Strict starshapedness of solutions to the horizontal p-laplacian in the Heisenberg group
aa r X i v : . [ m a t h . A P ] N ov STRICT STARSHAPEDNESS OF SOLUTIONS TO THE HORIZONTALP-LAPLACIAN IN THE HEISENBERG GROUP
MATTIA FOGAGNOLO AND ANDREA PINAMONTI
Dedicated to Alberto Farina on the occasion of his 50th birthday
Abstract.
We examine the geometry of the level sets of particular horizontally p -harmonicfunctions in the Heisenberg group. We find sharp, natural geometric conditions ensuring thatthe level sets of the p -capacitary potential of a bounded annulus are strictly starshaped. Introduction
The study of the geometric properties of the level sets of solutions to elliptic or parabolicboundary value problems is a classical but still very fertile field of research. Let us focus,without aiming to be complete, on the works most deeply linked with the object of the presentpaper. Starting in the classical ambient R n , consider bounded open sets Ω ⊂ Ω and let u be a p -harmonic function in Ω \ Ω attaining in some sense the value 1 on ∂ Ω and 0 on ∂ Ω . It is then quite natural to ask whether some geometric properties such as convexityor starshapedness of Ω and Ω are preserved by (the superlevel sets of) u . To the authors’knowledge, a first answer in the much easier case of p = 2, and in the space R dates back tothe 30s, when in [20] it was showed that the super level sets of u are starshaped if Ω and Ω are. Later, in [16, 18, 17] it was substantially shown that the same phenomenon occurs for theconvexity issue. The nonlinear case was arguably first considered in [34], where it was shownthat the p -capacitary potentials in starshaped rings are starshaped. It is important to pointout that the proof provided in such paper relies on a suitable symmetrization technique, thatdespite being powerful enough to treat very general equations [38], does not seem to provide the strict starshapedness (a notion to be described in a while) of u from that of Ω and Ω . We closethis historical excursus mentioning the fundamental [26], where, in addition to the challengingextension of the aforementioned convexity results to the nonlinear setting, the author fine-tunesthe maximum principle techniques we are adopting in the present work.Before venturing in a description of our main result, let us observe that the kind of issuewe just briefly discussed has recently gained attention also in the context of sub-Riemanniangeometries, that is actually the setting for this article. Indeed, in [10] it was shown that in thelinear situation p = 2 starshapedness of Ω and Ω is not only preserved by u , but actuallyimproved to strict starshapedness. The nonlinear generalization appeared in [14]. However,as in the standard Euclidean situation, if p = 2 it is not clear whether strict starshapednessis preserved. A C -domain containing the origin is said to be starshaped (with respect to theorigin) in the Heisenberg geometry if its outer unit normal ν satisfies h ν, Z i ≥
0, while it isstrictly starshaped if such inequality is strict. We are denoting by Z the dilation-generatingvector field. To fix the ideas, we may think of Z as the natural replacement for the Euclideanposition vector in the classical notion of starshapedness.The aim of the present paper is to explore the issue of strict starshapedness for the superlevelsets of p -capacitary potentials in the Heisenberg group H n . Leaving the definitions and a briefintroduction to the Heisenberg groups to the next section, let Ω ⋐ Ω ⋐ H n be C domains containing the origin, and consider, for p >
1, the solution u to ∆ H n p u = 0 in Ω \ Ω u = 1 on Ω u = 0 on ∂ Ω , (1.1)where by ∆ H n p we indicate the horizontal p -Laplacian, that is the natural analogue in H n of theclassical p -Laplacian in R n . Our main result substantially establishes that the strict starshaped-ness of Ω and Ω is preserved by u . Theorem 1.1.
Let u be a C -weak solution to (1.1) with p > for Ω ⊂ H n and Ω ⊂ H n bounded sets with C boundaries that are strictly starshaped with respect to the origin O ∈ H n and such that Ω ⋐ Ω . Assume also that Ω satisfies an uniform exterior gauge ball condition and Ω satisfies an uniform interior gauge ball condition . Then, { u ≥ t } ∪ Ω is a boundedset with C -boundary that is strictly starshaped with respect to O for any t ∈ [0 , . Moreover, |∇ u | 6 = 0 in Ω \ Ω . We are actually going to fully prove Theorem 1.1 for p = Q , with Q = 2 n + 2 since, with thetechniques adopted, the modifications needed to cover the case p = Q are straightforward, andillustrated in Remark 2.4 below.The above result will follow from a somewhat more general principle, Theorem 3.2 below,asserting that, if Ω and Ω satisfy the above conditions, then at any point where the classicalgradient of the solution u to (1.1) exists we have h∇ u, Z i uniformly bounded away from zero.Theorem 1.1 becomes then an immediate corollary of such statement. To the authors’ knowledge,it is not clear whether weak solutions to (1.1) actually do enjoy classical C -regularity fora general p >
1. On the other hand, it has been established in [12] for p belonging to aneighbourhood of 2. It has been moreover recently discovered in [33] that at least the horizontalgradient is H¨older continuous for any p > horizontal gradientof u does not vanish. This is actually sharp, since in the explicit, symmetric situation where Ω and Ω are two concentric gauge balls , i.e. defined with respect to the well known Koranyi normof H n , the level sets of u remain gauge balls, that in particular display a characteristic pointwhere the horizontal gradient vanishes, while the vertical derivative does not. This is also thereason why we cannot, with the available technology, infer higher regularity of u and its levelsets from the conclusion of Theorem 1.1. Indeed, it is known from the arguments in [36] thatthe nonvanishing of the horizontal gradient implies the smoothness of u , but it is not knownwhether the nonvanishing of the vertical derivative alone suffices to this aim.All in all, the issue of regularity for horizontally p -harmonic functions in Heisenberg groupshas been and still is a fervid field of research, and we think our work could serve as an additionalmotivation to carry on with that. In addition to the aforementioned contributions about thistopic, we cite the papers [30, 28, 5, 37]. Let us recall briefly that the optimal C ,α -regularity ofstandard p -harmonic functions is well known, and established independently in [11, 39].Let us now pass to discuss the last main assumption involved in Theorem 1.1, namely thetangent gauge ball condition we ask Ω and Ω to be subject to. Such property constitutesclearly a natural analogue of the round tangent ball conditions in the Riemannian geometry,that is indeed heavily used to deal with barrier arguments, as those performed here. On theother hand, differently from classical situations, a tangent gauge ball to the boundary of a setΩ ⊂ H n is not ensured no matter the regularity of ∂ Ω. This is explicitly shown in [29].The proof of Theorem 1.1 is inspired by the barrier argument used in the proof [26, Lemma 2].Indeed, such result can be substantially rephrased as the standard Euclidean version of Theorem
TRICT STARSHAPEDNESS OF SOLUTIONS TO THE HORIZONTAL P-LAPLACIAN 3 p -Laplacian. Closingthese brief comments on the proof, we observe that these ideas, at least from an heuristic pointof view, seem to be exportable to more general Carnot groups. On the other hand, as oftenoccurs, technical challenges could arise when dealing with such a generalization. We could comeback on this topic in future works.Let us comment on possible geometric applications and perspectives of results such as Theorem1.1. Problem (1.1) for the standard p -Laplacian, with Ω dilated away at infinity, has beenrecently utilized in [15] and [1] as a substitute for the Inverse Mean Curvature Flow [23, 32]to infer geometric and analytical inequalities for hypersurfaces in R n . In the earlier [15] thenonvanishing of |∇ u | was assumed in order to establish suitable monotonicity formulas. Inparticular, by the aforementioned [26, Lemma 2] , the results of [15] hold true for strictlystarshaped domains. Theorem 1.1 can thus be read as a first indication about the viability ofthese techniques in the sub-Riemannian setting. More generally, the preserving of starshapednessalong suitable evolutions of hypersurfaces is a highly desirable and thoroughly studied propertyin Geometric Analysis, let us cite for the mere sake of example [21, 41, 4, 35].The present paper is structured as follows. In Section 2 we recall and discuss the prepara-tory material we are going to need for the proof of Theorem 1.1. More precisely, we reviewdefinitions and basic properties of the Heisenberg group, we discuss hypersurfaces that fulfilstarshapedness and tangent gauge balls conditions, and recall some fundamental facts abouthorizontally p -harmonic functions. In Section 3 we work out the proof of Theorem 1.1, that asalready mentioned will follow from the slightly more general Theorem 3.2.2. The Heisenberg group and horizontally p -harmonic functions We summarize below some properties of the Heisenberg group that we will need throughoutthe paper. We follow here the presentation given in [31] and we address the interested reader to[3] for a complete overview.2.1.
The Heisenberg group.
Let n ≥
1. We denote by H n the Lie group ( R n +1 , · ), where thegroup product between z = ( x , . . . , x n , y , . . . , y n , t ) and ˜ z = (˜ x , . . . , ˜ x n , ˜ y , . . . , ˜ y n , ˜ t ) is definedby z · ˜ z = x + ˜ x , . . . , x n + ˜ x n , y + ˜ y , . . . , y n + ˜ y n , t + ˜ t − n X i =1 x i ˜ y i + ˜ x i y i !! . We denote by z − the inverse of z ∈ H n with respect to the group law defined above. The Liealgebra g of left invariant vector fields of H n is spanned by the vector fields X i = ∂∂x i + 2 y i ∂∂t , Y j = ∂∂y j − x j ∂∂t , T = ∂∂t i, j = 1 , . . . , n. It is easy to see that the Lie algebra of H n is stratified of step 2, i.e. denoting by V = span { X i , Y i , i = 1 , . . . , n } , V = span { T } it holds g = V ⊕ V and [ V , V ] = V , where [ V , V ] = span { [ v, w ] , v, w ∈ V } . As usual we refer to V as the horizontal layer and to X i and Y j as horizontal vector fields . M. FOGAGNOLO AND A. PINAMONTI
Given an open subset U ⊆ H n , and a function f ∈ C ( U ), we denote by ∇ H n f the horizontalgradient of f defined as ∇ H f = ( X f, . . . , X n f, Y f, . . . , Y n f )while we denote with ∇ the classical gradient in R n +1 .Given any two vector fields Z and W on H n we are going to consider its classical scalar productin R n +1 and to denote it by h Z, W i . Similarly, we indicate the R n +1 -norm of Z simply by | Z | .Various distances are usually considered in relation with Heisenberg groups, but we limit our-selves to the one induced by the Koranyi homogeneous norm. Given z, w ∈ H n with coordinatesas above, we define the Koranyi homogeneous norm as follows ρ ( z ) = ρ ( x , . . . , x n , y , . . . , y n , t ) = n X i =1 x i + y i ! + t and then the Koranyi, or gauge , distance between z and w is simply ρ ( w − · z ). Accordingly, wedenote by B ρ ( z, R ), for R >
0, the open ball with respect to such metric, that is B ρ ( z, R ) = { w ∈ H n | ρ ( w − z ) < R } . We refer to B ρ ( z, R ) as the gauge ball of center z and radius R .The proof of our main result, i.e. Theorem 1.1, requires suitable dilations of subsets andfunctions. To this end, let us recall the dilation δ λ : H n → H n of parameter λ ∈ R defined by δ λ ( x , , . . . , x n , y , . . . , y n , t ) = ( e λ x , . . . , e λ x n , e λ y , . . . , e λ , y n , e λ t ) . With such definition at hand, we introduce the following notations for dilated subsets and dilatedfunctions. For a subset A ⊂ H n and λ ∈ R , we define its dilation as A λ = δ λ ( A ) = { δ λ ( z ) | z ∈ A } . (2.1)Moreover, for a function f : A → R , we define its dilated f λ : A − λ → R by f λ ( z ) = f ( δ λ ( z )) . Starshaped sets and gauge ball properties.
The following is the definition we adoptfor starshaped sets in the Heisenberg group, well posed for sets with C -boundary. We addressthe reader to [13] and [14] for more extensive discussions on this geometric property in the moregeneral context of Carnot groups and for some equivalent definitions. Definition 2.1 (Starshaped and strictly starshaped sets in Heisenberg groups) . An open boundedset Ω ⊂ H n with C boundary is starshaped with respect to the origin if it contains the originand h ν, Z i ( z ) ≥ for any z ∈ ∂ Ω , where ν is the Euclidean exterior unit normal to ∂ Ω , and Z = ( x , . . . , x n , y , . . . , y n , t ) is the dilation-generating vector field. The set Ω is called strictly starshaped if inequality (2.2) holds with strict sign at any z ∈ ∂ Ω . It is worth observing the well-known relation between the vector field Z appearing above andthe geometry of H n . As the name of dilation-generating vector field suggests, the flow of Z isgiven by the dilation map H n × R → H n given by ( z, λ ) → δ λ ( z ). In other words, we have ddλ δ λ ( z ) = Z ( δ λ ( z )) , δ ( z ) = z (2.3)for any z ∈ H n , λ ∈ R . TRICT STARSHAPEDNESS OF SOLUTIONS TO THE HORIZONTAL P-LAPLACIAN 5
We now recall the definition of boundaries satisfying an interior or exterior gauge-ball condi-tion.
Definition 2.2 (Interior and exterior gauge-ball property) . Let Ω ⊂ H n be an open boundedset. Then, we say that Ω satisfies the exterior gauge ball property at a point z ∈ ∂ Ω if thereexist z ∈ H n \ Ω and R > such that B ρ ( z , R ) ⊂ H n \ Ω and B ρ ( z , R ) ∩ ∂ Ω = { z } . We saythat Ω satisfies a uniform exterior gauge ball property if R is uniform for z ∈ ∂ Ω .Similarly we say that Ω satisfies the interior ball condition at a point z ∈ ∂ Ω if there exist z ∈ Ω and R > such that B ρ ( z , R ) ⊂ Ω and B ρ ( z , R ) ∩ ∂ Ω = { z } . Again, we say that Ω satisfies a uniform interior gauge ball property if R is uniform for z ∈ ∂ Ω . It is worth pointing out that, strikingly differently from the Euclidean case, one can easilyfind in H n sets with smooth boundary not satisfying exterior or interior gauge ball conditions.This is ultimately due to the lack of strict, uniform Euclidean convexity of gauge balls, that is,their boundaries display points where some principal curvature (with respect to the flat metricof R n +1 ) vanishes. An explicit example of domains with smooth boundary not satisfying gaugeball conditions is shown in [29]. Indeed, in such paper it is shown, precisely in [29, Proposition2.4], that an interior gauge ball condition at a point z ∈ ∂ Ω, for some open set Ω, sufficesto prove a Hopf boundary point lemma for harmonic functions at z . On the other hand, theauthors provide in [29, Counterexample 2.3] a set with paraboloidal boundary in H with theHopf property failing on the vertex, that in particular does not admit a gauge ball touchingfrom the inside.It is also important to remark that while it is very easy to find sets with characteristicboundary points satisfying gauge ball condition (gauge balls themselves provide such examples),non-characteristic points of C , boundaries enjoy exterior and interior touching ball condition.This is shown in [19, Theorem 8.4]. In particular, the conditions of Definition 2.2 are strictlyweaker than being non-characteristic. Finally, we address the interested reader to [8, 24, 25, 40]and references therein for some discussions on the importance of the gauge-ball property fromthe regularity standpoint.In the proof of Theorem 1.1, we are using the following natural property of (strictly) star-shaped sets. It substantially consists in a refinement, holding true under the additional assump-tions of exterior or interior gauge ball conditions, of similar properties described in the moregeneral context of Carnot groups in [14, Proposition 4.2]. Proposition 2.3.
Let Ω ⊂ H n be a strictly starshaped open bounded set with C -boundary.Assume that Ω satisfies the exterior gauge ball condition at z ∈ ∂ Ω , and let B ρ ( z , R ) theexterior tangent gauge ball at z . Then, there exists λ > such that for any < λ ≤ λ we have δ λ ( z ) ∈ B ρ ( z , R ) .Analogously if Ω admits an interior tangent gauge ball B ( z , R ) at z ∈ ∂ Ω , then there exists λ > such that any − λ < − λ < we have δ − λ ( z ) ∈ B ρ ( z , R ) . In both cases, the constant λ depends continuously on R , on the center of the ball and on the point of tangency.Proof. Let B ρ ( z , R ) be a gauge ball of center z and radius R , and let z ∈ ∂B ρ ( z , R ). If h ν, Z i >
0, where ν is the interior normal to B ρ ( z, R ), as in the first case of the statement we areproving, then this means that Z points towards the interior of the gauge ball, and thus, since δ λ satisfies (2.3), we infer the existence of λ > δ λ ( z ) ∈ B ρ ( z , R ) for any 0 < λ < λ .From the continuity of the map ( z, λ ) → δ λ ( z ), following from (2.3), we also deduce that λ changes continuously with respect to the data. The statement about interior tangent balls isshown the very same way. (cid:3) M. FOGAGNOLO AND A. PINAMONTI
Preliminaries on p -harmonic functions in the Heisenberg group. Let 1 < p < ∞ ,we denote the horizontal p -Laplacian with ∆ H n p . It acts on a C function f of H n as∆ H n p f = n X i =1 X i (cid:0) |∇ H n f | p − X i f (cid:1) + Y i (cid:0) |∇ H n f | p − Y i f (cid:1) . (2.4)Consequently, we say that a C -function f : U ⊂ H n → R is horizontally p -harmonic in an openset U if ∆ H n p f = 0 in U . Explicit solutions.
We immediately exhibit explicit horizontally p -harmonic functions, that willserve us both as model solutions to (1.1) and to construct the barriers functions employed inthe proof of Theorem 1.1. We have that, if p = Q , for any w ∈ H n , the function v w ( z ) = ρ − Q − pp − ( w − z ) (2.5)is horizontally p -harmonic for any z ∈ H n \ { w } . This was established in [7, Theorem 2.1], wherethe authors showed that, up to a normalizing constant, the function G : H n × H n \ Diag H n defined by G ( z, w ) = v w ( z ) constitutes the fundamental solution for the horizontal p -Laplacianwith singularity at Diag H n = { ( z, z ) ∈ H n × H n } . In particular, as pointed out in the samepaper, the (unique) solution to problem (1.1) in the model situation where Ω = B ρ ( O, r ) andΩ = B ρ ( O, R ) for some
R > r , is given by u r,R ( z ) = ρ ( z ) − Q − pp − − R − Q − pp − r − Q − pp − − R − Q − pp − (2.6)for any p = Q . Remark . In the case p = Q , the analogue of (2.5), again according to [7], is given by v w ( z ) = log ρ ( w − · z ) . (2.7)In particular, the proof of the analogue of Theorem 1.1 in the case p = Q is obtained simply bymodelling the barrier functions employed in our proof on (2.7) rather than on (2.6). We addressthe interested reader to [27] for more details on how to handle this situation in the case of R n with the standard notion of p -Laplacian.Let us recall now some functions spaces suited to define the weak solutions to (2.4). We follow[6] and [9]. We let, for an open subset U ⊆ H n , for p ≥
1, the horizontal (1 , p ) -Sobolev space HW ,p ( U ) be defined as the metric completion of C ( U ) in the norm || f || HW ,p ( U ) = ˆ U | f | p + |∇ H n f | p dz. Analogously, we define the space HW ,p ( U ) as the metric completion of C ( U ) under the samenorm.We say that f ∈ HW ,p ( U ) is horizontally weakly p -harmonic if n X i =1 ˆ U (cid:0) |∇ H n f | p − X i f (cid:1) X i ϕ + (cid:0) |∇ H n f | p − Y i f (cid:1) Y i ϕ dz = 0for any ϕ ∈ C ( U ). From now on, we will frequently indicate horizontally weakly p -harmonicfunctions simply as p -harmonic, since no confusion can occur.By arguing exactly as in the Euclidean case, one recovers the fundamental Comparison Prin-ciple even for horizontally weakly p -harmonic functions in the Heisenberg group. We addressthe reader to [9, Lemma 2.6] for a statement in the more general context of quasilinear equa-tions in Carnot groups. It actually holds also comparing subsolutions to supersolutions of the TRICT STARSHAPEDNESS OF SOLUTIONS TO THE HORIZONTAL P-LAPLACIAN 7 p -Laplacian, but being here concerned only with p -harmonic functions, we state it in the sim-plified version for solutions. Proposition 2.5 (Comparison Principle for p -harmonic functions) . Let U ⊂ H n be an open set,and let u, v ∈ HW ,p ( U ) be p -harmonic functions. Then, if min( u − v, ∈ HW ,p ( U ) , then u ≥ v on the whole U . The above result roughly asserts that if two p -harmonic functions u and v satisfy u ≥ v onthe boundary of U then the same inequality holds true in the interior on U . Actually, this isexactly what happens when boundary data are attained with some regularity.Finally, let us recall that as an immediate consequence of the Harnack inequality for p -harmonic functions in Heisenberg groups [6, Theorem 3.1] we get the following special form ofa Strong Maximum/Minimum Principle, highlighted also in [9, Theorem 2.5]. Proposition 2.6 (Strong Maximum Principle for p -harmonic functions) . Let U ⊂ H n be anopen bounded subset, and let u ∈ HW ,p ( U ) be p -harmonic. Then, u cannot achieve neither itsmaximum nor its minimum in U .Existence and uniqueness for (1.1) . In the following statement we resume an existence-uniquenesstheorem for problem (1.1), recalling also a suitable definition of weak solutions. It is well knownthat such solution exists, and can be proved exactly as in the Euclidean case, considered infull details in [22, Appendix I], see in particular Corollary 17.3 there, and compare also with[9, Section 3]. The uniqueness immediately follows from the Comparison Principle recalled inProposition 2.5.
Theorem 2.7 (Existence and uniqueness of weak solutions to problem (1.1)) . Let Ω and Ω and Ω ⊂ H n be open bounded subsets of H n satisfying Ω ⋐ Ω . Then, there exists an uniqueweak solution u to (1.1) , that is u ∈ HW ,p (Ω \ Ω ) is horizontally weakly p -harmonic and,letting ϑ ∈ C ∞ (Ω ) satisfy ϑ ≡ on Ω , we have u − ϑ ∈ HW ,p (Ω \ Ω ) . It is important to point out that, again as a straightforward application of the ComparisonPrinciple, if ˜ u is another such function satisfying the properties in the statement of Theorem2.7 relatively to another boundary datum ˜ ϑ fulfilling the same assumptions asked for ϑ , then u coincides with ˜ u on Ω \ Ω . This is observed with some more details for example in [22, p. 115].For what it concerns the continuous attainment of the boundary datum, we point out that in[9, Theorem 3.9] continuity up to the boundary for Dirichlet problems involving the horizontal p -Laplacian is proved for domains with boundary with a so-called corkscrew on any point of theboundary.Let us finally observe that as a consequence of Propositions 2.5 and 2.6 we have 0 < u < \ Ω . Indeed, first observe that, since u = ϑ + f for some f ∈ HW ,p we can find byapproximating f a sequence { u k } k ∈ N of functions in C (Ω \ Ω ) approximating u in HW ,p -norm, and satisfying u k − ϑ ∈ C c (Ω \ Ω ). In particular, for any k ∈ N , u k satisfies min( u k , ∈ HW ,p (Ω \ Ω ), and thus, passing to the limit as k → ∞ , we infer that the same holds for u . Thus, being u p -harmonic, we get from the Comparison Principle recalled in Proposition 2.5that u ≥ p -harmonic function 1 − u , we alsofind that u ≤ \ Ω . However by the Strong Maximum Principle of Proposition 2.6, theinequalities 0 ≤ u ≤ Corollary 2.8.
Let u be the solution to (1.1) , in the sense of Theorem 2.7. Then, we have < u < in Ω \ Ω . M. FOGAGNOLO AND A. PINAMONTI Proof of Theorem 1.1
It is quite straightforward, but fundamental for our arguments, to observe that if a function f is p -harmonic, then so does the dilated f λ defined as f λ ( z ) = f ( δ λ ( z )). Lemma 3.1 (Dilation-invariance of p -harmonicity) . Let U ⊆ H n , and f ∈ HW ,p ( U ) be a p -harmonic function. Then, the function f λ ( x ) belongs to HW ,p ( δ − λ ( U )) and it is p -harmonic.Proof. It is obvious from the definition of δ λ ( U ) that f λ is well defined on such set. In order toprove the other assertions, the main computation is the following. We have, for j = 1 , . . . , n , X j ( f λ ( z )) = e λ (cid:20) ∂f∂x j ( δ λ ( z )) + 2 e λ y ∂f∂t ( δ λ ( z )) (cid:21) = e λ ( X j f ) λ ( z ) , (3.1)and analogously Y j ( f λ ( z )) = e λ (cid:20) ∂f∂y j ( δ λ ( z )) − e λ x ∂f∂t ( δ λ ( z )) (cid:21) = e λ ( Y j f ) λ ( z ) . (3.2)The inclusion of f λ in HW ,p ( δ λ ( U )) is a direct consequence of (3.1) and (3.2), while the p -harmonicity is shown as follows. We have, again as a consequence of the above relations n X i =1 ˆ δ − λ ( U ) (cid:0) |∇ H n f λ | p − X i f λ (cid:1) X i ϕ + (cid:0) |∇ H n f λ | p − Y i f λ (cid:1) Y i ϕ dz == e λ ( p − n X i =1 ˆ δ − λ ( U ) h |∇ H n f | p − λ ( X i f ) λ i X i ϕ + h |∇ H n f | p − λ ( Y i f ) λ i Y i ϕ dz = e λ ( p − − Q n X i =1 ˆ U (cid:2) |∇ H n f | p − ( X i f ) (cid:3) ( X i ϕ ) λ + (cid:2) |∇ H n f | λ p − ( Y i f ) (cid:3) ( Y i ϕ ) λ dz = e λ ( p − − Q n X i =1 ˆ U (cid:2) |∇ H n f | p − ( X i f ) (cid:3) X i ϕ λ + (cid:2) |∇ H n f | λ p − ( Y i f ) (cid:3) Y i ϕ λ dz = 0for any ϕ ∈ C c ( δ λ ( U )). The last step follows from the p -harmonicity of f in U , since ϕ λ clearlybelongs to C c ( U ). (cid:3) We are finally in position to prove the statement in turn implying Theorem 1.1.
Theorem 3.2.
Let u be a weak solution to (1.1) with p > for Ω ⊂ H n and Ω ⊂ H n boundedsets with C boundaries that are strictly starshaped with respect to the origin O ∈ H n and suchthat Ω ⋐ Ω . Assume also that Ω satisfies an uniform exterior gauge ball condition and Ω satisfies an uniform interior gauge ball condition . Then, there exists a positive constant M suchthat h∇ u, Z i < − M < at any point where ∇ u exists.Proof. As already declared, we prove the result for p = Q , addressing the reader to Remark2.4 for indications about the straightforward extension to the case p = Q . Consider, for any z ∈ ∂ Ω the gauge ball B ρ ( z , R ) contained in H n \ Ω and touching ∂ Ω in z . Similarly, for z ∈ ∂ Ω , consider B ρ ( z , R ) contained in Ω and touching ∂ Ω in z . These tangent gauge balls,with uniform radius R , exist by assumption, see Definition 2.2. On B ρ ( z , R ), define a function v satisfying v = 1 on B ρ ( z , R/
2) and v ( · ) = α ρ ( z − · ) − Q − pp − + β, TRICT STARSHAPEDNESS OF SOLUTIONS TO THE HORIZONTAL P-LAPLACIAN 9 on B ρ ( z , R ) \ B ρ ( z , R/ α and β are chosen so that v = 0 on ∂B ρ ( z , R )and v = 1 on ∂B ρ ( z , R/ B ρ ( z , R ) a function v satisfying v = 1on B ρ ( z , R/
2) and v ( · ) = α ρ ( z − · ) − Q − pp − + β, on B ρ ( z , R ) \ B ρ ( z , R/ α and β are chosen so that v = 0 on ∂B ρ ( z , R )and v = 1 on ∂B ρ ( z , R/ α = R Q − pp − Q − pp − − , β = − Q − pp − − . Observe now that the function v and v are smooth up to the boundary in B ρ ( z , R ) \ B ρ ( z , R/
2) and B ρ ( z , R ) \ B ρ ( z , R/
2) respectively, and they both enjoy nonvanishing gradientin these sets. Actually, a direct computation shows that |∇ v | ( w ) ≥ C, |∇ v | ( w ) ≥ C (3.3)for any w ∈ B ρ ( z , R ) \ B ρ ( z , R/
2) and any w ∈ B ρ ( z , R ) \ B ρ ( z , R/ C does not depend on w nor on w . Such gradients being nonvanishing, combined with ∂B ρ ( z , R )and ∂B ρ ( z , R ) being regular level sets of v and v , imply, on the one hand, thatlim B ρ ( z,R ) ∋ w → z ∇ v |∇ v | ( w ) = ν Ω ( z ) , (3.4)where z ∈ ∂ Ω and ν Ω ( z ) is the Euclidean outward unit normal to Ω , and on the other handthat lim B ρ ( z,R ) ∋ w → z ∇ v |∇ v | ( w ) = − ν Ω ( z ) , (3.5)where this time z ∈ ∂ Ω and ν Ω ( z ) is the Euclidean outward unit normal to Ω . In getting (3.4)and (3.5), we again used the tangency property of the gauge balls with respect to the boundariesof Ω and Ω . Observe now that there exists K > h ν, Z i ( z ) ≥ K (3.6)for any z ∈ ∂ Ω ∪ ∂ Ω , as follows from the strict starshapedness of Ω and Ω , their boundednessand the C -regularity of their boundaries. Now, Proposition 2.3, the limits (3.4) and (3.5), theuniform lower bounds on the gradients of v and v (3.3), and (3.6), imply that v ( δ λ ( z )) − v ( z ) λ = v ( δ λ ( z )) λ ≥ CK, (3.7)for any z ∈ ∂ Ω and analogously v ( δ − λ ( z )) λ ≥ CK for any z ∈ Ω , for any 0 < λ < λ . Importantly, observe that λ can be made independent of z ∈ ∂ Ω ∪ ∂ Ω , as it immediately follows from the continuity properties of such parameter statedin Proposition 2.3 and the compactness of ∂ Ω ∪ ∂ Ω .As observed in Corollary 2.8, 0 < u < 2) in light of the second inequality in (3.8) and againby construction of v . Then, the comparison principle for p -harmonic functions recalled inProposition 2.5 combined with (3.7) implies1 − u ( δ λ ( z )) λ ≥ Lv ( δ λ ( z )) λ ≥ LCK (3.9)for any 0 < δ < λ .Arguing very similarly in comparing the functions u and Lv in the annulus B ρ ( z, R ) \ B ρ ( z, R/ 2) with z ∈ ∂ Ω , we get, using the first inequality in (3.8) and the definition of v ,that u ( δ − λ ( z )) λ ≥ Lv ( δ − λ ( z )) λ ≥ LCK, (3.10)again for any 0 < λ < λ .Consider then the function u λ ( w ) = u ( δ λ ( w )) on Ω − λ \ Ω . Recall that by Ω λ we denote thecontraction of Ω through dilations, as defined in (2.1). By Lemma 3.1, u λ is p -harmonic andobserve that on ∂ Ω we have u λ λ ≤ λ − LCK = uλ − LCK by (3.9), and on ∂ Ω − λ we have 0 = u λ λ ≤ uλ − LCK, by (3.10). Thus, applying the Comparison Principle to the p -harmonic functions u λ /λ and u/λ − LCK/ w ∈ Ω \ Ω , that u ( δ λ ( w )) − u ( w ) λ ≤ − LCK (3.11)for any 0 < λ < λ . Assume now that ∇ u exists at w . Then, we havelim λ → + u ( δ λ ( w )) − u ( w ) λ = (cid:28) ∇ u ( w ) , ddλ δ λ ( w ) (cid:12)(cid:12) λ =0 (cid:29) = h∇ u, Z i ( w ) , (3.12)where in the last step we used (2.3). We thus conclude, coupling (3.11) with (3.12), that h∇ u, Z i ( w ) < − LCK < w ∈ Ω \ Ω . Observe that the upper bound in (3.13) does not depend on the particular point w where ∇ u exists, and thus it completes the proof of Theorem 3.2. (cid:3) Let us finally briefly show how Theorem 1.1 follows as a corollary. Proof of Theorem 1.1. The uniform negative upper bound for h∇ u, Z i holding true at any pointof standard diffentiability for u implies that if in addition such function is C , we have h∇ u, Z i < \ Ω . In particular, ∇ u never vanishes in the open annulus, and thus the sets { u ≥ t } ∪ Ω for t ∈ (0 , 1) are bounded by the C submanifolds { u = t } with exterior pointingunit normal at any w t ∈ { u = t } given by ν t = −∇ u/ |∇ u | computed at such point. 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