Asymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds

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by  Jean-Baptiste Casteras

ASYMPTOTIC DIRICHLET PROBLEM FOR -HARMONIC AND
A
MINIMAL GRAPH EQUATIONS IN CARTAN-HADAMARD
5 MANIFOLDS
1
0
2 JEAN-BAPTISTECASTERAS,ILKKAHOLOPAINEN,ANDJAIMEB.RIPOLL

n
a Abstract. WestudytheasymptoticDirichletproblemforA-harmonicequa-
J
tionsandfortheminimalgraphequationonaCartan-HadamardmanifoldM

1 whosesectionalcurvaturesareboundedfrombelowandabovebycertainfunc-
2 tionsdependingonthedistancetoafixedpointo∈M. Weare,inparticular,
interestedinfindingoptimal(orclosetooptimal)curvatureupperbounds.

]
G
D
h. 1. Introduction
t
a In this paper, we are interested in the asymptotic Dirichlet problem for -
m A
harmonic functions and for the minimal graph equation on a Cartan-Hadamard
[ manifold M of dimension n 2. We first recall that a Cartan-Hadamardmanifold
≥
is a simply connected, complete Riemannianmanifoldhaving nonpositivesectional
1
curvature. Itiswell-known,sincetheexponentialmapexp : T M M isadiffeo-
9v morphism for every point o M, that M is diffeomorphico tooRn.→One can define
∈
4 an asymptotic boundary ∂ M of M as the set of all equivalence classes of unit
∞
2 speed geodesic rays on M (see Section 2.1 for more details). The so-called geo-
5 metric compactification M¯ of M is then given by M¯ = M ∂ M equipped with
0 the cone topology. We also notice that M¯ is homeomorphic∪to∞a closed Euclidean
.
1
ball; see [19]. The asymptotic Dirichlet problem on M for some operator is then
0 Q
the following: Given a continuous function f on ∂ M does there exist a (unique)
5 ∞
1 function u C(M¯) such that [u] = 0 on M and u∂∞M = f? We will consider
∈ Q |
: this problem for two kinds of operators: the minimal graph operator (or the mean
v
curvature operator) defined by
Xi M
u
r (1.1) [u]=div ∇ ,
a M 1+ u2
|∇ |
and the -harmonic operator (of type p)p
A
(1.2) [u]= div ( u),
x
Q − A ∇
where : TM TM is subject to certain conditions; for instance (V),V
V p, 1A<p< →, and (λV)=λλp−2 (V) for all λ R 0 . ThehAp-Laplaciia≈n
| | ∞ A | | A ∈ \{ }
isanexampleofan -harmonicoperator(seeSection2.3fortheprecisedefinition).
A
We also note that u satisfies [u] = 0 if and only if G := (x,u(x)) x Ω is a
minimal hypersurface in the pMroduct space M R. { | ∈ }
×
2000 Mathematics Subject Classification. 58J32,53C21,31C45.
Key words and phrases. minimalgraphequation, Dirichletproblem,Hadamardmanifold.
J.-B.C.supportedbytheCNPq(Brazil)project501559/2012-4;I.H.supportedbytheAcademy
ofFinland,project252293; J.R.supportedbytheCNPq(Brazil)project302955/2011-9.
1
2 JEAN-BAPTISTECASTERAS,ILKKAHOLOPAINEN,ANDJAIMEB.RIPOLL
WewillnowgiveabriefoverviewoftheresultsknownfortheasymptoticDirich-
let problem on Cartan-Hadamard manifolds. The first result for this problem in
the case of the usual Laplace-Beltrami operator was obtained by Choi. In [11],
he solved the asymptotic Dirichlet problem assuming that the sectional curvatures
satisfyK a2 <0andthatM satisfiesa”convexconicneighborhoodcondition”,
i.e. given≤x −∂ M, foranyy ∂ M, y =x, there existV M¯, a neighborhood
∞ ∞ x
of x, and V∈ M¯, a neighbo∈rhood of y6 such that V and⊂V are disjoint open
y x y
sets of M¯ in te⊂rms of the cone topology and V M is convex with C2 boundary.
x
∩
Anderson [5] proved that the convex conic neighborhood condition is satisfied for
manifolds of pinched sectionalcurvature b2 K a2 <0 and therefore he was
− ≤ ≤−
able to solve the asymptotic Dirichlet problem for the Laplace-Beltrami operator
(see also [6] for a different approach). We point out that the asymptotic Dirichlet
problem was solved independently by Sullivan [40] using probabilistic arguments.
Ancona in a series of papers [1], [2], [3], and [4], was able to replace the curvature
lowerboundbyaboundedgeometryassumptionthateachballuptoafixedradius
is L-bi-Lipschitz equivalent to an open set in Rn for some fixed L 1; see [1].
≥
Finally, we give the following theorem by Hsu where the most general curvature
bounds under whichthe asymptotic Dirichletproblemforthe Laplacianis solvable
are given. Here and throughout the paper r(x) stands for the distance between
x M and a fixed point o M.
∈ ∈
Theorem 1. [28, Theorems 1.1 and 1.2] Let M be a Cartan-Hadamard manifold.
Suppose that:
- there exist a positive constant a and a positive and non-increasing function h with
∞
th(t)dt< such that
0 ∞
R h r(x) 2e2ar(x) Ric and K a2,
x x
− ≤ ≤−
or (cid:0) (cid:1)
- there exist positive constants r , α>2, and β <α 2 such that
0
−
α(α 1)
r(x)2β Ric and K −
− ≤ x x ≤− r(x)2
for all x M, with r(x) r . Then the Dirichlet problem at infinity for the
0
∈ ≥
Laplacian is solvable.
The asymptotic Dirichlet problem has been studied for more general operators
thantheLaplacian. Thefirstresultinthisdirectionhasbeenobtainedin[25]forthe
p-Laplacianunderapinchednegativesectionalcurvatureassumptionbymodifying
thedirectapproachofAndersonandSchoen[6]. In[27]HolopainenandVa¨h¨akangas
havebeenabletorelaxtheassumptiononthecurvature(seeTheorem3foramore
precise statement of these curvature assumptions). Of particular interest is the
case of the minimal graph operator. In [12], Collin and Rosenberg were able to
constructharmonicdiffeomorphismsfromthecomplexplaneContothehyperbolic
plane H2 disproving this way a conjecture of Schoen and Yau [37]. This result has
beengeneralizedby G´alvez andRosenberg[20] to anyHadamardsurfaceM whose
curvatureisboundedfromabovebyanegativeconstant. Afundamentalingredient
intheirconstructionsistosolvetheDirichletproblemonunboundedidealpolygons
with boundary values on the sides of the ideal polygons. These unexpected
±∞
results have raised interest in (entire) minimal hypersurfaces in the product space
ASYMPTOTIC DIRICHLET PROBLEM 3
M R, where M is a Cartan-Hadamardmanifold (see for example, [15], [18], [30],
×
[32], [35], [36], [39]).
Very recently in [8], the authors generalized (most of) the solvability results to
a larger class of operators of the form
Q
(1.3) [u]=div( ( u2) u),
Q P |∇ | ∇
with subject to the following growth conditions. Let : (0, ) [0, ) be a
P P ∞ → ∞
smooth function such that
(1.4) (t) P t(p−2)/2
0
P ≤
for all t>0, with some constants P >0 and p 1, and that := ′/ satisfies
0
≥ B P P
1 B
0
(1.5) < (t)
− 2t B ≤ t
forall t>0 with someconstantB > 1/2. Furthermore,assumethatt (t2) 0
0
− P →
as t 0+ and define (X 2)X =0 whenever X is a zero vector.
Fo→llowing [8] we calPl a|re|latively compact open set Ω ⋐ M -regular if for any
continuous boundary data h C(∂Ω) there exists a unique Qu C(Ω¯) which is
∈ ∈
-solution in Ω and u∂Ω=h. In addition to the growth conditions on , assume
Q | P
that
(A) thereisanexhaustionofM byanincreasingsequenceof -regulardomains
Q
Ω , and that
k
(B) locallyuniformlyboundedsequencesofcontinuous -solutionsarecompact
Q
in relatively compact subsets of M.
It is well-known that the conditions above are satisfied by the minimal graph op-
erator and the p- Laplacian (see [16], [22] and [39]).
The maintheoremin[8]isa solvabilityresultforthe asymptotic Dirichletprob-
lem for operators that satisfy (1.4), (1.5), and conditions (A) and (B) under
Q
curvature assumption
2 2
b r(x) K(P) a r(x)
− ≤ ≤−
on M, where P TxM is a(cid:0)2-pla(cid:1)ne and a,b: [0, (cid:0) ) (cid:1)[0, ), b a, are smooth
⊂ ∞ → ∞ ≥
functions satisfying suitable assumptions. Here, instead of giving the precise as-
sumptions on functions a and b, we state the following two solvability results as
special cases of their main theorem (Theorem 1.6 in [8]).
Theorem2. [8,Theorem1.5]LetM beaCartan-Hadamardmanifoldofdimension
n 2. Fix o M and set r()=d(o, ), where d is the Riemannian distance in M.
≥ ∈ · ·
Assume that
φ(φ 1)
r(x)2(φ−2)−ε Sect (P) − ,
− ≤ x ≤− r(x)2
for some constants φ > 1 and ε > 0, where Sect (P) is the sectional curvature of
x
a plane P T M and x is any point in the complement of a ball B(o,R ). Then
x 0
⊂
the asymptotic Dirichlet problem for the minimal graph equation (1.1) is uniquely
solvable for any boundary data f C M( ) .
∈ ∞
Theorem 3. [8, Corollary 1.7] Let M(cid:0) be a C(cid:1)artan-Hadamard manifold of dimen-
sion n 2. Fix o M and set r() = d(o, ), where d is the Riemannian distance
≥ ∈ · ·
in M. Assume that
(1.6) r(x)−2−εe2kr(x) Sect (P) k2
x
− ≤ ≤−
4 JEAN-BAPTISTECASTERAS,ILKKAHOLOPAINEN,ANDJAIMEB.RIPOLL
for some constants k > 0 and ε > 0 and for all x M B(o,R ). Then the
0
∈ \
asymptotic Dirichlet problem for the operator (defined as in (1.3)) is uniquely
Q
solvable for any boundary data f C M( ) .
∈ ∞
TheDirichletproblematinfinityfo(cid:0)r -har(cid:1)monicfunctionhasbeenconsideredin
A
[41] and[42]. In[42], Va¨h¨akangaswasable to generalizethe result obtainedin [27]
(forthep-Laplacian)tothe -harmoniccase. In[41],bygeneralizingamethoddue
A
toCheng[10],hesolvedtheasymptoticDirichletproblemfor -harmonicequations
A
of type p provided the radial sectional outside a compact set satisfy
φ(φ 1)
K(P) −
≤− r2(x)
for some constant φ>1 with 1<p<1+φ(n 1) and
−
K(P) C K(P′)
| |≤ | |
for some constant C, where P and P′ are any 2-dimensional subspaces of T M
x
containingthe (radial)vector r(x). Itis worthobservingthat no curvaturelower
∇
bounds are needed here.
The goalofthispaperisthreefold. Firstofall,wearelookingforanoptimal(or
atleastcloseto optimal)curvature upper bound under whichasymptotic Dirichlet
problems for equations (1.1) and (1.2) are solvable provided an appropriate curva-
turelowerboundholds. Secondly,weareusingPDEmethods,likeCaccioppoli-type
inequalities (Lemma 18), Moser iteration scheme (Lemma 20), and Young comple-
mentary functions to study the minimal graph equation. As far as we know such
methodsarenotfrequentlyusedinthecontextoftheminimalgraphequation. Last
butnotleast,wewantto publicizethe resultsandmethods ofthe stillunpublished
preprint [42] of Va¨h¨akangas. Our main results are the following two theorems.
Theorem 4. Let M be a Cartan-Hadamard manifold of dimension n 2. Assume
≥
that
2ε¯
logr(x) 1+ε
(1.7) K(P) ,
− r(x)2 ≤ ≤−r(x)2logr(x)
(cid:0) (cid:1)
for some constants ε > ε¯> 0, where K(P) is the sectional curvature of any plane
P T M that contains the radial vector r(x) and x is any point in the comple-
x
⊂ ∇
mentof aball B(o,R ). Then theasymptotic Dirichlet problem for the -harmonic
0
A
equation (1.2) is uniquely solvable for any boundary data f C ∂ M provided
∞
∈
that 1 p<nα/β.
≤ (cid:0) (cid:1)
Theorem5. LetM beaCartan-Hadamardmanifold ofdimension n 3satisfying
≥
the curvature assumption (1.7) for all 2-planes P T M, with x M B(o,R ).
x 0
⊂ ∈ \
Then the asymptotic Dirichlet problem for the minimal graph equation (1.1) is
uniquely solvable for any boundary data f C ∂ M .
∞
∈
In Theorem 4 above α and β are so-called s(cid:0)tructu(cid:1)ral constants of the operator
;see2.3fordetails. WenoticethattheLaplace-Beltramioperatorcorrespondsto
A
thecasep=2andα=β =1,andthereforeiscoveredbyTheorem4indimensions
n > 2. Thus we obtain a generalization to higher dimensions of a recent result
by Neel [31]. The curvature upper bound (1.7) appears also in a recent paper
[34]where Ripolland Telicheveskysolvedthe asymptotic Dirichlet problemfor the
minimalgraphequationonrotationallysymmetricHadamardsurfaces. Noticethat
ASYMPTOTIC DIRICHLET PROBLEM 5
dimension n = 2 is excluded in Theorem 5. However, we believe that the result
holds also in the 2-dimensional setting.
We point out that our curvature assumptions are in a sense optimal. Assume
that
1
K(P )
x ≥−r(x)2logr(x)
and let us consider first the case of an -harmonic operator of type p n. The
A ≥
standard Bishop-Gromov volume comparison theorem gives
Vol(∂B ) C(ρlogρ)n−1
ρ
≤
for some constant C and for all ρ r large enough. It is then easy to see that
0
≥
∞ dρ ∞ dρ
C =
(Vol(∂B )1/(p−1) ≥ ρlogρ ∞
Zr0 ρ Zr0
whichimpliesthatM isso-calledp-parabolicandhenceeverybounded -harmonic
A
function (with of type p) is constant; see e.g. [24] and [13]. On the other hand,
A
in [33] Rigoli and Setti proved the following nonexistence theorem:
Theorem 6. Let M be a complete manifold and u C1(M) be a solution of
∈
ϕ( u) u
div |∇ | ∇ =0,
u
|∇ |
where ϕ C1((0, )) C0([0, )) satisfies the following conditions:
∈ ∞ ∩ ∞
(1) ϕ(0)=0,
(2) ϕ(t)>0, for all t 0,
≥
(3) ϕ(t) Atδ, for all t 0,
≤ ≥
for some positive constants A and δ. Assume that
(Vol(∂Bρ)1δ)−1 / L1( ),
∈ ∞
then M is ϕ-parabolic i.e. u is constant.
Using this theorem, we also see that the curvature upper bound would be sharp
for the minimal graph equation in dimension n = 2. Notice that δ = 1 for the
minimal graph equation. We close this introduction with some comments on the
necessityofcurvaturelowerbounds. Indeed,Ancona’sandBorb´ely’sexamples([4],
[7]) show that a (strictly) negative curvature upper bound alone is not sufficient
forthe solvability ofthe asymptoticDirichlet problemfor the Laplaceequation. In
[23], Holopainen generalized Borb´ely’s result to the p-Laplace equation, and very
recently, Holopainen and Ripoll [26] extended these nonsolvability results to the
operator (as defined in (1.3)), in particular, to the minimal graph equation.
Q
The planofthe paperisthe following: Section2isdevotedtopreliminaries. We
recallsomewell-knownfacts onCartan-Hadamardmanifolds, Jacobiequations, -
A
harmonicfunctions,theminimalgraphequation,andYoungfunctions. InSection3
weproveTheorem4. Weadoptthesamestrategyastheoneusedin[42]. Itisbased
onaMoseriterationprocedureinvolvingaweightedPoincar´einequality. Finally,in
Section4weproveTheorem5adaptingtothe minimalgraphequationthe method
used in Section 3 for -harmonic functions. In this case since this equation does
A
not satisfy (2.2), some extra difficulties appear.
Acknowledgement. We would like to thank Joel Spruck for his help to obtain the
decay estimate for logW in Lemma 22.
|∇ |
6 JEAN-BAPTISTECASTERAS,ILKKAHOLOPAINEN,ANDJAIMEB.RIPOLL
2. Preliminaries
2.1. Cartan-Hadamard manifolds. WerecallthatCartan-Hadamardmanifolds
are complete simply connected Riemannian n-manifolds, n 2, with nonpositive
≥
sectional curvature. Let M be a Cartan-Hadamard manifold, ∂ M the sphere at
∞
infinity,andM¯ =M ∂ M. Recallthatthesphereatinfinityisdefinedastheset
∞
∪
ofall equivalence classesof unit speed geodesic raysinM; twosuch raysγ andγ
1 2
areequivalentifsup d γ (t),γ (t) < . Foreachx M andy M¯ x there
exists a unique unitt≥sp0eed1geodes2ic γx,y:∞R M such ∈that γx,y =∈x an\d{γx},y = y
(cid:0) (cid:1) → 0 t
for some t (0, ]. If v T M 0 , α>0, and R>0, we define a cone
x
∈ ∞ ∈ \{ }
C(v,α)= y M¯ x : ∢(v,γ˙x,y)<α
{ ∈ \{ } 0 }
and a truncated cone
T(v,α,R)=C(v,α) B¯(x,R),
\
where ∢(v,γ˙x,y) is the angle between vectors v and γ˙x,y in T M. All cones and
0 0 x
open balls in M form a basis for the cone topology on M¯.
2.2. Jacobi equation. We use the curvature upper bound in order to prove a
weighted Poincar´einequality and to estimate from above the norm of the gradient
ofanangularfunction. Thecurvaturelowerbound,inturn,isusedtoestimatethe
volume form from above. All of these estimates will be given in terms of solutions
to a 1-dimensionalJacobi equation. If k: [0, ) [0, ) is a smooth function, we
∞ → ∞
denote by f C∞ [0, ) the solution to the initial value problem
k
∈ ∞
(cid:0) (cid:1) f (0)=0,
k
(2.1) f′(0)=1,
 k
 f′′ =k2f .
k k
It follows that the solution fk isa nonnegative smooth function. Concerning the
curvature upper bound in (1.7) we have the following estimates:
Proposition7. [11,Prop. 3.4]Supposethat f: [R , ) R, R >0,is apositive
0 0
∞ →
strictly increasing function satisfying the equation f′′(r)=a2(r)f(r), where
1+ε
a2(r) ,
≥ r2logr
for some ε > 0 on [R , ). Then, for any 0 < ε˜< ε, there exists R R such
0 1 0
∞ ≥
that, for all r R ,
1
≥
f′(r) 1 1+ε˜
f(r) r(logr)1+ε˜, + .
≥ f(r) ≥ r rlogr
2.3. -harmonic functions and Perron’s method. In this section we define
A
-harmonic and -superharmonic functions and recordtheir basic properties that
A A
will be relevant in the sequel. We refer to [22] for the proofs and for the nonlinear
potential theory of -harmonic and -superharmonic functions.
A A
Let Ω be an open subset of a Riemannian manifold M. Suppose that for a.e.
x Ω we are given a continuous map : T M T M such that the map x
x x x
∈ A → 7→
(X ) is a measurable vector field whenever X is. We assume further that there
x x
A
ASYMPTOTIC DIRICHLET PROBLEM 7
are constants 1 < p < and 0 < α β < such that for a.e. x Ω, for all
v,w T M, v =w, and∞for all λ R ≤0 we∞have ∈
x
∈ 6 ∈ \{ }
(v),v αv p;
x
hA i≥ | |
(v) β v p−1;
x
(2.2) |A |≤ | |
(v) (w),v w >0;
x x
hA −A − i
(λv)=λλp−2 (v).
x x
A | | A
We denote the set of suchoperatorsby p(Ω) andwe say that is of type p. The
A A
constants α and β are called the structure constants of .
A function u W1,p(Ω) is called a (weak) solution ofAthe equation
∈ loc
(2.3) div ( u)=0
x
− A ∇
in Ω if
(2.4) ( u), ϕ =0
x
hA ∇ ∇ i
ZΩ
for all ϕ C∞(Ω). If u Lp(Ω), it is equivalent to require (2.4) for all ϕ
∈ 0 |∇ | ∈ ∈
W1,p(Ω) by approximation. Continuous solutions of (2.3) are called -harmonic
0 A
functions (of type p). By the fundamental work of Serrin [38], every solution of
(2.3) has a continuous representative. In the special case (h) = hp−2h, -
x
A | | A
harmonic functions are called p-harmonic and, in particular, if p = 2, we obtain
the usual harmonic functions.
A function u W1,p(Ω) is a subsolution of (2.3) in Ω if
∈ loc
div ( u) 0
x
− A ∇ ≤
weakly in Ω, that is
( u), ϕ 0
x
hA ∇ ∇ i≤
ZΩ
for all nonnegative ϕ C∞(Ω). A function v is called supersolution of (2.3) if v
∈ 0 −
is a subsolution. Finally, a lower semicontinuous function u: Ω ( ,+ ] that
→ −∞ ∞
isnotidentically+ inanycomponentofΩiscalled -superharmonic ifforevery
openD ⋑Ωandfor∞everyh C(D¯)thatis -harmonAicinD,h uon∂D implies
∈ A ≤
h u in D.
≤
A fundamental feature of (sub/super)solutions of (2.3) is the following well-
known comparison principle: If u W1,p(Ω) is a supersolution and v W1,p a
∈ ∈
subsolution of (2.3) in Ω such that max(v u,0) W1,p(Ω), then u v a.e. in
− ∈ 0 ≥
Ω. The existenceof -harmonicfunctionsisgivenbythe followingresult. Suppose
that Ω ⋐ M is a relAatively compact (nonempty) open set and that θ W1,p(Ω).
∈
Then there exists a unique -harmonic function u in Ω, with u θ W1,p(Ω).
A − ∈ 0
Given a function f C(∂ M) the Dirichlet problem at infinity for -harmonic
∞
functions consists in fi∈nding a function u C(M¯) such that (u) = A0 in M and
∈ A
u = f. In order to solve the Dirichlet problem for the -harmonic functions,
|∂∞M A
we will use Perron’s method. Let p(M), with p (1, ). We begin by
A ∈ A ∈ ∞
recalling the definition of the upper class of a function f ∂ M.
∞
∈
Definition 8. A function u: M ( , ] belongs to the upper class of
f
→ −∞ ∞ U
f: ∂ M [ , ] if
∞
→ −∞ ∞
(1) u is -superharmonic in M,
A
(2) u is bounded from below and,
8 JEAN-BAPTISTECASTERAS,ILKKAHOLOPAINEN,ANDJAIMEB.RIPOLL
(3) liminfu(x) f(x ), for all x ∂ M.
0 0 ∞
x→x0 ≥ ∈
The function
H =inf u: u
f f
{ ∈U }
is called the upper Perron solution and H = H the lower Perron solution.
f − −f
Theorem 9. One of the following is true :
(1) H is -harmonic in M,
f
A
(2) H in M,
f
≡∞
(3) H in M.
f
≡−∞
Next we define -regular points at infinity.
A
Definition 10. A point x ∂ M is called -regular if
0 ∞
∈ A
lim H (x)=f(x )
f 0
x→x0
for all f C(∂ M).
∞
∈
It is easy to see that the Dirichlet problem at infinity for -harmonic functions
A
is uniquely solvable if every point at infinity is -regular.
A
2.4. Minimal graph equation. Let Ω M be an open set. We say that a
function u W1,1(Ω) is a (weak) solution o⊂f the minimal graph equation (1.1) if
∈ loc
u, ϕ
(2.5) h∇ ∇ i =0
1+ u2
ZΩ |∇ |
for every ϕ C∞(Ω). Note that thpe integral above is well-defined since
∈ 0
1+ u2 u a.e.,
|∇ | ≥|∇ |
and therefore p
u, ϕ u ϕ
h∇ ∇ i |∇ ||∇ | ϕ < .
ZΩ (cid:12)(cid:12) 1+|∇u|(cid:12)(cid:12)2 ≤ZΩ 1+|∇u|2 ≤ZΩ|∇ | ∞
In fact, it is equivaplent to require (2.p5) for every ϕ W˚1,1(Ω). Indeed, let ϕ
∈ 0 ∈
W˚1,1(Ω) and let (ϕ ) be a sequence in C∞(Ω) such that ϕ ϕ in L1(Ω).
0 j 0 ∇ j → ∇
Supposing that (2.5) holds for all such ϕ , we get
j
u, ϕ u, ϕ u, ϕ u, ϕ ϕ
j j
h∇ ∇ i = h∇ ∇ i h∇ ∇ i = h∇ ∇ −∇ i
(cid:12)(cid:12)ZΩ 1+|∇u|2(cid:12)(cid:12) (cid:12)(cid:12)ZΩ 1+|∇u|2 −ZΩ 1+|∇u|2(cid:12)(cid:12) (cid:12)(cid:12)ZΩ 1+|∇u|2 (cid:12)(cid:12)
(cid:12)(cid:12) p (cid:12)(cid:12) (cid:12)(cid:12) pu ϕ ϕj p (cid:12)(cid:12) (cid:12)(cid:12) p (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) |∇ ||∇ −∇ | ϕ (cid:12)ϕj (cid:12) 0 (cid:12)
≤ 1+ u2 ≤ |∇ −∇ |→
ZΩ |∇ | ZΩ
as j 0. The following lepmma guarantees the existence of (strong) solutions of
→
(1.1) with given boundary values.
Lemma 11. Suppose that Ω ⋐ M is a smooth relatively compact open set whose
boundary has nonnegative mean curvature with respect to inwards pointing unit
normalfield. Then for eachf C2,α(Ω¯)thereexistsauniqueu C∞(Ω) C2,α(Ω¯)
∈ ∈ ∩
that solves the minimal graph equation (1.1) in Ω with boundary values u∂Ω =
|
f ∂Ω.
|
ASYMPTOTIC DIRICHLET PROBLEM 9
Proof. Thislemmafollowsfromwell-knowntechniquesusedinthecontinuitymethod
of elliptic PDE theory and therefore we just sketch the argument. Set
V = t [0,1]: u C2,α(Ω¯) such that [u]=0 in Ω and u∂Ω=tf ∂Ω .
{ ∈ ∃ ∈ M | | }
We have V = since 0 V. Moreover, by the implicit function theorem, V is
6 ∅ ∈
open in [0,1]. Given t V, let u be a solution of (1.1) such that u∂Ω = tf ∂Ω.
∈ | |
Since constant functions are solutions of (1.1), we have supΩ¯|u| ≤ max∂Ω|f| by
the comparison principle (see e.g. [8, Lemma 1]. Also, since ∂Ω has nonnegative
mean curvature with respect to inwards pointing unit normal field, we may use
classical logarithmic type barriers to prove that max u C where C is a
∂Ω
|∇ | ≤
constant that depends only on f and on Ω (see e.g. [15, Section 4] for details). By
[36, Lemma 3.1] we have maxΩ¯|∇u| ≤ C for some constant independent of u and
t. Ho¨lder estimates and theory of linear elliptic PDEs imply that the C2,β norm
of u is bounded by a constant depending only on f and Ω for some 0 < β < 1.
Then, if t V converges to t [0,1] and u is a solution of (1.1) such that
n n
∈ ∈
u ∂Ω = t f ∂Ω, then (u ) contains a subsequence converging in the C2 norm on
n n n
Ω¯ t|oasolutio|nu C2(Ω¯)of (1.1)inΩsuchthatu∂Ω=tf ∂Ω. Regularitytheory
implies that u C∈∞(Ω) C2,α(Ω¯). It follows that|t V, s|o that V is closed and
hence V =[0,1∈]. ∩ ∈ (cid:3)
From now on we will mainly consider solutions of (1.1) that are at least C2-
smooth.
2.5. Young functions. Let φ: [0, ) [0, ) be a homeomorphismand let ψ =
∞ → ∞
φ−1. Define Young functions Φ and Ψ by setting, for each t [0, ),
∈ ∞
t
Φ(t)= φ(s)ds
Z0
and
t
Ψ(t)= ψ(s)ds.
Z0
Then we have the Young inequality
ab Φ(a)+Ψ(b)
≤
for all a,b [0, ). The functions Φ and Ψ are said to form a complementary
∈ ∞
Young pair. Furthermore, Φ (and similarly Ψ) is a continuous, strictly increasing,
and convex function satifying
Φ(t)
lim =0
t→0+ t
and
Φ(t)
lim = .
t→∞ t ∞
Such Young functions are usually called N-functions (nice Young functions) in the
literature; see e.g. [29] for a more general definition of Young functions.
Following [42] we consider complementary Young pairs of a special type. Sup-
pose that a homeomorphism G: [0, ) [0, ) is a Young function that is a
∞ → ∞
diffeomorphism on (0, ) and satisfies
∞
1 1
(2.6) dt< ,
G−1(t) ∞
Z0
10 JEAN-BAPTISTECASTERAS,ILKKAHOLOPAINEN,ANDJAIMEB.RIPOLL
and
tG′(t)
(2.7) lim =1.
t→0 G(t)
ThenG(1/p)p, withp 1,isalsoaYoung functionandwecandefine F: [0, )
· ≥ ∞ →
[0, ) so that G(1/p)p and F(1/p) form a complementary Young pair. The space
∞ · ·
ofsuchfunctions F willbe denotedby . Note thatif F , then λF and
p p p
F ∈F ∈F
F(λ) for everypositive constantλ. It is provedin [42] that is non-empty.
p p
· ∈F F
More precisely, we have the following:
Proposition 12. [42, Proposition 4.3] Fix ε (0,1). There exists F such
0 p
∈ ∈ F
that
(2.8) F(t) tp+ε0exp 1 log e+ 1 −1−ε0
≤ −t t
for all t [0, ). (cid:16) (cid:0) (cid:0) (cid:1)(cid:1) (cid:17)
∈ ∞
Weomitthedetailsofthe proofofProposition12andreferto[42]. Herewejust
sketchtheconstruction. ThefunctionF isobtainedbyfirstchoosingλ (1,1+ε )
0
∈
and a homeomorphism H: [0, ) [0, ) that is diffeomorphic on (0, ) and
∞ → ∞ ∞
satisfies
log1 −1 loglog1 −λ if t is small enough;
H(t)= t t
((cid:0)tp/ε0 (cid:1) (cid:0) (cid:1) if t is large enough,
and then setting G(t) = tH(s)ds. Then G and G˜, G˜(t) = G(t1/p)p, are Young
0
functions. Let F˜ be the complementary Young function to G˜ and, finally, define F
R
by setting F(t)=cF˜(tp) for a suitable positive constant c.
Since G is convex, we have G(t) ct for all t 1. Therefore G−1(t) ct for all
t large enough and, consequently, ≥∞1/G−1 =≥ . Taking into accoun≤t (2.6) we
0 ∞
conclude that the function γ, defined by
R
t 1
γ(t)= ds,
G−1(s)
Z0
is a homeomorphism [0, ) [0, ) that is a diffeomorphism on (0, ). Hence
∞ → ∞ ∞
the same is true for its inverse
(2.9) ϕ=γ−1: [0, ) [0, ).
∞ → ∞
We collect the properties of ϕ to the following lemma.
Lemma13. [42,Lemma4.5]Thefunctionϕ: [0, ) [0, )isahomeomorphism
∞ → ∞
that is smooth on (0, ) and satisfies
∞
(2.10) G ϕ′ =ϕ,
◦
and
ϕ′′(t)ϕ(t)
(2.11) lim =1.
t→0+ ϕ′(t)2
Fromnowon,ϕwillbethefunctiondefinedin(2.9)suchthatthecorresponding
F satisfies (2.8). We define an auxiliary function ψ = (ϕ′)p−1ϕ. It is easy
p
∈ F
to see that ψ: [0, ) [0, ) is a homeomorphism that is smooth on (0, ). It
∞ → ∞ ∞
follows from (2.11) that
ψ′(t)
(2.12) lim =p.
t→0+ϕ′(t)p