Geometric approach to nonvariational singular elliptic equations

0.26 MB·English

by  Damião Araújo

Geometric approach to nonvariational
singular elliptic equations
2
1 Damião Araújo & Eduardo V. Teixeira
0
2

n
a Abstract
J

9
1
In this work we develop a systematic geometric approach to study fully nonlinear elliptic

] equations withsingular absorption termsaswellastheirrelated freeboundary problems. The
P magnitudeofthesingularityismeasuredbyanegativeparameter(g 1),for0<g <1,which
A −
reflectsonlackofsmoothness foranexisting solution alongthesingular interface betweenits
.
h positive and zero phases. We establish existence as well sharp regularity properties of solu-
t
a tions. Wefurther prove that minimal solutions are non-degenerate and obtain fine geometric-
m measurepropertiesofthefreeboundaryF=¶ u>0 . InparticularweshowsharpHausdorff
[ estimates which imply local finiteness of the p{erimet}er of the region u>0 and H n 1 a.e.
−
{ }
1 weakdifferentiability property ofF.
v
5
5
1 Introduction
0
4
.
1 Theaimofthispresentworkistostudyfinequalitativepropertiesofnonvariationalsingularelliptic
0
2 equationsoftheform
1
Xiv: (1.1) F(D2uu) ∼= uf−q ·c {u>0} ionn W¶ W ,
(cid:26)
r
a where W RN is a bounded Lipschitz domain, q =1 g , for 0<g <1, f is a continuous, non-
⊂ −
negative boundary datum and the governing operator F is assumed to be uniform elliptic, i.e.,
D F isapositivedefinitematrix. Thestudyofsingularequationsas (1.1)ismotivated
(i,j) 1 i,j N
byapplica≤tion≤sinanumberofproblemsinengineeringsciences. Infactthefreeboundaryproblem
(cid:0) (cid:1)
F(D2u) = g ug 1 in u>0
(1.2) − { }
u= (cid:209) u = 0 on ¶ u>0
(cid:26) | | { }
is used, for example, to model fluids passing through a porous body W . For instance, u could
represent the density of a gas, or else the density of certain chemical specie, in reaction with a
porouscatalystpellet,W .
1
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 2
The variational theory, F(M) = Trace(M), for the free boundary problem (1.2) is fairly well
understood, nowadays. It appears as the Euler-Lagrange equation in the minimization of non-
differentiablefunctionals:
1
(1.3) (cid:209) u(X) 2+u(X)g dX min.
2| | −→
Z
See, forinstance[15, 16, 2, 21]. Noticethat such aproblem isquitedifferent fromtheone treated
intheclassicalpaper[7]. Thelatterhas been recentlystudiedin thefullynonlinearsettingin[9].
The case g =1 in (1.3) represents the obstacle problem, [4]; the case g =0 relates to the cav-
itation problem, [1]. Fully nonlinear version of the obstacle problem has been considered in [13].
Nonvariational cavitation problem has been recently studied in [17]. The delicate intermediary
case, 0 < g < 1, addressed in this present work brings major novelty adversities as the equation
satisfied within the positive set u >0 is nonhomogeneous and blows-up along the a priori un-
{ }
known quenching interface F=¶ u>0 W - the so called free boundary of the problem. The
{ }∩
lackofvariationalorenergyapproachestooimpliessignificantdifficultiesintheproblemandnew,
nonvariational solutions have to be established. In fact, since the free boundary problem consid-
eredinthispaperhasnonvariationalcharacter,onecannotusethepowerfulmeasure-distributional
language to setup weak version of the problem. Instead we shall employ a perturbation scheme
andwillobtainuniformestimateswithrespecttotheapproximatingparametere . Asolutiontothe
fully nonlinear free boundary problem (1.2) will therefore be obtained as the limit of appropriate
approximatingconfigurations.
The first main problem to be addressed concerns the optimal regularity for solutions to Equa-
tion (1.1). Optimal estimates for heterogeneous equations, Lu = f(X,u) is in general a quite
delicate issue. For the singular setting studied in this present work, optimal estimates are even
more involved as they can be understood as invariant (tangential) equations for their own scaling.
g
WeshowinSection4ofthepresentworkthatsolutionsarelocallyofclassC1,2 g . Thisresultwas
−
onlyknowninthevariationalsetting,forminimizersofEuler-Lagrangefunctional,see[15,16, 2]
and [10, 11].
Thesecondprincipalresultdevileredinthisarticlestatesthatminimalsolutions,i.e.,solutions
g
obtained from Perron’s type method do grow precisely as dist(X,F)1+2 g , which corresponds to
−
the maximum growth rate allowed. Such a result implies a quite restrictive geometry for the free
quenching interface F. As consequence of our sharp gradient estimate, Theorem 4.1 and optimal
growth rate, Theorem 5.4, a minimal solution is trapped between the graph of two multiples of
g
dist(X,F)1+2 g , i.e.,
−
g g
c dist(X,F)1+2 g u(X) C dist(X,F)1+2 g , X u>0 .
· − ≤ ≤ · − ∈{ }
By means of geometric considerations, in Section 6 we establish a clean Harnack inequality for
solutions to (1.1) within free boundary tangential balls, B u >0 , B tangent to F. In Section
⊂{ }
7, under an extra asymptotic structural assumption on the governing operator F, we establish
Hausdorff estimates of the free boundary. In particular we show c u>0 W BV(W ), that is,
{ }∩ ′ ∈
u > 0 is locally a set of finite perimeter. We further show that the reduced free boundary has
{ }
H n 1 totalmeasure. ThelasttwoSectionscloseuptheprojectbyobtainingasolutiontothefully
−
nonlinearfree boundaryproblem(1.2)withthedesired analyticand geometricproperties.
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 3
2 Mathematical set-up
Throughout this paper W will be a fixed Lipschitz bounded domain in RN, f : ¶ W R is a
+
→
continuousboundarydatumand 0<g <1 isa fixedreal number. Weshalldenoteby Sym(N) the
spaceofallrealN N symmetricmatricesandF C1(Sym(N) 0 )willbeauniformlyelliptic
× ∈ \{ }
fullynonlinearoperator;thatis,weshallassumethatthereexisttwoconstants0<l L suchthat
≤
(2.1) F(M +N ) F(M)+L N + l N , M,N Sym(N).
−
≤ k k− k k ∀ ∈
Theultimategoalofthispaperis tostudyexistenceand finequalitativepropertiesofsolutions
tothesingularequation
(2.2) F(D2u)=g ug 1 c .
− u>0
· { }
From the equation itself, one notices that the Hessian of an existing solution blows-up along the
freeboundaryF=¶ u>0 W ;therefore, solutionscannot beofclassC2. Inthefullynonlinear
{ }∩
setting, the problem of optimal regularity for solutions to Equation (2.2) is a rather delicate issue
anditwillbeaddressedinSection4. Partofthesubtlenessofthisproblemcomesfromtheintrinsic
complexityof the regularitytheory for viscositysolutionsto uniform ellipticequations. We recall
thatitis wellknownthatsolutionstohomogeneousequation
(2.3) F(D2u)=0,
has a priori C1,m bounds for some m > 0 that depends only on N,l and L . Under concavity or
convexity assumption on F, a Theorem due to Evans and Krylov, states that solutions are C2,a .
Nevertheless,Nadirashviliand Vladuthaverecentlyshownthatgivenany0<h <1 itispossible
tobuildupauniformlyellipticoperatorF,whosesolutionstothehomogeneousequation(2.3)are
notC1,h , see[14], Theorem 1.1.
Therefore, in order to access the optimal regularity estimate available for the free boundary
problem (2.2), it is natural to assume that F has a prioriC2,t estimates for some small 0<t <1.
Suchahypothesiswillbeenforcedhereafterinthepaper,thoughallbutTheorem4.1donotdepend
onsuch condition.
Let us turn our attention to the singularly perturbed approach we shall use in order to grapple
with the lack of variational approaches available. In this paper we suggest the following singular
perturbationschemeto appropriatelyapproach thefreeboundaryproblem(1.2):
F(D2u) = b e (u), in W
(Ee ) u = f on ¶ W .
(cid:26)
The singular perturbation term b e is build up as follows: initially select your favorite function
r C¥ [0,1]and set
∈ 0
g
(2.4) a :=1+ .
2 g
−
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 4
Throughoutthewholepaper, a willalways bethefixed valuestated in(2.4). In thesequel, define
(2.5) Be (t)= t−eeaa s 0 r (s)ds,
0
Z
where 0 < s 0 < 21 is an arbitrary technical choice. Notice that Be is a smooth approximation of
c . Finally,weset
(0,¥ )
(2.6) b e (t)=g tg−1Be (t).
Suchaconstructioniscarefullycarriedoutastopreservethenaturalscalingofthedesiredequation
(2.2).
Wefinish thisSectionby listingthemainnotationsadoptedthroughoutthearticle:
Thedimensionof theEuclidean space theproblem is modeled in will be denoted by N 2.
• ≥
W will be a fixed bounded domain in RN. For a domain O RN, ¶ O will represent the
⊂
boundaryofthedomainO. c willstand forthecharacteristicfunctionoftheset S.
S
The N-dimensional Lebesgue measure of a set A RN will be denoted by LN(A). H n 1
−
• ⊂
willstandforthe(n 1)-Hausdorffmeasure.
−
, will be the standard scalar product in RN. For a vector x = (x , ,x ) RN, its
1 N
• h· ·i ··· ∈
Euclidean norm will be denoted by x := x ,x . The tensor product x y denotes the
| | h i ⊗
matrixwhoseentries aregivenby x y for1 i, j N.
i j
p≤ ≤
B (p) will be the open ball centered at p with radius r. Furthermore, we shall denote kB=
r
•
kB (p):=B (p),foranyk>0.
r kr
Constants C,C ,C , > 0 and c,c ,c ,c > 0 that depend only on dimension, g and
1 2 0 1 2
• ··· ···
ellipticity constants l , L will be call universal. Any additional dependence will be empha-
sized.
3 Existence of minimal solutions
In thissectionwecommenton theexistenceofa viscositysolutionto equation(Ee ). Moreimpor-
tantly,weshallestablishhereinastableprocesstoselectspecialsolutionsto(Ee ). Aswewillshow
inSection5,thefamilyofminimalsolutionsturnsouttosatisfythedesiredappropriategeometric
features. Such properties will allow us to establish Hausdorff estimates of the free boundary in
Section 7.
Notice that because of the lack of monotonicity of equation (Ee ) with respect to the variable
u, classical Perron’s method cannot be directly employed. The next theorem proved in [17], is an
adaptationofPerron’s method,which isby nowfairlywell understood.
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 5
Theorem 3.1. Let g be a bounded, Lipschitz function defined in the real line R. Suppose F uni-
formlyellipticandthattheequationF(D2u)=g(u)admitsaLipschitzviscositysubsolutionu and
⋆
a Lipschitzviscositysupersolutionu⋆ suchthatu =u⋆ = f C(¶ W ). Definethesetof functions,
⋆
∈
S:= w C(W ); u w u⋆ andwsupersolutionofF(D2u)=g(u) .
⋆
∈ ≤ ≤
Then, (cid:8) (cid:9)
v(x):= infw(x)
w S
∈
isa continuousviscositysolutionofF(D2u)=g(u)andv= f continuouslyon ¶ W .
ExistenceofminimalsolutiontoEquation(Ee )followsbychoosingu⋆=u⋆(e )andu⋆=u⋆(e )
solutionstothefollowingboundary valueproblems
F(D2u ) = z , in W F(D2u⋆) = 0, in W
⋆
and
u = f on ¶ W , u⋆ = f on ¶ W ,
⋆
(cid:26) (cid:26)
where
z :=supb e e g−1.
∼
Theexistencethefunctionsu andu⋆isconsequenceofstandardPerron’smethod. Byconstruction
⋆
u⋆ is viscosity subsolution of (Ee ) and u⋆ is a viscosity supersolution oh (Ee ). Note that u⋆,u⋆
∈
C0,1(W ) C(W ). Thusadirect applicationofTheorem3.1yieldsthefollowingexistenceresult:
∩
Theorem3.2(Existenceofminimalsolutions). LetW Rn beaLipschitzdomainand f C(¶ W )
∈ ∈
be a nonnegative boundary datum. Then, for each e > 0 fixed, equation (Ee ) has a nonnegative
minimalviscositysolutionue C(W¯).
∈
As previously mentioned, more importantly than assuring existence of a viscosity solution to
(Ee ), Theorem 3.2 provides a particular choice of solutions to such an equation. In comparison
with the variational theory, this choice is a replacement for the selection of minimizers of the
Euler-Lagrange functional (see for instance [19] for further details). Therefore, unless otherwise
stated, whenever we mention viscosity solution to (Ee ), we mean the minimal solution provided
byTheorem 3.2.
4 Sharp regularity estimates
The first main result we provein this paper is the optimal regularity estimate, uniform in e , avail-
able for solutions to (Ee ). We will show that ue is locally a C1,b function and we shall further
determine the optimal b > 0 in terms of the degree of singularity g . This key information has
onlybeenknownforvariationalsolutions,[15,10,11]andtheproofsmakedecisiveuseofenergy
considerations. In principle it is not even clear that one should expect the same regularity theory
fornonvariationalproblems.
Thus, we start off this Section by rather informal, heuristic considerations as to guide us
through the genuine results to be established later on. Let us analyze the limiting free bound-
ary problem (1.2). Suppose 0 is a free boundary point and, say, e is the unit outward normal
n
−
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 6
pointingtowardsthequenchingphase u=0 . IfuisC1,b at0,then,inasmallneighborhood,say,
, Br u >0 , r 1, u behaves like{ Xn1+}b . Therefore, the singular potential of the equation
∩{ } ≪ ∼
(1+b )(1 g)
in (1.2) is like Xn · − . In view of the regularity theory for heterogeneous fully nonlinear
∼
equationsF(D2u)= f(X),establishedin [5]and[20], weobtainthefollowingimplication
Xn(1+b )·(1−g) Lqweak implies u C1,1−q1.
∈ ∈
Thereasoningabovegivesthefollowingsystemofalgebraicequations
q (1+b )(g 1) = 1
− −
b = 1 1.
q
(cid:26) −
Solving for b , revels, b = g , which agrees with the optimal regularity estimate established for
2 g
thevariationaltheory. −
g
ThisSectionisdevotedtoestablishlocalC1,2 g regularityestimatesforsolutionsue toEquation
−
(Ee ),uniformine . RecallthatweareworkingunderthenaturalassumptionthatF hasaprioriC2,t
estimates. In fact we shall obtain a universal control on the gradient of ue near the free boundary
in terms of the value of ue . Since ue =0 along the free boundary, our estimate gives the desired
regularity through the interface F=¶ u>0 W . Actually the gradient estimates we obtain are
g { }∩
evenstrongerthan theaimedC1,2 g regularity. Here isitsprecisestatement:
−
Theorem 4.1 (Uniform optimal regularity). Given a subset W ⋐ W , there exists a constant C
′
depending on, f ¥ , g , W ′, dimension, ellipticity, but independent of e , such that, any family of
k k
viscositysolutions ue ofequation(Ee )satisfies,
{ }
(cid:209) ue (X) 2 Cue (X)g , X W ′.
| | ≤ ∀ ∈
g
In particular,ue ∈Cl1o,c2−g , uniformlyine .
Proof. For simplicity, we shall drop the subscript e in ue , writing simply u. We will analyze the
followingauxiliaryfunction
v:=y (ue ) (cid:209) ue 2, for y (t)=t−g .
| |
Our ultimate goal is to show that v is locally bounded in W , for bounds that do not depend on e .
Hereafter in the proof we select a positive function f C2(W ) that vanishes on ¶ W and satisfies
∈
(cid:209) f 2 =O(f ). The purpose of such a function is merely to localize our analysis. Define, in the
| |
sequel,
w :=f v in W ,
·
and letX W beamaximumpointofw in W¯, thatis
0
∈
f (X ) v(X )=maxw .
0 0
· W¯
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 7
Differentiatingw and Equation(Ee ),weobtain
(4.1) Dw =f v+f v , D w =f v+f v +f v +f v
i i i ij ij i j j i ij
and
(4.2) (cid:229) Fij(D2u)Dijuk =b e′(u)uk.
i,j
Let A := F (D2u(X )). By uniform ellipticity of the operator F, the matrix (A ) is strictly
ij ij 0 ij
positive. Also,sinceX isamaximumpoint,D2w (X ) isnon-positive. Therefore
0 0
0 (cid:229) A D w (X )
i,j ij ij 0
≥
(4.3)
= v(cid:229) A D f +2Tr (A )(cid:209) f (cid:209) v +f (cid:229) A D v (X ).
i,j ij ij ij i,j ij ij 0
⊗
It followsfrom(4.1)a(cid:2)nd from thefact that(cid:0)X isacritical p(cid:1)ointofw that (cid:3)
0
(cid:209) f (X )
(4.4) (cid:209) v(X )= v(X ) 0 .
0 − 0 f (X )
0
Combining(4.4), ellipticityof(A )and analyticpropertiesoff , wereach
ij
v(cid:229) A D f +2Tr (A )(cid:209) f (cid:209) v v(cid:229) A D f +2Tr (A )(cid:209) f (cid:209) v
i,j ij ij ij i,j ij ij ij
⊗ ≥ − ⊗
(cid:0) (cid:1) (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) 2 (cid:12)
= v (cid:229) A D f + Tr (A )(cid:209) f (cid:209) f
i,j ij ij f ij
− ⊗
(4.5) (cid:12) (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:209) f 2 (cid:12)
C(L )max D2f ,| | v
f
≥ − W¯ | |
(cid:26) (cid:27)
=: C v.
0
−
Letus turnourattentionontheterm(cid:229) A D v(X ). Differentiatingv, weobtain
i,j ij ij 0
(4.6) D v=y (u)u (cid:209) u 2+2y (u)(cid:229) u u .
i ′ i k ki
| |
k
Differentiatingaboveexpression,onereaches
D v = (y u u +y (u)u ) (cid:209) u 2+2y (u)u (cid:229) u u
ij ′′ i j ′ ij ′ i k kj
| |
k
+ 2y (u)u (cid:229) u u +2y (u)(cid:229) (u u +u u ).
′ j k ki kj kj k kij
k k
It followsfrom(4.4) and (4.6)that,at X , foreach i,
0
(cid:229) u u = 1 y (u)u (cid:209) u 2+vf i .
k ki −2y (u) ′ i| | f
k (cid:26) (cid:27)
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 8
Thus,combiningthesewith(4.1),wefind
(y (u))2 v y (u)
A D v = y (u) 2 ′ A (cid:209) u (cid:209) u (cid:209) u 2 2 ′ A (cid:209) u (cid:209) f
ij ij ′′ − y (u) ij ⊗ | | − f y (u) ij ⊗
(cid:20) (cid:21)
(4.7)
+y ′(u)Aijuij (cid:209) u 2+2y (u) Tr(D2u(Aij)D2u)+b e′(u) (cid:209) u 2 .
| | | |
(cid:0) (cid:1)
By ellipticityand definitionofthef , followstheestimates
A (cid:209) u (cid:209) u l (cid:209) u 2,
ij
⊗ ≥ | |
A (cid:209) u (cid:209) f L (cid:209) u (cid:209) f ,
ij
| ⊗ | ≤ | |·| |
Aijuij L F(D2u)=L b e (u),
| | ≤
Tr(D2u(A )D2u) 0,
ij
≥
(y (u))2
y (u) 2 ′ = ( g 2+g )u g 2.
′′ − y (u) − − −
(cid:20) (cid:21)
Hereit isimportantto noticethat g 2+g >0. Usingall theseaboveestimatesin(4.7)weobtain
−
L v
A D v ( g 2+g )l vu 2 (cid:209) u 2 2g (cid:209) u (cid:209) f
ij ij ≥ − − | | − u f | |·| |
g L u−g−1b e (u) (cid:209) u 2+2u−g b e′ (cid:209) u 2.
− | | | |
Ontheotherhand,fort >0
b e (t)=g tg−1Be (t) g tg−1
≤
and
b e′(t) = g (g 1)tg−2Be (t)+g tg−1Be′(t)
g (g −1)tg−2Be (t).
≥ −
Herewehaveused
t s e b
B′e (t)=e −b r −e b0 >0.
!
Thesetogethergiveus,
L v
A D v ( g 2+g )l v u 2 (cid:209) u 2 2g (cid:209) u (cid:209) f
ij ij ≥ − · − | | − u f | |·| |
L g 2u g 1 ug 1 (cid:209) u 2 2g (1 g )ug 2 u g (cid:209) u 2
(4.8) − − − · − | | − L−v − · − | |
= ( g 2+g )l v u 2 (cid:209) u 2 2g (cid:209) u (cid:209) f
− · − | | − u f | |·| |
L g 2u 2 (cid:209) u 2 2g (1 g )u 2 (cid:209) u 2.
− −
− | | − − | |
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 9
Combining(4.3), (4.5)and(4.8), takingintoaccount that (cid:209) f =O(f ),wereach
| |
C v f (l ( g 2+g )v C )u 2 (cid:209) u 2 2g L u 1v (cid:209) u (cid:209) f
0 1 − −
=≥ f (l (−g 2+g )v−C )u 2|(cid:209) u|2−2g L u 1u| g u|g·/2|√v|(cid:209) f (cid:209) u 2
1 − − −
− − | | − g | |·| |
f (l ( g 2+g )v C1)u−2 (cid:209) u 2 2C(f )g L u−2−1√vf (cid:209) u 2,
≥ − − | | − | |
whereC :=(g 2L +2g (1 g ))>0. Clearly wecan assume
1
−
(cid:209) u(X ) u(X )=0.
0 0
| | 6
Hence, as (cid:209) f 2 =O(f ), wederive
| |
(4.9) C0u−g+2 f (l ( g 2+g )v C1) 2C(f )g L u−2g +1 vf .
≥ − − −
Intheregion ue 1,F(D2ue )isuniformlybounded,independentlypofe . Thus,byAlexandroff-
| |≥
Bakeman-Pucci maximumprinciple,
ue C2,
| |≤
foraconstantC that doesnotdepend upone . By such considerationsand (4.9)follows
2
C l ( g 2+g )f (X )v(X ) C v(X )f (X ).
3 0 0 4 0 0
≥ − −
for universal constantsC ,C that does not depend upon ep. Clearly the above estimative implies
3 4
that
v(X)f (X) v(X )f (X ) C,
0 0
≤ ≤
i.e,
f u g (cid:209) u 2 C.
−
| | ≤
for a constantC that depends only on dimension, ellipticity, g , f ¥ and f , but is independent of
k k
e .
It is now classical to obtain ue g is locally bounded , uniformly in e . The proof of
k kC1,2 g
Theorem4.1 isconcluded. −
The uniform optimal regularity established in Theorem 4.1 gives, in particular, compactness
of the family of solutions to Equation (Ee ). It will also be important to our analysis the following
consequenceofTheorem4.1:
Corollary4.2. GivenasubdomainW ⋐W ,thereexistconstantsC andr >0dependingong ,W
′ 0 ′
anduniversalparameterssuchthatforX W andr r , thereholds
0 ′ 0
∈ ≤
sup ue ue (X0)+Cue (X0)g/2r+Cra
≤
Br(X0)
g
where, a =1+ .
2 g
−
NONVARIATIONAL SINGULAR ELLIPTIC EQNS 10
Proof. Definetheauxiliaryfunction
f(Y):=ue (Y) ue (X0) (cid:209) ue (X0) (Y X0).
− − · −
whereY B (X ). Clearly
r 0
∈
f(X )= (cid:209) f(X ) =0
0 0
| |
and therefore, fromTheorem 4.1we obtain
a
f(Y) f(X ) C Y X ,
0 0
| − |≤ ·| − |
whichimmediatelygives,by triangularinequality,
ue (Y) ue (X0)+ (cid:209) ue (X0) Y X0 +C Y X0 a .
≤ | |·| − | | − |
However,applyingoncemoreTheorem 4.1,wereach
ue (Y) ue (X0)+Cue (X0)g/2 Y X0 +C Y X0 a ,
≤ ·| − | | − |
and theproofofCorollary 4.2isconcluded.
5 Nondegeneracy of minimal solutions
g
In the previous Section we have shown that solutions to Equation (Ee ) are locally of classC1,2 g .
−
Inparticularsuchanestimateprovidesanupperboundonhowfastue growthsawayfrom,say,the
levelsurface ue e a , fora as in(2.4). Thatis,
{ ∼ }
ue (Z).[dist(Z, ue e a )]a .
{ ∼ }
The main result we shall prove in this Section states that minimal solutions do growth precisely
as dist(X0, ue e a )a , see Corollary 5.5 for the precise statement. In fact we shall establish a
{ ∼ }
strongernondegeneracypropertyofminimalsolutions,whichalso hasfundamentalimportancein
ourblow-upanalysis.
Tosimplifythestatementoftheresults,weintroducesomedefinitionsandnotations. Hereafter
weshallusesystematicallywefollowingnotations:
ue >k := x W ue (x)>k ,
{ } { ∈ | }
t >ue >l := x W t >ue (x)>l ,
{ de (X}) := d{is∈t(X,¶| ue >e a ), }
{ }
The nondegeneracy feature of minimal solutions are based on the construction of appropriate
viscosity supersolution whose value within an inter disk is much smaller than its value on the
boundaryofan outerdisk.
Proposition5.1. Assume,withnolossofgeneralitythat0 W . Given0<h ,thereexistsaradially
∈
symmetricfunctionq C1,1(W )anduniversalsmallconstants0<c <1and0<c <1suchthat
2 1
∈