2l .
All of the discussion so far carries over to more general funct ionals
having p-power growth. For a minimizer u of the ordinary p-energy, one
has the additional
MONOTONIC!TY PROPERTY. [r,a (u) ~ IEs,a (u) whenever 0 < r < s < dist(a, oM).
Using tllis along with a discussion of homogeneous minimizers and an
induction argument of H. Federer [ft one may, as in [SUd, show that the
singular set Su has not only m- p dimensional measure 0 . but even
Hausdorff dimension m- [p ]- 1 . Here Su = 0 in case m < [p J+ 1 and is
finite in case m = [p] + 1 . Using tr1e exampies of singularities discussed
56
in
one can easily show that the dimension estimate is
5. DENSITY
ln [HL 2 ] it was observed that
tf N
is sirnp/.._4'
-1 connected(i.e.
then one may delete the :5m:tl!ne::ss
=
IdS <
o),
£,
t/7(!
Comparison Lemma.
The proof
this is different
the
construction
[SU 1, 4.3] and involves choosing projections from a generically-situated
complex in the manner of [FFJ or [W]. The idea is easily described in case
N = Sin . One first verifies that the IRn•l -valued harmonic extension h of
tp satisfies the inequality in the conclusion
the Compar-ison Lemma.
course, h probably does not have
in Sin . To correct
we take
w=
h where TI8 : IR 0 ' 1
~ sn is an appropriate retraction and
a E IB!/z . To see that a suitable a can be found, one uses Fubini's theorem
to obtain an estimate
Next we examine some easy consequences
t~1e new Comparison
Lemma. We assume u t: L1·D (M.
is a p-energy minimizer and 1Br(a) c M.
(1) 1Er;2,a (u) s :A.IEr,a(u)
(2) 1Er/2,a (u)
s
(1
+
(3) El(a) S ( 1+ C) 1+D
(4) 'i7u
E L~oc
+
C:A.-P where C dependson!yon m,N, andp.
C)[IEr,a (u)]Pf1+p
2m-p .
for some q > p and dim(Su) < m-p.
To verify (1) we use Fubini's Theorem to choose s with
that
< s < r so
57
apply the new Comparison Lemma with <P(x) = u(sx) and note that u llBs
is minimizing and that N is bounded. Inequality (2) follows from (1) by
0
taking A.
= [[r,a (u)}-1il•D .
To obtain (3) we iterate (2) to see that
and then let j-!' oo. For (4) one may combine (2) with an appropriate
"boundary-Sobolev" inequality to get a reverse Holder-type inequality and
then argue in a marmer similar to [G, §5] (See [HKL 2] for details).
6. NATURE OF THE SINGULAR SET.
The best results on the behavior near singularities are known for
J
ordinary energy MI'Vu 12 dx when m = dim M = 3 . Here by [SUd and the
study of asymptotics by L. Simon in [Sd. there exists, for any singularity
a of an energy minimizer u a smooth harmonic map v: $ 2 ... N such that
0
lf, moreover N = ~ 2 • tr1en R. Gulliver and B. White [GW] have verified that
an easier asymptotics result [5 2] is applicable so that nds convergence is
as fast as a positive power of r . They also find a real analytic N and a
minimizer u : tB 3 _. N for which positive power decay to the tangent map
fails.
Harmonic maps from $ 2 to $ 2 are classified by rational functions
of z or z [J 2 , 1.5]. By the general bound (3) above, a homogeneous
degree 0 extension of such a map will not be minimizing if the degree of
this map is too large. Recently H. Brezis, J. ~1. Coron, and E. Lieb [BCL] have
shown that any such homogeneous minimizer is. in fact, of degree 1 and is
precisely in the form g(x/lxl) for some rotation g E 10(3) .
In general, there are many open problems on the nature of the singular
set of an energy minimizing map. One would like to extend some of the
58
above results to handle m > 3 or p ""' 2 or more general energy
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School of Mathematics
University of Minnesota
Minneapolis, MN 55455 USA