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, 1A2, 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¨häkangas 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 Gálvez 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 operator 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 assumptions 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 dimension 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¨häkangaswasable 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 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 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’sandBorbély’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 Borbély’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 onaMoseriterationprocedureinvolvingaweightedPoincaréinequality. 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 Poincaréinequality 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

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

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