A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
A VNER F RIEDMAN
Y ONG L IU
A free boundary problem arising in magnetohydrodynamic system
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 22, n
o3 (1995), p. 375-448
<http://www.numdam.org/item?id=ASNSP_1995_4_22_3_375_0>
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Magnetohydrodynamic System
AVNER FRIEDMAN - YONG LIU
1. - The
plasma problem
We shall consider a two-dimensional
problem arising
inmagnetohydro- dynamic (MHD) system, modeling
theplasma
confinement in the Tokamak machine.Let Q be a bounded domain in
]R2, representing
the cross-section ofperfectly electrically conducting
shell. As shown inFigure 1,
theplasma
fillsa subdomain
SZp
ofQ,
withboundary rp;
thecomplimentary
domain isthe vacuum
region 0~.
Figure
1Let
B, J
and P denote themagnetic induction,
the currentdensity
andthe fluid pressure,
respectively.
Inequilibrium,
thetriple (B, J, P)
satisfies the MHDsystem:
Pervenuto alla Redazione il 24 Marzo 1994 e in forma definitiva il 21 Novembre 1994.
where po is the
magnetic permeability
of the vacuum, v is the outward unitnormal, B2 - B . B,
and thesubscripts
p and vdenote, respectively,
thesubscripted quantities
from the interior of theplasma region Qp
and the vacuumregion
The first two
equations
in(1.1)
and(1.2)
are Maxwell’sequations,
thethird
equation
in( 1.1 )
is theequation
ofplasma
motion in thestationary
case,and
(1.4)
is the pressure balance condition.Since we are
considering
here the two-dimensionalproblem (where
Q isviewed as a cross-section X3 = 0 of an infinite
cylinder),
B,J
and Pdepend only
on the variables x 1, 3:2.Since div
B = 0,
we can introducelocally
a flux function uby
(assuming
that the vectorB
lies in the two-dimensional spaceJR2); since, by (1.3), au/aT
= 0 onrp
where T is the unittangent
vector, u is asingle-valued
function in the whole domain
Q, uniquely
determined up to an additive constant.It is easy to
verify
thatand
then, by ( 1.1 )
and(1.2),
Since
f x d
= JVu, the thirdequation
in(1.1)
becomesTherefore VP is
parallel
to Thisimplies
that Pdepends only
on u, thatis,
where g is an unknown function. This function cannot be determined
by
theMaxwell
equations;
it is a constitutive function for theplasma. Equations (1.9)
and
(1.10) imply
thatso that
(1.7)
can be written in the formThe
boundary
condition(1.3) implies
thatwhere r is the unit
tangent
vector. It follows that u is constant onFp
and ona S2,
and we determine uuniquely by taking
We also set
where ^I
is anunspecified
constant which we take to bepositive.
Finally,
condition(1.4)
can be written asand,
ingeneral, g(o)
> 0. This conditionformally
coincides with Bernoulli’s condition for two-fluid flow.Temam
[21, 22]
considered the model within this case P = 0 on
rp
and thesystem (1.8), (1.12), ( 1.14)-( 1.16)
reduces towhere
More realistic models with P > 0 on
rp
were consideredby Grad,
Kadish and Stevens[15]
andby Ushijima [23].
Here we shall assume the constitutive lawso that P > 0 on
rp.
In
[21]
A =2auo
isunspecified positive
constant whereas either the cons-tant ’1 or the flux
are
specified; furthermore,
the conditionis
imposed.
In[22]
A isprescribed
but the condition(1.20)
isdropped.
In our formulation of the
plasma problem
we shall assumethat q
and bare
given,
and that a is unknown(which
means that Au- + Au- = 0 holds inQp
with unknownpositive
constantA).
We alsoimpose
the additional condition(1.20),
where K isgiven.
Other formulations are also
possible;
forexample,
one mayprescribe
A and
drop
the condition(1.20).
Since we do not prove hereuniqueness
andcontinuous
dependence
of the solution on theparameter
K, the two formulations may not beequivalent. Furthermore, when
isunprescribed and, instead,
the condition(1.19)
isimposed,
weget
yet another different version of theproblem.
We have chosen the
present
formulation ofprescribing
~y, b and K because it istechnically
the easiest: it lends itself moredirectly
to a variational formulation.We
hope
to consider other formulations in a later work.In the next section we rewrite the
plasma problem
withslightly
normalizedconstants, and describe the main results and the structure of the paper.
Many
ofthe results of this paper are valid in n-dimensions. For this reason we shall assu- me that Q is n-dimensional where n > 2, unless the
contrary
isexplicitly
stated.2. - Mathematical
formulation;
the structure of the paperLet Q be a bounded domain in Rn with
C’ boundary aQ,
andlet 03B3
andJ.t2 be given positive
constants. Theplasma problem
is to find a function u, aclosed surface
rp lying
in S2 and apositive
constant A such thatand
here
Vu-(Vu+)
denotes the limit of Vu fromIn this paper we shall prove that for n = 2 there exists a solution to the
plasma problem
and thatFp
iscontinuously
differentiableand,
infact, analytic;
the open set
Qv
isconnected,
butQp
may have several componentsG 1,
...,GN .
There is howeveronly
onecomponent,
sayG 1,
in which u 0everywhere;
inall the other components u vanishes
identically.
For thesimpler plasma
modelwith
(1.17),
it is known[19] (see
also[13;
pp.529-531])
that for any variational solutionQp
has asingle component.
For n > 3 there still exists a solution of the
plasma problem
and u isLipschitz continuous,
but the freeboundary
condition(2.5)
is satisfiedonly
ina weak sense.
(We
do not establish here theregularity
of the freesurface).
Our
approach
is based on variational methods of Alt and Caffarelli[2],
andAlt,
Caffarelli and Friedman[3],
and the more recent papers of Caffarelli[5] [6] [7].
The structure of the paper is as follows:
In Section 3 we introduce a variational formulation of the
plasma problem
with a functional
J(u)
as in[3]
but with theintegral
constraint(2.6).
Weestablish several
simple properties
of theminimizers,
such assubharmonicity,
and a weak form of the free
boundary
condition. However theintegral
constraintdoes not allow us to
adapt
any of thedeeper
results from[3].
We canpartially
overcome this
difficulty by introducing,
in Section4,
another variationalproblem
which does not have any constraints in the admissible classes
but, instead,
hasa new and a bit more
complicated functional, J,,(u), namely:
where
and 11
« 1.Any
minimizer ofJ,,
is shown to be also a minimizer ofJ,
and for the rest of the paper westudy only
minimizers ofJ~ .
In Section 5 we derive Ca
regularity
of any minimizer u, and in Section 6 we provenondegeneracy.
These results are obtainedby adapting
methodsof
[3].
The next fundamentalresult, proved
in Section7,
is themonotonicity
theorem which asserts that
is monotone
increasing
in r. This is an extension of amonotonicity
theoremproved
in[3]
in case A = 0, and it can beused, precisely
as in[3],
to establishthe
Lipschitz continuity
of u.In Section 8 we consider
blow-up
sequences andidentify
their limits. In Section 9 webegin
the initialstudy
of the freeboundary, showing
that Au+ and Au- + Au- are measuressupported
on the freeboundary, absolutely
continuouswith
respect
to The results of Sections8,
9 are similar to those in[3];
see also
[2].
In
establishing
theregularity
of the freeboundary
in[3]
thefollowing comparison
result(Lemma
8.1 in[3])
was crucial: if u, u are minimizers andu > u on the
boundary
then u > u in the entire domain. This lemma is nolonger
true in our case(where
A >0).
We thereforeproceed
inentirely
differentway,
adapting
the recent Harnackinequality approach
of Caffarelli[5] [6]
forweak solutions.
We
begin,
in Section10, by introducing
weak solutions and subsolutions.In
[5]
Caffarelli constructed alarge
class of subsolutions which he used to establish theregularity
of "flat" free boundaries. In our case, where A > 0,we can
only
construct a much smaller class of subsolutions. Howeverby exploiting
additional ideas from[3]
we shall beable, nevertheless,
to establish(in subsequent sections)
theregularity
of the freeboundary.
In Section 11 we introduce
(as
in[6])
theconcept
ofe-monotonicity
andprove
(Theorem 11.3)
that(i) e-monotonicity, (ii) flatness,
and(iii)
fullmonotonicity
at theboundary
imply Lipschitz continuity
of the freeboundary
of a weak solution.In Section 13 we
improve
Theorem 11.3by showing
thatassumption (i)
is
always satisfied;
this is Theorem 13.2. Theproof
which is anadaptation
from Caffarelli’s paper
[6] requires
severalauxiliary estimates,
some of whichare
proved
in Section 12.(In
case A = 0 the condition(iii)
was not assumedin
[6];
it was in fact verifiedby using
the muchlarger family
of subsolutions available in case A =0).
In Section 14 it is shown that minimizers of
Jq
are weak solutions. At thispoint
we go back toadapt
some results from[3]
in order to show that theassumption (iii)
above is satisfiedby
any minimizer. Since for n = 2assumption (ii)
is also satisfied(as proved
in Section 8 of[3]),
we conclude that for n = 2 the freeboundary
isLipschitz
continuousand,
infact,
it is alsocontinuously
differentiable and
Vu±
is continuous up to theboundary.
In Section 15 weestablish the
G1+a
smoothness of the freeboundary;
theanalyticity
of the freeboundary
then followsby
knowntechniques.
The sets
Q+(u) = {x
E >0}, S2- (u) - int{x
EQ, u(x) 0}
are open and contains aneighborhood
of aS2. In Section 16we prove that is
connected,
whereas consists of a finite number ofcomponents G 1, ... , GN . In just
onecomponent,
sayG 1, u
0throughout.
In all other
components Gj, u =-
0; suchcomponents
do in fact exist for some domains Q.’
3. - A variational
problem
We introduce the admissible class
and the functional
where it-, it, are
positive
constantssatisfying
and
IA
denotes the characteristic function of a set A.PROBLEM
(J).
Find u c K such thatThis variational
problem
is ageneralization
of the one introducedby
Temam[21]
for the case /~+ = /~_ = 0.THEOREM 3.1. There exists a solution
of problem (J).
The
proof
is the same as for Theorem 1.1 of[3].
We denote a solutionby
u, andproceed
to establish some basicproperties.
THEOREM 3.2. The solution u is subharmonic in il.
PROOF. Take
any 03BE
ECol (12), >
0 and e > 0, and setand
Then u,
c K andwhere we have used the fact that sgn uc =
sgn Üg.
Since a, = 1
+ o(l)
andwe have
It follows that
and the
proof
iscomplete.
For the next result we
temporarily
assume that u is continuous in SZ. Thenare
nonempty
open subsets of Q andi2p
c C Q.THEOREM 3.3.
If
u is continuous inQ
thenwhere
and
0
PROOF. For
any
ECol (Up)
0with I -I sufficiently
small so that0
u -
c
0 in and define Ue asbefore, by (3.3), (3.2).
Then sgn u = sgn u0
and u, = u in
and,
since u, E K,Recalling (3.4)
weget
i.e., (3.5)
holds. Theproof
of(3.7)
is similar.The set
rp
= is called thestrict free boundary.
The next theoremshows that u
satisfies,
in ageneralized
sense, theequation
provided
the set{v,
=0}
has zero measure.THEOREM 3.4. Let
SZo
be any open subsetof
5~2.If
u is continuous in00
andmeasfx
E00, u(x)
=01
= 0then, for
any ~ Ewhere v is the outward normal.
Here 6 and 6 are such that
-s}
anda{u> 6}
arecontinuously differentiable; by
Sard’s theorem this is true for and 6.PROOF. For
simplicity
of notation we takeQo
= S2.Following
theproof
of Theorem 2.4 in
[3]
wedefine,
for any with smallI e 1, diffeomorphism
by r,(x)
= x + and setUg(Tg(X»
=u(x).
Next we introduce a functionThen u,
E K andSince
we
obtain,
aftersimple calculation,
Noting (cf. (3.4))
thatand that the linear term in c in the
preceding inequality
mustvanish,
weget
We now use the
assumption
that measlu
=0}
= 0 to conclude from the last relation thatNotice
that,
in{u
>61,
Au = 0 andwhere in the last
equation
we made use of theidentity
Similarly,
in andTherefore we deduce from
(3.9)
thatwhere in the last
equality
we made use of the fact that onlu
=6}
and on{u
=2013~} respectively;
this is the assertion(3.8).
REMARK 3.1. If u > 0 in
S2o
andmeas{x
E > 0 then we haveIVu+12
=112 2
=J-L2
onrp
nS2o
in the sense that4. - Another variational
principle
The constraint in the definition of the admissible functions poses serious difficulties
to establishing
furtherproperties
for any minimizer u(by trying
to extend theanalysis
of[3]).
We need to have a more "flexible" class of ad- missible functions. This however willrequire
a modification of the functional J.Let q
be apositive
real number and letWe introduce the admissible class
and a new functional
PROBLEM
(J.).
Find u EKo
such thatTHEOREM 4.1.
If q
is smallenough
then any continuousfunction
u in S2which is a solution to
problem (J,~),
is also a solution toproblem (J).
. PROOF. All we need to prove is that any solution to
problem (J,~)
satisfiesSuppose
Then
and,
since u iscontinous,
is open.Take
whereIf lel
is smallenough
then sgnuc = sgn u, and weeasily get
It follows that
and, therefore,
o
Since
np
c n we conclude that 7y >a 1 (SZ)
whereAi(Q)
is the firsteigenvalue
of A in Q. Thus if we
choose q Ai(Q)
then(4.5)
cannot hold.Suppose
next thatObserve that
Therefore
Since
J(u)
> 0, we conclude that0
consequently Proceeding
as before wederive, analogously
to(4.6),
~
0
Multiplying by
u andintegrating
overS2p
results inRecalling (4.8), (4.9),
we obtainwhich is a contradiction if q
1 /(2C).
From now on we
study only
the minimizers ofJq,
and assumethat q
issmall
enough
so that any continuous minimizer is also a solution toproblem
(J).
5. - Ca
regularity
THEOREM 5.1. There exists a solution to
problem (J,,).
Indeed,
theproof
is the same as for Theorem 1.1 of[3].
THEOREM 5.2.
Any
solution u toproblem (J,,)
is subharmonic in Q.The
proof
isthe
same as for Theorem 3.2provided
we choose ae =I 1 11 ’"
so that
By
Lemma 10.1 in[13;
p.90]
there is a version of the subharmonic function u such thatand this
implies
that u is upper semicontinuous. It follows that the set0
Lip
=lu 0}
is openand, by
theproof
of Theorem3.3,
By
theisoperimetric inequality
for theprincipal eigenvalue,
the volume of each0
component Di
off2p
is bounded from belowby
a constantdepending only
onA. Hence there is
only
a finite number of suchcomponents Di.
Sinceby elliptic
estimateswhere ci > 1. It follows that
Since u is
subharmonic, by
the maximumprinciple u
in Q.Thus, altogether,
THEOREM 5.3.
Any
solution u toproblem
is inCO(O) for
any0a 1.
PROOF. Let
Br
cc Q be any ball of radius r and denoteby Vr
the solution toExtend vr by
u toS2BBr . Then, by
theminimality
of u,It follows that
where we have used
(5.1 ).
Proceeding
as in[20;
Th.5.3.6] (cf. [2])
one can use the last estimate toestablish the bound
so that
and
then, by [20;
Th.3.5.2], u
is inTo prove that u E C" near the
boundary,
we flatten theboundary, i.e.,
fora
given
xo E aSZ and a small ballBp(xo)
we introduce aC2 diffeomorphism
y = from
Bp(xo)
f1 Q onto the half ballLet vr
be harmonic function inwith v,,
= u on Consider thefunction i3r
= vr inBr (0)
and extend it toB~(0) by
Similarly
define u = u and extend it in the same way toB; (0).
As beforewe have
11 .
(vr
has been extended into Qby u),
and this leads towhere
where is a
positive
definite matrix with bounded measurable coefficients.The
technique
of[20]
can theagain
be used todeduce
that ii E forsome 0 ri 1, and therefore u E
(xo)
nS2)
for some PI > 0.REMARK 5.1. The C" norm of u in any
compact
subset G of Qdepends only
on M, 6and tt2
where M >sup lul
a
From Theorems 5.3 and 4.1 we have:
COROLLARY 5.4.
Any
minimizerof (J,7)
is also a minimizerof (J).
NOTATION. In order to indicate the
dependence
ofSZ"
andS2p
on u, weintroduce the notation:
DEFINITION. The
free boundary F(u)
is definedby
REMARK 5.2. We claim:
Indeed,
ifzo g F(u)
while xo E then there is aneighborhood
N ofxo such that u 0 in N. Since u is
subharmonic,
either u 0 in N or u =- 0 in N. In either case N c so thatFinally
if xo E aS2 thenu(xo)
> 0 andagain
xog aS2-(u).
REMARK 5.3. In Section 15 it will be shown that for some domains Q the set
lu
=0}
hasnonempty
interior. This situation does not occur ifU2
= 0and n = 2; see
[ 19].
6. - Lower and upper estimates on averages
The results of this section are similar to results obtained in
[3]
in case theadmissible class in
Ko (defined
in(4.2))
and the variational functional isJ(v);
the fact that we are
dealing
here withJq(v)
instead ofJ(v)
causesjust
minorchanges.
We shall refer to[3]
whenever thechanges
are obvious.THEOREM 6.1
(Nondegeneracy).
For any 0 /c 1 there exists apositive
constant c
depending only
andlz2
such thatif Br
C Q andthen u+ = 0 in
The
proof
is similar to theproof
of Theorem 3.1 in[3];
the termcancels out the term where v is the modification of u, as defined in
[3].
COROLLARY 6.2.
If Br
C Qwith
center in thefree boundary a ju
>O} f1 SZ,
then
where c
depends only
onTHEOREM 6.3. There exists a
positive
constant cdepending only
on112
and such that
if Br
C Q with center in{u
=01
thenPROOF. The
proof
is similar to theproof
of Theorem 4.1 in[3];
theonly
difference occurs in the
proof
of Lemma 4.2 of[3],
which asserts thatTo prove
(6.2)
define vby v
= u inS2BBr
andFrom and the uniform bound on u
(see (5.1 ))
we obtainThe left-hand side is
equal
towhere Au is a measure
supported
in{u 0}.
Hencewhere we have
again
used the uniform bound(5.1 ).
We can nowcomplete
theproof
of(6.2) precisely
as in Lemma 4.2 of[3].
7. - The
monotonicity
theoremN
THEOREM 7.1
(The monotonicity theorem).
Set N =max{n - 1, 2}
and let2 + N
a 1.Suppose
u is a minimizerof (J,~ ).
Take anypoint
xo in thefree boundary a {u
>O}
n Q and denoteby B,
the ball in Q with center xo and radiusr.
Then there exists apositive
constantdepending only
onp,2,
a andthe
of
u such that thefunction
where is monotone
increasing
in r.In the
special
case where U:i: are bothharmonic,
this result is due toAlt,
Caffarelli and Friedman[3];
in their case, Ka = 0. The mainpart
of theproof
of Theorem 7.1 follows
closely
theproof
in[3];
however we shall alsoneed,
for thepresent
case, the additional lemma:LEMMA 7.2. Under the
assumptions of
Theorem7.1,
there exists a constantdepending only
on a,lt2
and the a-Holder normof
u, such thatPROOF. Denote
by Sr
thesupport
of ut onS’r
=By Corollary 6.2,
where denotes the Hausdorff measure of
By nondegeneracy
there exists apoint
yr ES’r
with > cr(c
>0).
S ince u E C" we then
have,
for any x E(0 1)
Therefore
where xm is the volume of the m-dimensional unit ball.
It follows that
Denote
by Vov
thegradient
of a function von Sr
and introduce constants(which depend
onr)
I-and
Then, (cf. [3;
p.441])
and, by
an estimate of Friedland andHayman [12]
on the functionswhere
Hence, by (7.3),
By
the definition ofa±
and of0.1 (in (7.4)),
so
that, by (7.6),
If n = 2
then,
since{3; =
1, we havewhere we have
again
used(7.6);
hence(7.8)
is valid also if n = 2.Finally
ifH(Sr-)
= 0 then(7.8)
istrivially
true.Multiplying (7.8) by r2-n
andintegrating
from 0 to p, the assertion(7.2)
follows.PROOF OF THEOREM 7.1. Consider the function
Then a.e.
We first wish to prove that
Without loss of
generality
we may assume thatsince, otherwise, (7.10)
followsimmediately
from(7.9).
Proceeding
as in theproof
of(5.2)
in[3]
weget
The
proof actually requires
that(7.12) Au+
and Au- + Au- arenonnegative
measures,a fact that will be established in Lemma 9.2
(by
aproof
which isindependent
of Theorem
7.1).
One uses(7.12)
and theregularization procedure,
as in[3]
to establish the above
inequalities. (Actually
a careful look at theproof
showsthat
equalities hold). Also,
as in[3; (5.6)],
so that
and
Substituting (7.13)
and(7.14)
into(7.9)
results inSince, by [3; (5.7)],
and since we have assumed that
(7.11) holds,
the assertion(7.10)
follows.We now substitute the
inequality (7.2)
of Lemma 7.2 into(7.10)
to deducethat _
from which the assertion of Theorem 7.1 follows.
_
THEOREM 7.3.
Any
solution uof problem
isLipschitz
continuous inn.
The
proof
in the interior ofQ,
based on Theorems 7.1 and6.3,
is similar to the secondproof
of Theorem 5.3 in[3]. (The
fact that Theorem 7.1 differs somewhat from themonotonicity
formula in[3]
causesonly
minorchanges).
Inthat
proof
the Harnackinequality
and ascaling argument
were used. Note that the Harnackinequality
forpositive
solutions of Au + cu = 0(c
>0)
is valid[14]
and the constant in the
inequality depends only
on the sup-norm of c(and
thecompact subdomain).
Since thescaling changes
theequation
Au + Au = 0 intoan
equation
AV +p2AV
= 0 with psmall,
we can use the Hamackinequality (as
in
[3])
with a constantindependent
of thescaling,
and this allows theproof
ofTheorem 7.3 to
proceed
as in[3].
To prove the
Lipschitz continuity
near theboundary
of SZ we recallthat,
by
Theorem5.3, u
is continuous in Q. Since u = q > 0 onaL2, u
ispositive
insome
Q-neighborhood
N of 90. Hence Du = 0 in Nand, by elliptic regularity,
u E U
aSZ)
for any 0 a 1.8. -
Blow-up
limitsSuppose u
is a solution toproblem (J~ )
andThe sequence , ..
is called a
blow-up
sequence withrespect
to Since C in any bounded set andUk(O)
= 0, anyblow-up
sequence has asubsequence
forwhich
--~
UO(X) uniformly
in bounded sets,uo is called a
blow-up
limit.LEMMA 8.1. As k - oo,
locally
in theHausdorff
metric,The
proof
is the same as for[3;
Lemma6.1].
Set 18
DEFINITION 8.1. A function u is called a minimizer of
(JO)
in or aglobal
minimizer, if for anyB,
C JRn and for any v EH1(Br)
with v = u onLEMMA 8.2.
Any
blow up limit uo is a minimizerof (JO)
inPROOF. Since u is a minimizer of
(J.),
that v = u outside
Bpk ,
further,
forany v
suchor,
by
thechange
of variables X +Choosing,
as in theproof
of Lemma 5.4 of[2], ~
= V +(1 - UO)
where~ E and
letting
k - oo in(8.2),
we then obtain(cf. [2])
the assertion(8.1)
for B =Bi.
Theproof
for any.Br
is the same.THEOREM 8.3.
Suppose
D cc il, C D withcilu
>o}
n Q. Then ...where c is a constant
depending
on theLipschitz coefficient of
u in D.The
proof
is the same as for[3;
Theorem6.3].
Consider a
blow-up family
and let
Note that the us
satisfy
a uniformLipschitz
bound and therefore Ka is inde-pendent
of c.By monotonicity, 1.
isincreasing
function of p.Consequently
there exists a
nonnegative
constant Q such thatFor any
converging blow-up
sequence Uek -~ uo we have thatuo(x) uniformly
in boundedsubsets, Vuo(x)
a.e., andby
theLebesgue
bounded convergence
theorem,
where
Br
=Br(O). Combining
this with(8.3)
we conclude:LEMMA 8.4. For any
blow-up
limit uoof {uE}
there holds:Since uo is a minimizer of
(JB),
Lemma 6.6 of[3]
is thenvalid, namely:
LEMMA 8.5. In Lemma
8.4,
(i)
then in JRn;(ii) if
u > 0 and n = 2 thenuo(x) = J.t2(X . e)+ - e)-
inJR2
where e isa constant unit vector, are
positive
constants, andit 2 2_ ~u2 -
2 2
9. -
Properties
of the freeboundary
THEOREM 9.1. There exists a
positive
constant c E(0, 1)
suchthat, for
any ballB,
C il with center inalu
>ol
n Q,PROOF. The left-hand
inequality
can beestablished,
as in[3], by
thenondegeneracy
andLipschitz continuity
of the solution. To obtain theright-hand inequality
let v be the solution ofThen v > u in
Br
andBy
Poincare’sinequality
Since u v, the
inequalities
0 u- -I
and u- + v- 2u-hold,
sothat
Choosing 6 =
andusing
the fact that u- isbounded,
weget
2
with another constant c > 0. The
remaining part
ofproof
is the same as forTheorem 7.1 in
[3].
We introduce the distributions
LEMMA 9.2.
A+(A-) is a
Radon measuresupported
ina f u
>O}
fl S2- , -, -
PROOF.
Proceeding similarly
to[2;
Remark4.2]
we have for any~ c- > 0 and c > 0,
A similar relation holds for the test function
Since we obtain
Noting
thatit follows that
by
thecontinuity
of u. Hence Au- + Au- > 0 in the sense of distributions.Similarly Du+ >
0 in the sense of distributions. Since ut arecontinuous,
the lemma follows(cf. [2]).
THEOREM 9.3. For any open set D C C K2 there exist
positive
constants C, c such thatfor
anyB,
C D with center in9ju
>0}
PROOF. We
proceed
as in[2;
Theorem4.3].
To prove the upper bound in(9.2)
we first deduce from theproof
of Lemma 9.2 with suitable~’s
thatfor a.e. r.
Using
theLipschitz continuity
of u, it follows thatThe
proof
of(9.3)
isprecisely
as in[2;
Theorem4.3];
for thelower
bound weneed to use
Corollary
6.2.THEOREM 9.4
(Representation Theorem).
(ii)
There exist Borelfunctions q.’
such thatthat is,
for
any(iii)
For any open set D there existpositive
constants c, Cdepending
on D and the
Lipschitz
constantof
u such thatfor
any ballBr(z)
C D withx E a {u > 0} f1 SZ,
Indeed this follows as in
[3;
Th.7.3];
we use here Theorem 9.3.Since
a{u
>0}
f1 SZ has finite measure, the set A = Q n{u
>0}
hasfinite
perimeter locally
in S2, thatis,
ou> -VIA
is a Borel measure and the totalvariation litul
is a Radon measure; see[2]
and the references therein to[11;
4. 5 .11, 4. 5 .5 ] .
We denoteby
>0 }
the reducedboundary
ofa { u
>oini2,
that
is,
the set ofpoints
for which there exists aunique
unit normal. Then[2]
[ 11; 4.5.6]
THEOREM 9.5
(Identification theorem).
Let Xo Eared {u
>0}
with(i) If,
in Lemma8.5,
(J > 0 and n = 2, thenwhere
e(xo)
is a unit vector, iii arepositive
constants, and(ii) If, in
Lemma8.5, (J
= 0 thenand
Here
vu(zo)
is the outward normal toa{u
>0}
at ~o.The
proof
is the same as for Theorem 7.4 in[3].
As in
[2],
Theorem 9.1implies
thatand
then,
forH n-
I a.a. xo C >0},
theassumptions (9.6), (9.7)
aresatisfied. Therefore for
Hn-1
I a.a. x Ea{u
>0}
the freeboundary
in aneigh-
borhood of xo is
approximately
astraight
line if n = 2.10. - Weak solutions and subsolutions
In order to
study
theregularity
of the freeboundary
for solutions u ofproblem (J,~),
we need to introduce theconcept
of weak solutions and weak subsolutions.In the
sequel
we shall denote the dotproduct x -
y alsoby (x, y).
DEFINITION 10.1. Let
G(s)
be a continuousstrictly
monotoneincreasing
function of s E R with
G(s) >
s if s > 0, and let Q be a bounded open set in R~. Consider a function u continuous in Q andsatisfying:
where A is a
positive
constant.Then we say that u satisfies the weak
free boundary
conditionif for any xo E
F(u)
for whichF(u)
has a one-sidedtangent
ball at xo(i.e.,
there exists a ball
Bp(y)
such that xo E andBp(y)
is contained either inor in
0-(u)),
and
~Q
> 0, a =G(~3),
where v is the unit radial direction ofaBP(y)
at xopointing
intoQ+(u).
The function u is called a weak solution. We shall often writeuv
= CY, uj =(3.
DEFINITION 10.2. A function v continuous in Q is called a subsolution if instead of
(i), (ii)
and(10.1)
it satisfies:and,
for any xo EF(v)
at whichF(v)
has atangent
ballBp(y)
from orfrom
for some
(3
> 0 and a >G(~3).
DEFINITION 10.3. A
point
xo EF(v)
at whichF(v)
has atangent
ball from0+(v) (see
Definition10.2)
will be called aregular point.
LEMMA 10.1. Let v u be two continuous
functions
inQ,
v u inQ+(v),
v a subsolution and u a weak solution.
If
everypoint
inF(v)
isregular,
thenF(v)
nF(u) = 4>.
The
proof,
whicheasily
follows from Definitions 10.1-10. 3 and the strong maximumprinciple,
is the same as that of Lemma 6 in[5].
LEMMA 10.2. Let S2 be a bounded domain and let E
CO(i2)
nL°°(SZ) (i
=1, 2) satisfy:
-
for
0, e > 0. Then thefunction
cp =max{y>, satisfies:
PROOF. Introduce the solution w to
Then > 0 for i =
1, 2, i.e.,
is subharmonic. For any ball B C S2 let h be the harmonic function in B withboundary values p - w
on aB. Thenh > (pi - w on aB
and,
since w issubharmonic, h > ~pa - w
in B; hence h> ~p - w.
We concludethat p - w
issubharmonic,
andconsequently,
We next state a
comparison principle
for afamily
of subsolutions.LEMMA 10.3.
An
whereAn
is theprincipal eigenvalue of A for
the Dirichlet
problem
in a boundeddomain
Q. Let vt, 0 t 1, be afamily
of subsolutions in Q,
continuous inQ
x[0, 1],
and let u be a weak solution continuous in Q. Assume thatis nonempty, u on aS2, and vt u on
(c)
everypoint
xo EF(vt)
isregular,
and(d)
thefamily K2’(vt)
is continuous, that is,for
any c > 0,i2+(vt,)
CNe(Q+(Vt2» if ~tl
1 -t2 ~ 8(ë),
where8(ê)
ispositive
andN,(A)
denotes the
ê-neighborhood ,of
the set A. Then vt 5 u in K2for 0
t 1.The case A = 0 is Lemma 7 in
[5].
The present case of A > 0requires
additional
analysis.
PROOF. Set T = in
SZ}. By (a)
and thecontinuity
of vt, T isnonempty
andclosed,
and it remains to show that T is open. Letto E T. By (b)
and thestrong
maximumprinciple
we have that vto u ini2+(vt,).
Sinceevery
point
of isregular,
it followsby
Lemma 10.1 thatQ+(Vto)
iscompactly
contained inK2’(u)
U9Q,M(a;)
> --_0+(u) (we
used hereassumption (b)).
We can therefore choose c > 0 smallenough
such thatBy assymption (d),
there is then a 6 > 0 such thatThus u is harmonic in
Q+(vt) while vt
issubharmonic, and, by
the maximumprinciple (using (b)),
vt u inif t -
6. Sinceobviously vt u
init remains to show that
Observe that
(10.3) implies
that-
to)
6. For such t’s we introduce the function for any small It satisfies(by (10.5))
It will be convenient to work with the functions
By
Lemma 10.2on so that -T on and
consequently Wt
= 0in some
n-neighborhood
of henceWithout loss of
generality
we may assume that is connected(otherwise
we restrict
Wt
to eachcomponent
ofi2-(u)). Extending Wt by
0 intowe then
have, by (10.7),
Since A
Au,
we can choose a subdomain G of Q such that and AAG.
Denoteby Wj’° u-mollifiers
of Thenand,
for smallenough o,,
Consider the
principal eigenfunction p
of A in G:By
Sard’stheorem,
>p}
is smooth for almost every p > 0. We choose p smallenough
so that alsoThen, by
Green’s formula and(10.9), (10.10),
Taking p --~
0 and then a ---+0,
we obtainLetting, finally,
T --; 0 weget
and,
since in whichcompletes
theproof
ofWe shall need later on to work with