STABLE MINIMAL GRAPHS IN THE HEISENBERG GROUP Hn GIOVANNACITTIANDMATTEOGALLI ABSTRACT. WeprovethatastrictlystableminimalCh2intrinsicgraphGislocallyarea-minimizing, 7 i.e. givenany C1 graph S withthesameboundary, Area(G)<Area(S)unless G =S. Asacon- h 1 sequenceweshowtheexistenceandtheuniquenessofC∞minimalgraphswithprescribedsmall 0 boundarydatum. 2 n a J CONTENTS 2 2 1. Introduction 1 ] 2. Preliminaries 3 G 3. Stabilityoperatorforminimalgraphs 6 D 4. Anexistenceresult 8 . 5. Areaminizingproperty 10 h t References 11 a m [ 1 1. INTRODUCTION v 4 The Heisenberg group, denoted as Hn can be identified with R2n+1, with the choice at every 1 pointofanhorizontal2n-dimensionaldistributionH,suchthat[H,H]=H hasdimension1, 2 2 6 (1.1) THn=H ⊕H ,[H,H]=H and[H,H ]=0. 2 2 0 0 . Starting from the seminal paper [21] by Garofalo and Nhieu, where general properties and 1 existenceof sets with minimum perimeter are proved in Carnot groups, a remarkable efforthas 0 7 been devoted to the development of an exhaustive theory for minimizers of the area functional 1 inthesub-Riemannian setting. Howeverthetheoryisstillveryfarfrombeingcomplete. : v Due to the lack of symmetry of the space, surfaces can have very different expression if ex- i X pressed as graph on different variables. Identifying via theexponential coordinates the Heisen- berg group with its algebra, we can consider graphs defined on the horizontal distribution H r a with values in H . Graphs of this type with prescribed mean curvature f, called t-graphs, are 2 criticalpointsofthefunctional F(u)= |∇u+F~|+ fu, ˆ ˆ Ω Ω Date:January24,2017. 2000MathematicsSubjectClassification. 53C17,49Q20. Key words and phrases. Sub-Riemannian geometry, contact manifolds, stable area-stationary surfaces, Schauder estimates. TheresearchleadingtotheseresultshasreceivedfundingfromthePeopleProgramme(MarieCurieActions)ofthe EuropeanUnion’sSeventhFrameworkProgrammeFP7/2007-2013/underREAgrantagreementn.607643. 2 G.CITTIANDM.GALLI onadomainΩ,where F~ isavectorfieldand f ∈L∞(Ω). In[9]Chengetal. provedtheexistence of C1 minimizing t-graphs in Hn that at our knowledge, this is the only existence results for prescribedmeancurvaturegraphs. Ontheothersidewhilechoosinggraphswhosenormalateverypointbelongstothehorizontal distribution we obtain theso called intrinsic graphs introduced by Franchi, Serapioni and Serra Cassano in [14] and [15]. It has been proved, in [1] and [11], that in this case there are non- linearvectorfields (X ,···,X ) 1,u n,u suchthattheareafunctionalofthegraphofucanbewrittenas A(u)= 1+|∇uu|2 dL2n. ˆ Ω p We remark that this sub-Riemannian area functional is not convex. This is why standard existence technique do not apply. However existence of minima of this functional have been obtainedbySerraCassanoandVittone,[30],buttheyhaveonly BV regularity. Inparticularitis not clear if theysatisfytheassociatedfirst variation equation which express thecuruvature of a surface: 2n−1 Xu(u) (1.2) H(u)=dA(u)= Xu i =0. i i=1 (cid:18) 1+|∇u(u)|2(cid:19) X This is a second order quasilinear subelliptic equaption, but due to the expression of the vector fields X , the regularity of the solution is not understood, and there is still a big gap between i,u this existence result for minima and their regularity properties. Regularity results in the three- dimensionalcase,withspecialattentiontothefirstHeisenberggroupH1 hasbeenobtainedonly for Lipschitz continous solutions. Among them, we would like to stress [10], [7] , [2], [19], [20]and[17]. Indimensionbiggerthenthreetheonlyregularity result inHn, n>1,is proved by Capogna et al. in [8], always forLipschitz continuous solutions. Themain obstacleto study directlythetheproblem H =0, inΩ v (1.3) (v=φ, in∂Ω isthefactthattheoperator (1.4) L (v)=dH(u)(v) u does not satisfy the maximum principle for general functions u. Indeed in the subriemannian settingthestabilityoperator L v takestheform u 2n−1 2n−1 L v= Xu(a (∇uu)Xuv)+ (∂ u)a (∇uu)Xuv u i ij j 2n i2n−1 i i,j=1 i=1 X X Xu u 2n−1 + ∂ 2n−1 + Xu(∂ ua (∇uu)) v, 2n i 2n i2n−1 1+|∇uu|2 (cid:18) (cid:18) (cid:19) i=1 (cid:19) X (see Proposition 3.1), and nothping is known on the sign of the zero order term. In addition Schauderestimatesarenotavailableforgeneralsubriemannianvectorfields. STABLEMINIMALGRAPHSINHn 3 Inthispaperweinvestigatepropertiesofstrictlystablecriticalpoints,whicharegraphsG with u vanishing mean curvature in an open set Ω such that, for all compact test functions v ∈ C2(Ω), h v6≡0,theindexform I(v,v)=− L (v),v u is strictly positive, where L (v) is defined in (1.(cid:10)4). Our(cid:11)main result is stated in Theorem 5.3 u below: Let u∈ C2(Ω˜) and let u be a strictly stable critical point of the area functional h inadomainΩ⊂Ω˜. Thenthereexistsatubularneighborhood U ofΩsuchthat forany C1 graphS⊂U,∂Ω=∂S,wehaveA(G )<A(S)orG =S. h u u This result extends to the present setting the one proved in Riemannian manifolds by White [31], using regularity theorems from geometric measure theory. Grosse-Brauckmann [23], see also [28, §109], show that another way to prove this result is to foliate a neighbourhood of a strictlystableextremalwithstationarysurfaces. Thefieldofnormalstotheleavesisacalibration, andthestatementfollowsbythedivergencetheorem. When u is a strictly stable critical point u, L (v) is positive, hence invertible. Hence it is u possible to show an existence result in a neighborhood of any stable point. In particular, since u=0hasthispropertywededucethat Thereexistsǫ >0such thatif||φ|| ¶ǫ ,thentheproblem(1.3)has an 0 L∞(∂Ω) 0 uniquesolution v∈C∞(Ω). ThedelicateaspectoftheproofoftheexistencetheoremisthelackofSchauder estimatesat the boundary. As we mentioned before, they are not known in the subriemannian setting and the only result in this direction is due to Jerison [24]. However his proof can not be repeated ingeneral Liegroups, sinceit isbasedonFouriertransform. Onthecontrary, internal Schauder estimates are well known in this context, aftertheresults of [29] (we also quotea more recent contribution[6]). Henceweproveanadhocversion ofSchauder estimateswith apenalization ontheboundaryoftheset,whicharesufficienttoobtaintheresult. Usingthisexistenceresult,weareabletofollowthesameideaof[23]andconstructafoliation byminimalgraphsinatubularneighborhoodofastrictlystableminimalgraph. Thepositivityof the operator L for strictly stable function uimplies that theoperator L satisfies the maximum u u principle(seealsoProposition5.1). AsaconsequencewewillestablishtheproofofTheorem5.3. As a corollary we will deduce that this critical point of the area functional, found above for ||φ|| ¶ǫ ,isindeedastableminimumwithprescribedboundarydatum. L∞(∂Ω) 0 The paper is organized as follows. In Section 2 we provide the necessary background on the sub-Riemannian Heisenberg group and intrinsic graphs with prescribed mean curvature. In Section3weintroducethestability operator. Insection3weprovetheSchauder estimates and our existence result. The local area-minimizing property of a strictly stable minimal intrinsic graphwillappearinSection5. 2. PRELIMINARIES Inthissectionwegathersomeresultstobeusedinlatersections. 4 G.CITTIANDM.GALLI 2.1. The Heisenberg group Hn. Thestructure of the Heisenberg group Hn can be modeled on R2n+1 usingthefollowingbasisofleft-invariant vectorfields ∂ ∂ ∂ X = ,i=1,...,n−1, X = −x ,i=n,...,2n−2, i ∂x i ∂x i−n+1∂x i i 2n ∂ ∂ ∂ ∂ X = , X = −z , X = . z ∂z 2n−1 ∂x ∂x 2n ∂x 2n−1 2n 2n Thevectorfields{X ,X ,...,X }generatethehorizontaldistributionH,while T iscalledthe z 1 2n−1 Reeb vector field and it is transverse to H. A vector field X is called horizontal if X ∈ H. A horizontalcurveisaC1 curvewhosetangentvectorliesinthehorizontaldistribution. Notethat [X ,X ]=[X ,X ]=T, i=1,...,n−1, z 2n−1 i i+n−1 while theother commutators vanish, so that H is a bracket-generating distribution and Hn has vanishing pseudo-hermitian Webster curvature and pseudo-hermitian torsion, see [13] or [16]. For this reason the Heisenberg group is the model example of pseudo-hermitian manifolds and playthesamerolethattheEuclideanspacehaswithrespecttoaRiemannianmanifold. 2.2. Theleftinvariantmetric. WeshallconsideronHntheRiemannianmetric g= ·,· sothat {X ,X ,...,X ,T}isanorthonormalbasisateverypoint. Therestrictionof g toH coincides z 1 2n−1 (cid:10) (cid:11) withtheusualsub-Riemannian metricinHn inducedbythevectorfieldsX ,X ,...,X . z 1 2n−1 ForanytangentvectorU onHn wedefineJ(U)=D T,where DistheLevi-Civitaconnection U associatedtotheRiemannian metric g. Thenwehave J(X )=X ,J(X )=−X ,J(X )=X ,J(X )=−X , z 2n−1 2n−1 z i i+n−1 i+n−1 i for i = 1,...,n−1, and J(T) = 0, so that J2 = −Id when restricted to H. The involution J :H →H providesacomplexstructureonHn,see[4]. 2.3. Geometry of surfaces in Hn. Given a C1 surface Σ immersed in Hn we define the sub- RiemannianareaofΣby (2.1) A(Σ)= |N |dΣ, ˆ h Σ whereN istheunitnormalvectorwithrespecttothemetric g,N istheorthogonalprojectionof h N toH anddΣistheRiemannianareaelementofΣ. ThesingularsetΣ consistsofthosepoints 0 p where H coincides with the tangent plane T Σ of Σ. We define the horizontal unit normal p p vectorν (p)andthecharacteristicvectorfield Z(p)by h N (p) (2.2) ν (p):= h , Z(p):=J(ν )(p) h |N (p)| h h forall p∈Σ−Σ . Since Z isorthogonaltoν andhorizontal,wegetthat Z istangenttoΣ. 0 p h p 2.4. EuclideanLipschitzgraphsinHn. LetW ={(0,x)∈Hn:x ∈R2n},weconsiderthegraph Σ = {(z,x) : z = u(x),x ∈ Ω} ⊂ Hn of an Euclidean Lipschitz function u : Ω ⊂ W → R. Let {E} beabasisofthehorizontaltangentspace TΣ∩H,where i i=1,...,2n−1 E =(X u)X +X , i=1,...,2n−1. i i z i STABLEMINIMALGRAPHSINHn 5 Wedenoteby{Xu} abasisofthehorizontaltangentvectorsprojectedtoΩ i i=1,...,2n−1 Xu=X , i=1,...,2n−2 i i ∂ ∂ Xu = +u(x) . 2n−1 ∂x ∂x 2n−1 2n Thegradient∇u isdefinedas ∇u=(Xu,...,Xu ) 1 2n−1 and∇uuiswell-definedandcontinuousinΩ,sinceG inaso-calledintrinsicgraph,[15]and[3], u infacttheintegral curves of X starting fromΩmeet G inexactlyone point. Thearea formula z u (2.1)foragraphG canbeexpressedas u (2.3) A(G )= (1+|∇uu|2)1/2dL2n, u ˆ Ω whereL2n denotestheLebesguemeasureonΩ,[1,Proposition2.22]. 2.5. Graphswithprescribedmeancurvature. LetG thegraphofanEuclideanLipschitzfunc- u tionu:Ω⊂W →R. G hasprescribedmeancurvature f ifitisacriticalpointofthefunctional u (2.4) A(G ∩B)− f, u ˆ E ∩B u for any bounded open set B in the cylinder {(z,x) ∈ Hn : x ∈ Ω}. Here we have denoted by E = {(z,x) ∈ Hn : x ∈ Ω,z < u(x)} the subgraph of G . We remark that our definition is the u u counterpartoftheonesgiveninthree-dimensionalsub-Riemannian contactmanifolds,[20]and [17],andintheEuclideansetting,[25,(12.32)andRemark17.11]. 2.6. Distance generatedby vectorfields. Notethatthevectorfields Xu,...,Xu satisfyHor- 1 2n−1 mander’sfiniterankconditioninR2n. Consequentlytheygiverisetoacontroldistanced ,whose u metric balls B ofradius r have volume comparable to r2n+1, where2n+1is thehomogeneous r dimension of the space (R2n,d ). The distance d coincides with the sub-Riemannian distance u u restrictedtotheverticalplanecontainingΩ,see[27]. 2.7. Function spaces. Let f be a continuous function on E ⊂ Hn. We say that f ∈ Ck(E) if h Y (...(Y (f))) exists continuous, for any Y ,...,Y ∈ H. It is easy to check that f ∈ C2k(E) 1 k 1 k h implies f ∈Ck(E). Given0<α<1,wedefineC0,α(Ω)asthespacecomposedbyallfunctions f :Ω→Rhaving d finitethefollowingnorm |f(x)− f(y)| ||f|| :=sup|f|+ sup dα , α,d,Ω x,y d (x,y)|α Ω x,y∈Ω,x6=y u where d :=min(d (x,∂Ω),d (y,∂Ω)) x,y u u and d denotes the distanceinduced by thevector fields Xu, i =1,...,2n−1. We also say that u i f ∈C2+α(Ω)ifthenorm d 2n−1 2n−1 ||f|| :=sup|f|+ supd|Xuf|+ ||d2XuXuf|| <+∞. 2+α,d,Ω i i j α,d,Ω Ω i=1 Ω i,j=1 X X 6 G.CITTIANDM.GALLI 2.8. First variation of the Area functional. We first recall that, if u is a graph, Lipschitz with respecttotheEuclideanmetric,theareaofthegraph G is u A(G )= 1+|∇uu|2 dL2n u ˆ Ω p (seeforexample[18],[12],[8]). Moreover,ifuisacriticalpointofthefunctional(2.4)wehave that 2n−2 X uX φ+Xu u(Xu )∗φ i i 2n−1 2n−1 (2.5) i=1 − fφ dL2n=0 ˆ P 1+|∇uu|2 Ω (cid:26) (cid:27) foranyφ∈C1(Ω). Inordertosimpplifynotationswewilldenote 0 X u Xu u (2.6) A(∇uu)= i fori=1···2n−2 A (∇uu)= 2n−1 . i 2n−1 1+|∇uu|2 1+|∇uu|2 As a consequence thpe definition of mean curvature for the graph Gpreads (see for example u [18],[12],[8]): 2n−1 Xuu 2n−1 (2.7) H := Xu i = XuA(∇uu) u i i i 1+|∇uu|2 i=1 (cid:18) (cid:19) i=1 X X Wecanalsoconsidertheoperatorofthepminimalsurfaceequationinthenon-divergenceform 2n−1 (2.8) M = a (∇uu)XuXu(u), u ij i j i=1 X where p p (2.9) a :R2n→R, a (p)=δ − i j . ij ij ij 1+|p|2 3. STABILITY OPERATOR FOR MINIMAL GRAPHS Inthissectionwedefinethestabilityoperatorforsubriemannianminimalgraphs. Proposition3.1. LetG ={z=u(x):x ∈Ω}beaC2 graphinHn andlet v∈C2(Ω). Weconsider u h h avariationof G oftheform G ={z=u(x)+sv(x): x ∈Ω}. Thenthesecond variationofthe u u+sv areaofG is u d2 A(G )= vL vdL2n, ds2 u+sv ˆ u (cid:12)s=0 Ω (cid:12) where (cid:12) (cid:12) 2n−1 a (∇uu) 2n−1 L v= Xu( ij Xuv)+ (∂ u)a (∇uu)Xuv u i j 2n i2n−1 i 1+|∇uu|2 i,j=1 i=1 (3.1) X X pXu u 2n−1 + ∂ 2n−1 + Xu(∂ ua (∇uu)) v, 2n i 2n i2n−1 (cid:18) (cid:18) 1+|∇uu|2(cid:19) i=1 (cid:19) X andthefunctionsa aredefinedpin(2.9). ij STABLEMINIMALGRAPHSINHn 7 Proof. Itisastandardcomputationshowthat d H =2n−1Xu+sv Xiu+sv(v) −2n−1Xu+sv Xiu+sv(u+sv)Xuj+sv(u+sv)Xjv ds u+sv i 1+|∇u+sv(u+sv)|2 i (1+|∇u+sv(u+sv)|2)3/2 i=1 (cid:18) (cid:19) i,j=1 (cid:18) (cid:19) X X +v∂ pX2un−1(u+sv) +Xu v∂2n(u+sv) 2n 2n−1 1+|∇u+sv(u+sv)|2 1+|∇u+sv(u+sv)|2 (cid:18) (cid:19) (cid:18) (cid:19) 2n−1 p Xu+sv(u+sv)Xu+sv(u+sv)v∂pu − Xu+sv i 2n−1 2n . i (1+|∇u+sv(u+sv)|2)3/2 i=1 (cid:18) (cid:19) X Whenweevaluatethisexpressionats=0toobtainthefollowingstatement: d 2n−1 Xu(v) 2n−1 Xu(u)Xu(u)Xuv H = Xu i − Xu i j j ds(cid:12)s=0 u+sv i=1 i(cid:18) 1+|∇u(u)|2(cid:19) i,j=1 i(cid:18)(1+|∇u(u)|2)3/2(cid:19) (cid:12) X X (cid:12)(cid:12) +v∂ pX2un−1(u) +Xu v∂2n(u) 2n 2n−1 1+|∇u(u)|2 1+|∇u(u)|2 (cid:18) (cid:19) (cid:18) (cid:19) 2n−1 pXu(u)Xu (u)v∂ u p − Xu i 2n−1 2n i (1+|∇u(u)|2)3/2 i=1 (cid:18) (cid:19) X Observingthat d 2n−1 2n−1 2n−1 H = a (∇uu)XuXuv+ Xu(a (∇uu))+∂ ua (∇uu) Xuv ds u+sv ij i j j ij 2n i2n−1 i (cid:12)s=0 i,j=1 i=1 (cid:18) j=1 (cid:19) (cid:12) X X X (cid:12) Xu u 2n−1 (cid:12) + ∂ 2n−1 + Xu(∂ ua (∇uu)) v 2n i 2n i2n−1 (cid:18) (cid:18) 1+|∇uu|2(cid:19) i=1 (cid:19) X 2n−1 p 2n−1 = Xu(a (∇uu)Xuv)+ ∂ ua (∇uu)Xuv i ij j 2n i2n−1 i i,j=1 i=1 X X Xu u 2n−1 + ∂ 2n−1 + Xu(∂ ua (∇uu)) v 2n i 2n i2n−1 (cid:18) (cid:18) 1+|∇uu|2(cid:19) i=1 (cid:19) X weobtainthethesis. p (cid:3) Wecannowintroducethefollowingdefinition Definition. We call stability operator L associated to a minimal graph G the operator L v u u u definedinequation(3.1). We consider a Euclidean Lipschitz minimal graph G with vanishing mean curvature in the u HeisenberggroupHnandasmoothdomainΩ. WesaythatG isstrictlystableifforall v∈C2(Ω) u h withcompactsupport, v6≡0,theindexform I(v,v)=− vL vdL2n ˆ u Ω isstrictlypositive,where L isthestabilityoperatordefinedin(3.1). u Lemma3.2. LetusnotethatthereexistsaconstantC suchthat,ifuisaminimumand||u|| ¶C, C2 thenuisstrictlystable 8 G.CITTIANDM.GALLI Proof. 2n−1 a (∇uu) 2n−1 vL v= vXu ij Xuv + v∂ ua (∇uu)Xuv ˆ u ˆ i j ˆ 2n i2n−1 i i,j=1 (cid:18) 1+|∇uu|2 (cid:19) i=1 X X pXu u 2n−1 + v ∂ 2n−1 + Xu(∂ ua (∇uu)) v ˆ 2n i 2n i2n−1 1+|∇uu|2 (cid:18) (cid:18) (cid:19) i=1 (cid:19) X Integratingbypartthefirsttermp,andusingthefactthat[X ,X ]=∂ ,,weget 1 n+1 2n 2n−1 a (∇uu) a (∇uu) vL v=− Xuv ij Xuv− v∂ u ij Xuv ˆ u ˆ i j ˆ 2n j 1+|∇uu|2 1+|∇uu|2 i,j=1 X 2n−1 p p + v∂ ua (∇uu)Xuv ˆ 2n i2n−1 i i=1 X Xu u Xu u − X 2n−1 vX v+ X 2n−1 vX v ˆ n+1 1 ˆ 1 n+1 1+|∇uu|2 1+|∇uu|2 (cid:18) (cid:19) (cid:18) (cid:19) 2n−1 p p − ∂ ua (∇uu)Xuvv− (∂ u)2a (∇uu)v2 ˆ 2n i2n−1 i ˆ 2n i2n−1 i=1 X Bythestructureoftheequationwehave 2n−1 a (∇uu) Xuv ij Xuv¾ |Xuv|2¾ |v|2 ˆ i j ˆ i ˆ i,j=1 1+|∇uu|2 X ByHölderinequalityandSobolevepmbeddingTheoremwehave vXuv¶ |Xuv||v|¶C |Xuv|2 ˆ i ˆ i Sˆ i sothat − vL v¾(1−C ||u||2 ) |Xuv|2. ˆ u S C2 ˆ i Itfollowsthat− vL v isstrictlypositiveif||u||2 issufficientlysmall. (cid:3) u C2 ´ 4. AN EXISTENCE RESULT Inthissectionweproveafirstexistenceresultforsolutionsoftheproblem(1.3). Preciselywe showthat Proposition 4.1. Let u ∈ C2(Ω˜) and let G be critical point of the area functional in Hn, n > 1. h u We assume that G is strictly stable in a domain Ω ⊂ Ω˜. Then there exist ǫ > 0, such that if u 0 ||φ−u|| ¶ǫ ,thentheproblem(1.3)hasanuniquesolution v∈C∞(Ω). L∞(∂Ω) 0 Let us explicitly note that, if u is a solution of class C2 of (1.3), then it is of class C∞. Using h mollifierswecanmimictheproofof[8,Theorem1.2]andwecaneasilyconcludethefollowing Lemma 4.2. Let G = {z = u(x) : x ∈ Ω} be a C2 graph in Hn with mean curvature H = g ∈ u h u C∞(Ω),n>1. Thenuisasmoothfunction. STABLEMINIMALGRAPHSINHn 9 Proof. Wedenotebyu thestandardmollificators ǫ u (x)= u(y)ϕ (x−y)ǫ−ndy. ǫ ˆ ǫ R2n Itissimpletoverifythat (i) X f →X f uniformlyoncompactsubsetsofΩ,forǫ→0. u¯ ǫ u¯ (ii) X2f →X2f uniformlyoncompactsubsetsofΩ,forǫ→0. u¯ ǫ u¯ For every ε, the function u satisfies the representation formula (4.2) in [8]. Letting ǫ go to 0, ǫ thesameformulaissatisfiedbyuHencewecannowproceedasin[8,Section4],andobtainthe smoothnessresult. (cid:3) Findingasolutionoftheproblem(1.3)isequivalent toprovetheinvertibilityofthemap F :C2,α(Ω)→C0,α(Ω)×C(∂Ω) d d definedby F(w)=(H ,w ), w ∂Ω As it is well known, thelocal invertibity property of F can be studied through its differential (cid:12) (cid:12) dF(u)= L , so that we focus onthis linear operator. Thecontinuiuty ofits inverse is expressed u bytheSchauderestimatesattheboundary,insuitable Cα spaces: Proposition4.3. Let L definedby (3.1),whereu:Ω⊂→R isasmoothfunction. Let f ∈Cα(Ω) u d andletv∈C2+α(Ω)aboundedfunctionsatisfying L v= f inΩ. Thenv∈C2+α(Ω)andthereexists loc u d C >0(independentof v)suchthat ||v|| ¶C ||v|| +||d2f|| . C0,α(Ω) L∞(Ω) C0,α(Ω) d d (cid:16) (cid:17) TheproofofProposition4.3isstandardwiththetechniquesdevelopedinthelasttwodecades, since L isasub-ellipticsecondorderlinearoperatorwithsmoothcoefficients. InteriorSchauder u estimatesfor L canbefoundin[6]. Hereweneedsimilarestimates,butweneedtoprovidean u explicitestimateoftheconstantC intermsofthedistancefromK totheexteriorofΩ. Due to the existence of a fundamental solution for L , it is also possible to obtain these esti- u mates,mimickingtheclassicalargumentpresentedintheEuclideansettingin[22,§4]. ProofofLemma4.1. Weconsiderthemap F :C2,α(Ω)→C0,α(Ω)×C(∂Ω)definedby d d F(w)=(H ,w ), w ∂Ω where H isthemeancurvatureofthegraphG . The(cid:12)differentialof F inuis w w (cid:12) dF (v)=(L v,v ). u u ∂Ω Thekernelof dF doesnotcontainanon-trivial functi(cid:12)on,sinceotherwisethefirsteigenvalue of u (cid:12) L inΩwouldbesmallerthanorequalto0. Ontheotherhand,theproblem u L v= f, inΩ u (v=φ, in∂Ω has a solution for all f ∈ C0,α(Ω) and φ ∈ C(Ω). From Proposition 4.3 the inverse of dF is d u continuous. By the Implicit Function Theorem, there is a diffeomorphism from a neighborhood ofuin C2,α(Ω)into aneighborhood of(H ,u )in C0,α(Ω)×C(∂Ω). Inparticular, there exists d u |∂Ω d 10 G.CITTIANDM.GALLI ǫ >0sothat,foreveryφ suchthat||φ−u|| ¶ǫ ,thegivenproblemhasasolution v such 0 L∞(∂Ω) 0 thatthegraph G of v isarea-stationary withzeromeancurvatureandG =ψ. v v ∂Ω Finallythefunction v isofclassC∞ forCorollary4.2. (cid:3) (cid:12) (cid:12) 5. AREA MINIZING PROPERTY Let us note that, if u is strictly stable, ten the first eigenvalue λ (Ω) of the operator L is 1 u positive,infactifλisaneigenvalueof L witheigenfunction v∈C2(Ω)∩C (Ω)wehave u h 0 0<− vL vdL2n=λ v2dL2n. ˆ u ˆ Ω Ω AsaconsequencethefollowingMaximumprincipleholds: Lemma5.1. Weconsidertheoperator L v definedin(3.1). Supposethatλ (Ω)>0,whereλ (Ω) u 1 1 denotes the first eigenvalue of L on a bounded C2,α domain Ω ⊂ Rn. Assume that Lv ¶ 0 and inf v>0,theninf v>0. ∂Ω Ω Proof. First we observe that the minimum of v can not be zero, otherwise v ≡ 0. This can be achieved mimic the classical proof of the Hopf’s maximum principle (see for example [22, Theorem 3.5] or [26, § 3]) and replacing the role of Euclidean balls with the sets δ O , r >0, r 1 introducedin[5,p.1161]. Nowwesupposethat vachievedanegativeminimumintheinteriorofΩ,thenwecanextend thetheoperator L inasmallneighborhoodΩ˜ ofΩandwecansupposethatthefirsteigenvalue u λ (Ω˜) is positive as well as therestriction to Ωof theeigenfunction v associated to λ (Ω˜). We 1 1 1 consider thefunction w = v/v on Ω, that must has a negative minimum insome interior point 1 p. Henceatthepoint p 2n−1 0¾ L (v)= L (wv )= L (v )w+v Xu(a (∇uu)Xuw)¾−λ (Ω˜)v w u u 1 u 1 1 i ij j 1 1 i,j=1 X andwecanconcludethatw(p)¾0and v(p)¾0incontradictionwithourassumption. (cid:3) Lemma5.2. Letu∈Lip(Ω˜)∩C2(Ω)andletG beaminimalgraphinHn,n>1. Weassumethat h u G isstrictlystableinadomainΩ⊂Ω˜. Thenthereexistǫ >0,atubularneighborhoodU ofG and u 0 u afamily G ofsurfaceswithvanishingmeancurvaturesuchthatG =G and G isafoliationof u u u u ǫ 0 ǫ U forǫ∈[−ǫ ,ǫ ]. 0 0 Proof. Asbeforeweconsiderthemap F(w)=(H ,w ). w ∂Ω We have proved that there is a diffeomorphism fro(cid:12)m a neighborhood of u in C2,α(Ω) into a (cid:12) d neighborhood of(H ,u ) in C0,α(Ω)×C(∂Ω). Inparticular, thereexists ǫ >0so that,forall u |∂Ω d 0 ǫ ∈ (−ǫ ,ǫ ), there is a function u such that the graph G of u is area-stationary with zero 0 0 ǫ u ǫ ǫ meancurvatureandG =u +ǫ. uǫ ∂Ω ∂Ω Letuscheckthattheunionofthegraphs G provideafoliationofatubularneighborhoodof (cid:12) (cid:12) uǫ G inHn. ThegraphsG(cid:12) provid(cid:12)eavariationofG inHn. Ifwecomputethevariational function u u u ǫ v:= d u ,then v satisfies dǫ ǫ=0 ǫ (cid:12) (cid:12)