Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions
UUniqueness of critical points of solutions to the mean curvatureequation with Neumann and Robin boundary conditions ∗ Haiyun Deng † , Hairong Liu & Long Tian School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China; School of Science, Nanjing Forestry University, Nanjing, 210037, China
Abstract:
In this paper, we investigate the critical points of solutions to the prescribedconstant mean curvature equation with Neumann and Robin boundary conditions respectivelyin a bounded smooth convex domain Ω of R n ( n ≥ R n ( n ≥
3) by the projection of higher dimensionalspace onto two dimensional plane.
Key Words: prescribed constant mean curvature equation, critical point, uniqueness, non-degeneracy.
In this paper we consider the following mean curvature equationdiv( ∇ u √ |∇ u | ) = H in Ω , (1.1)with Neumann and Robin boundary conditions respectively, where H is a positive constant, Ω isa bounded smooth convex domain in R n ( n ≥ u has one unique critical point. In 1992 Alessandrini and Magnanini [1] considered the geometricstructure of the critical set of a solution to semilinear elliptic equation in a nonconvex domain Ω , whose boundary is composed of finite simple closed curves. They deduced that the critical set is ∗ The work is supported by National Natural Science Foundation of China (No.11401307, No.11401310), High leveltalent research fund of Nanjing Forestry University (G2014022) and Postgraduate Research & Practice InnovationProgram of Jiangsu Province (KYCX17 0321). The second author is sponsored by Qing Lan Project of JiangsuProvince. † Corresponding author E-mail: [email protected], Tel.: +86 15877935256. a r X i v : . [ m a t h . A P ] J un ade up of finitely many isolated critical points. In 2012 Arango and G´omez [6] considered thecritical points of the solutions for quasilinear elliptic equations with Dirichlet boundary conditionin strictly convex domains and nonconvex domains respectively. If the domain is strictly convexand u is a negative solution, they proved that the critical set has exactly one non-degeneratecritical point. On the other hand, they obtained the similar results of the semilinear case for aplanar annular domain, whose boundary has nonzero curvature. In 2017 Deng, Liu and Tian [15]investigate the geometric structure of interior critical points of solutions u to a quasilinear ellipticequation with nonhomogeneous Dirichlet boundary conditions in a simply connected or multiplyconnected domain Ω in R . They develop a new method to prove Σ ki =1 m i + 1 = N or Σ ki =1 m i = N, where m , · · · , m k are the respective multiplicities of interior critical points x , · · · , x k of u and N is the number of global maximum points of u on ∂ Ω. All the above results involved the criticalpoints of solutions to elliptic equations with Dirichlet boundary condition in the planar domains.In addition, a number of other authors have investigated this problem and some other relatedproblems (see [2, 3, 4, 5, 12, 16, 17, 20, 21, 22, 29]). However, the critical set K has not been fullyconsidered for some general cases, especially for higher dimensional spaces, Neumann and Robinboundary value problems.For higher dimensional cases, there has few results about the critical points of solutions toelliptic equations. In 1998 Cabr´e and Chanillo [9], under the assumption of the existence of asemi-stable solution, showed that the solution of Poisson equation −(cid:52) u = f ( u ) has exactly onenon-degenerate critical point in a smooth bounded convex domain of R n ( n ≥ R n ( n ≥ . Concerning the Neumann and Robin boundary value problems, the critical points of solutionsto elliptic equations seems to be less considered. In 1990, Sakaguchi [27] proved that the solutionsof Poisson equation with Neumann and Robin boundary conditions respectively exist exactly onecritical point. The goal of this paper is to obtain some results about the critical set of solutions tomean curvature equation with Neumann and Robin boundary conditions respectively in boundedsmooth convex domains Ω of R n ( n ≥ Theorem 1.1.
Let Ω be a bounded smooth convex domain in R . Suppose that H is a positiveconstant and that u is a solution of the following boundary value problem (cid:40) div( ∇ u √ |∇ u | ) = H in Ω , ∂u∂ −→ n = c on ∂ Ω , (1.2) where −→ n is the unit outward normal vector of ∂ Ω and c is a positive constant. Then u has exactlyone critical point p in Ω and p is a non-degenerate interior minimal point of u . Theorem 1.2.
Let Ω be a bounded smooth convex domain in R . Suppose that H is a positiveconstant. Let u be a solution of the following boundary value problem (cid:40) div( ∇ u √ |∇ u | ) = H in Ω , ∂u∂ −→ n + αu = 0 on ∂ Ω , (1.3)2 here α is a positive constant. Then u has exactly one critical point p in Ω and p is a non-degenerate interior minimal point of u . In addition, we will give partial results about the critical points of solutions in higher dimen-sional spaces.
Theorem 1.3.
Let Ω be a bounded smooth convex domain of rotational symmetry with respect to x n axis in R n ( n ≥ . Suppose that u = u ( | x (cid:48) | , x n ) is an axisymmetric solution of equation (1.2)or (1.3), where | x (cid:48) | = (cid:113) x + x + · · · + x n − . Then u has exactly one critical point p in Ω and p is a non-degenerate interior minimal point of u . The rest of this paper is organized as follows. In Section 2, we introduce the local Chen &Huang’s comparison technique. In Section 3, firstly, we prove that every interior critical point of u is non-degenerate. Then, by the strong maximum principle and Hopf lemma, we show that u does not have maximum points in Ω and that u cannot have minimal points on ∂ Ω. Moreover, weprove the sufficient and necessary condition for existence of the saddle points of u . In Section 4,firstly, we show the uniqueness of the interior minimal points of u by continuity argument. Then,by the sufficient and necessary condition for existence of the saddle points and the non-degeneracyof interior critical points in Section 3, we prove the uniqueness of the interior critical points of u . In Section 5, our main idea is to study the projection of higher dimensional spaces onto twodimensional plane. So we need to consider the domains Ω of revolution formed by taking a strictlyconvex planar domain about one axis. We deduce that u has exactly one critical point p in abounded smooth strictly convex domain of R n ( n ≥
3) and p is a non-degenerate interior minimalpoint of u . In order to obtain the non-degeneracy and uniqueness of the critical points of u in planardomains. In this section, we will recall the key local Chen & Huang’s comparison technique in [13].For the sake of completeness, in our setting, we will give a complete proof of Lemma 1 in [13]. Lemma 2.1.
Suppose that u, v satisfy the same constant mean curvature equation (1.1). Withoutloss of generality, we suppose that u, v have a second order contact at Z = ( x , x , u ( x , x )) with ( x , x ) = (0 , . Then by changing coordinate ( x , x ) into ( ξ, η ) linearly, the difference u − v around ( ξ, η ) = (0 , is given by u − v = Re( ρ · ( ξ + ηi ) k ) + o (( ξ + η ) k ) , (2.1) where k ≥ , ρ is a complex number and ξ + ηi is the complex coordinate.Proof. Since u, v satisfy the same constant mean curvature equation. Then we have0 = (1 + u x + u x )( u x x + u x x ) − ( u x u x x + u x u x x + 2 u x u x u x x ) − H (1 + |∇ u | ) = (1 + u x ) u x x + (1 + u x ) u x x − u x u x u x x − H (1 + |∇ u | ) , (2.2)and 0 = (1 + v x ) v x x + (1 + v x ) v x x − v x v x v x x − H (1 + |∇ v | ) . (2.3)3ow we define p ( t ) , q ( t ) , m ( t ) , r ( t ) , s ( t ) for 0 ≤ t ≤ p ( t ) = (1 − t ) v x x + tu x x , q ( t ) = (1 − t ) v x x + tu x x , m ( t ) = (1 − t ) v x x + tu x x ,r ( t ) = (1 − t ) v x + tu x , s ( t ) = (1 − t ) v x + tu x , and consider the following function W = W ( p ( t ) , q ( t ) , m ( t ) , r ( t ) , s ( t )) = (1 + s ) p + (1 + r ) m − rsq − H (1 + r + s ) . Let w = u − v, therefore by (2.2) minus (2.3) we have0 = W ( p (1) , q (1) , m (1) , r (1) , s (1)) − W ( p (0) , q (0) , m (0) , r (0) , s (0)) = (cid:90) W t dt = a w x x + 2 a w x x + a w x x + b w x + b w x , where a = (cid:82) (1 + s ) dt, a = − (cid:82) rsdt, a = (cid:82) (1 + r ) dt,b = (cid:82) [2( rm − sq ) − H √ r + s r ] dt,b = (cid:82) [2( sp − rq ) − H √ r + s s ] dt. Then w satisfies the following equation Lw := a w x x + 2 a w x x + a w x x + b w x + b w x = 0 , where a − a a < Lw = 0 . Next, the rest of proof issame to that in [13]. We transform ( x , x ) into ( ξ, η ) such that ξ (0 ,
0) = 0 , η (0 ,
0) = 0 and at(0 , Lw = ( ∂ ∂ξ + ∂ ∂η + b (cid:48) ∂∂ξ + b (cid:48) ∂∂η ) w = 0 . (2.4)Since the coefficients of Lw and w itself are analytic in ( x , x ) as well as in ( ξ, η ) , then we get thefollowing Taylor expansion around ( ξ, η ) = (0 ,
0) of Lw : Lw = (cid:110) (1 + α ξ + β η + O ( ξ + η )) ∂ ∂ξ + (1 + α ξ + β η + O ( ξ + η )) ∂ ∂η +2( α ξ + β η + O ( ξ + η )) ∂ ∂ξ∂η + ( τ + δ ξ + λ η + O ( ξ + η )) ∂∂ξ (2.5)+( τ + δ ξ + λ η + O ( ξ + η )) ∂∂η (cid:111) w. By Theorem I in [7], we have w ( ξ, η ) = (cid:80) ∞ j =0 P k + j ( ξ, η ) , (2.6)where P k ( ξ, η ) is a non-zero homogeneous polynomial in ( ξ, η ) of degree k . By the assumption of u and v have a second order contact at (0 , , we have k ≥ . By (2.5) and (2.6), the equation (2.4)yields 0 = ( ∂ ∂ξ + ∂ ∂η ) P k + { terms of order ≥ k − } . By the uniqueness of the power expansion, we show that P k is a harmonic homogeneous polynomial.Then P k ( ξ, η ) = Re( ρ · ( ξ + ηi ) k ) , (2.7)where ρ is a complex number, then (2.6) and (2.7) imply (2.1).4 emma 2.2. (see[13, Lemma 2]) Suppose that u = u ( x , x ) is a non-constant solution of thefollowing homogeneous quasilinear elliptic equation Lu = a u x x + 2 a u x x + a u x x + b u x + b u x = 0 in Ω , where the coefficients a ij and b i ( i, j = 1 , are analytic. Then every interior critical point of u isan isolated critical point. Remark 2.3.
By the above two lemmas and the implicit function theorem, we can know that thenodal set N ∩ Ω of ( u − v ) consists of at least three smooth arcs intersecting at (0 , and dividing Ω into at least six sectors. Moreover, the nodal set N of ( u − v ) is globally a union of smooth arcs. In this section, firstly, we investigate the non-degeneracy of critical points in a planar boundedsmooth convex domain Ω by using the local Chen & Huang’s comparison technique. Then we provethe sufficient and necessary condition for existence of the saddle points by using the geometricproperties of approximate surfaces at the non-degenerate critical points.
Lemma 3.1.
Suppose that u is a solution to (1.3). Then u < in Ω and ∂u∂ −→ n > on ∂ Ω .Proof. According to the assumption of div( ∇ u √ |∇ u | ) = n (cid:80) i,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j = H > , ∂u∂ −→ n + αu = 0 on ∂ Ω , where a ij = √ |∇ u | ( δ ij − u xi u xj |∇ u | ). By the strong maximum principle, u obtains its maximumon ∂ Ω . In fact, by the positive definiteness of the matrix A = ( a ij ), if u obtains the maximumat x ∈ Ω , then B = ( D ij u ( x )) is seminegative definite. Hence the matrix AB is seminegativedefinite with a nonpositive trace, it implies that (cid:80) ni,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j ≤ , which is a contradiction.Thus there exists a point x ∈ ∂ Ω such that u ( x ) = max Ω u. Suppose that u ( x ) ≥ . By Hopflemma we have ∂u ( x ) ∂ −→ n > . This contradicts with the fact that ∂u ( x ) ∂ −→ n + αu ( x ) = 0 , thus u < . Therefore ∂u∂ −→ n = − αu > ∂ Ω . Lemma 3.2.
Let u be a solution to (1.2), or (1.3). Then u has at least one critical point in Ω . Proof.
Since ∂u∂ −→ n > ∂ Ω , the Hopf lemma implies that u cannot have minimal points on ∂ Ω. Onthe other hand, since u is an analytic non-constant function, therefore u must obtain its minimumat some p ∈ Ω with ∇ u ( p ) = 0 . Then u has at least one point p with ∇ u ( p ) = 0 . Lemma 3.3.
Let u be a solution to (1.2), or (1.3). Then u is a Morse function, i.e., the Gaussiancurvature K ( p ) := det( D u ( p )) (cid:54) = 0 for any critical point p. Morse and semi-Morse function are described in [8, 11]. In order to prove Lemma 3.3, we needthe following lemma. 5 emma 3.4.
For constant H is from (1.1) and any constant h , there exists a number T (0 < T < ∞ ) such that the following initial value problem X (cid:48)(cid:48) ( t ) = H (1 + | X (cid:48) ( t ) | ) , − T < t < T,X (0) = h,X (cid:48) (0) = 0 , (3.1) has a unique C ∞ -solution X ( t ) , which satisfies the following X ( t ) = X ( − t ) , − T < t < T, (3.2) X ( t ) ≥ h, − T < t < T, (3.3) X (cid:48) ( t ) ≥ , ≤ t < T. (3.4) Proof.
Since the solution of problem (3.1) is X ( t ) = h + H (1 − √ − H t ) for | t | < T = H . Sothe results naturally hold.
Proof of Lemma 3.3.
We set up the usual contradiction argument. Suppose that p ∈ Ω is a pointsuch that ∇ u ( p ) = 0 and the Gaussian curvature K ( p ) = 0 . Without loss of generality, by using asuitable parallel translation and a rotation of coordinates, we may suppose that p = 0 and [ D ij u (0)] = diag[ H, . (3.5)By Lemma 3.4 for h = u (0) , we get a unique solution to (3.1), denote by v. Let v ( x ) (= v ( x , x )) = X ( x ) , thus v satisfies div( ∇ v √ |∇ v | ) = H, in ( − T, T ) × R , [ D ij v (0)] = diag[ H, ,v (0) = u (0) = h and ∇ v (0) = ∇ u (0) = 0 . (3.6)By (3.6), we know that ( u − v ) vanishes up to second order derivatives at 0. Moreover, wecan know that ( u − v ) is not identically zero. In fact, v = v ( x ) = X ( x ) . On the other hand,Lemma 3.1 shows that ∂u∂ −→ n > , we can suppose that unit outward normal vector −→ n = (0 , ∂u∂ −→ n = ∇ u · −→ n = u x > . So ( u − v ) is not identically zero. The unique continuation theorem ofsolutions for elliptic equations shows that ( u − v ) never vanishes up to infinite order at 0. UsingLemma 2.1, we get ( u − v )( x ) = P k ( x ) + o ( | x | k ) as | x | → k ≥ , where P k ( x ) is a homogeneous polynomial of degree k and P k ( x ) is notidentically zero. In addition, Lemma 2.2 shows that every interior critical point of ( u − v ) isisolated. Furthermore, Remark 2.3 shows that the nodal sets of ( u − v ) consist of k smooth arcs insome neighborhood U of the origin, and that all smooth arcs intersect at (0 ,
0) and divide U into2 k ( k ≥
3) sectors.Firstly, we investigate the case of Neumann boundary condition (1.2). In order to prove theresult, we should divide the proof into two cases, i.e., T is large enough and not large enoughrespectively. Consider I + = (cid:110) x ∈ Ω ∩ { ( − T, T ) × R } ; u ( x ) − v ( x ) > (cid:111) , (3.8)6nd I − = (cid:110) x ∈ Ω ∩ { ( − T, T ) × R } ; u ( x ) − v ( x ) < (cid:111) . (3.9)Therefore, it follows from Lemma 2.1 and Remark 2.3 thatBoth I + and I − have at least three components and eachof them meets the boundary ∂ (Ω ∩ { ( − T, T ) × R } ) . (3.10)Case 1: If T is large enough, i.e., Ω ∩ { ( − T, T ) × R } = Ω. Since Ω is convex, by Lemma 3.1and Lemma 3.4 we have that ∂u∂ −→ n > v (cid:48) ( x ) ≥ , v (cid:48)(cid:48) ( x ) ≥ ≤ x < T , then we knowthat ∂ ( u − v ) ∂ −→ n preserves sign in some fixed arc and ∂ ( u − v ) ∂ −→ n changes sign alternatively in two adjacentarcs. The sign distribution for directional derivative of ( u − v ) on ∂ Ω is shown in Fig. 1.
Fig. 1.
The sign distribution for directional derivative of ( u − v ) on ∂ Ω. Now we put γ + = (cid:110) x ∈ ∂ Ω; ∂ ( u − v ) ∂ −→ n ( x ) > (cid:111) , (3.11)and γ − = (cid:110) x ∈ ∂ Ω; ∂ ( u − v ) ∂ −→ n ( x ) < (cid:111) . (3.12)Therefore, it never occurs that a component of ( I − ∩ Ω) meets ∂ Ω exclusively in γ + . Suppose bycontradiction that γ is a component of ( I − ∩ Ω) which meets ∂ Ω exclusively in γ + . By Lemma2.1, we see that ( u − v ) satisfies (3.7), then the strong maximum principle implies that a negativeminimum of ( u − v ) in γ is attained at p ∈ γ + and ∂ ( u − v ) ∂ −→ n ( p ) ≤
0. This contradicts with thedefinition of γ + . By the same way, we know that it never occurs that a component of ( I + ∩ Ω)meets ∂ Ω exclusively in γ − . But these facts contradict with (3.10)Case 2: If T is not large enough, i.e., Ω \{ Ω ∩ { ( − T, T ) × R } (cid:54) = ∅ . Choose a number (cid:101) T suchthat (cid:101) T < T, which is sufficiently near to T. Set (cid:101)
Ω = ( − (cid:101) T , (cid:101) T ) × R . We only should replace Ω byΩ ∩ (cid:101) Ω and we can use the same method of case 1.Secondly, we consider the case of Robin boundary condition (1.3). We can use the same methodin the situation of Neumann boundary condition. Indeed, we select (cid:101) T with { v ( x ) = 0 } = { x = ± (cid:101) T } . Thus ∂ (Ω ∩ (cid:101) Ω) consists ofat most two components of (cid:110) ∂ ( u − v ) ∂ −→ n + α ( u − v ) > (cid:111) and at most two components of (cid:110) ∂ ( u − v ) ∂ −→ n + α ( u − v ) < (cid:111) . Lemma 3.5.
Let u be a solution to (1.2), or (1.3). Then u does not have maximum points in Ω . roof. Since div( ∇ u √ |∇ u | ) = H >
0, the strong maximum principle implies that u cannot obtainmaximum in Ω . In fact, let operator L be the mean curvature operator. Suppose that u hasmaximum points in Ω and that x is a maximum point, then D u ( x ) = D u ( x ) = 0 , D u ( x ) ≤ D u ( x ) ≤ Lu ( x ) = D u ( x ) + D u ( x ) ≤
0. However, this is contradict with
Lu > . This completes the proof of Lemma 3.5.Next, we show the sufficient and necessary condition for existence of the saddle points.
Lemma 3.6.
Let u be a solution to (1.2), or (1.3). Then u has at least two minimal points in Ω ,if and only if there exists a point p such that ∇ u ( p ) = 0 and K ( p ) < . Proof. (i) Firstly, we prove the section of “if”. Suppose that p is a point such that ∇ u ( p ) =0 and K ( p ) < . Therefore there exists an open neighborhood U of p in which the nodal sets of( u − u ( p )) consist of at least two smooth arcs intersecting at p and divides U into at least foursectors. Next, we consider the following super-level set U + := { x ∈ Ω | u ( x ) > u ( p ) } . Lemma 3.5 implies that each component of U + has to meet the ∂ Ω . Therefore we know thatthe following sub-level set U − := { x ∈ Ω | u ( x ) < u ( p ) } has at least two components. By ∂u∂ −→ n > , then u has at least two minimal points in Ω.(ii) Secondly, we prove the section of “only if”. Since ∂u∂ −→ n > ∂ Ω and Ω is convex, so wecan extend the solution u to R , denoting by u ( x ) = u ( y ) + dist( x, y ) · ∂u∂ −→ n ( y ) , (3.13)where y is the unique point on ∂ Ω such that dist( x, y ) = dist( x, Ω) . Hence we know that u ∈ C ( R ) and ∇ u (cid:54) = 0 in R \ Ω . Next, we consider the sub-level set N z = { x ∈ R | u ( x ) < z } . Therefore we have that ∂N z has only one curve for sufficiently large z. (3.14)Now, we suppose by contradiction that u has at least two minimal points and there does notexist point p such that ∇ u ( p ) = 0 and K ( p ) < . By Lemma 3.3, Lemma 3.5 and (3.13), we knowthat each critical point of u is a minimal point and ∇ u (cid:54) = 0 in R \ Ω respectively. On the otherhand, Lemma 2.2 shows that every critical point is isolated and the number of critical points isfinite. Then we suppose that there exists a finite sequence of minimal points of u , denotes by { p , p , . . . , p k } such that ∇ u ( x ) (cid:54) = 0 for all x ∈ R \{ p , p , . . . , p k } . (3.15)Let z = max { u ( p i ) | ≤ i ≤ k } . Therefore we know that the boundary ∂N z of the sub-level set N z is C curve for z > z and { ∂N z } is diffeomorphic to each other. According to the assumption,since K ( p i ) > , then the approximate surface is an elliptic paraboloid in a neighborhood of criticalpoint p i (If K ( p ) < , then the approximate surface is a hyperbolic paraboloid in a neighborhoodof critical point p ). The elliptic paraboloid and hyperbolic paraboloid as shown in Fig. 2 and Fig.3, respectively. 8 ig. 2. The elliptic paraboloid.
Fig. 3.
The hyperbolic paraboloid. If z is near to z , then { ∂N z } has at least two curves. This contradicts with the fact (3.14). Thiscompletes the proof of Lemma 3.6 . In this section, firstly, we show the uniqueness of the interior minimal points of u in Ω bycontinuity argument. Then, by the sufficient and necessary condition for existence of the saddlepoints and the non-degeneracy of interior critical points in Section 3, we prove the uniqueness ofthe critical points.For t ( t ∈ [0 , v t to the following problems: (cid:40) div( ∇ v √ t |∇ v | ) = H, in Ω , ∂v∂ −→ n = c, on ∂ Ω , or (cid:40) div( ∇ v √ t |∇ v | ) = H, in Ω , ∂v∂ −→ n + αv = 0 , on ∂ Ω . (4.1)Let u t = tv t for t > , then u t respectively satisfy (cid:40) div( ∇ u √ |∇ u | ) = tH, in Ω , ∂u∂ −→ n = c, on ∂ Ω , or (cid:40) div( ∇ u √ |∇ u | ) = tH, in Ω , ∂u∂ −→ n + αu = 0 , on ∂ Ω . (4.2)By the analysis of Lemma 3.2, we know that the solution v t of (4.1) has at least one minimalpoint in Ω . According to Lemma 3.3 and Lemma 3.6, we can easily get the following lemmas for v t . Lemma 4.1.
For any t ∈ (0 , . Let v t be a solution to (4.1). Then v t is a Morse function, i.e.,the Gaussian curvature K t ( p ) := det( D v t ( p )) (cid:54) = 0 for any critical point p. Lemma 4.2.
For any t ∈ [0 , . Then v t has at least two minimal points, if and only if there existsa point p such that ∇ v t ( p ) = 0 and K t ( p ) < . Next, we will prove that the Gaussian curvature of v t is positive at any critical point for thecase of t = 0 . Lemma 4.3.
Let v t be the solution of (4.1). Suppose that ∇ v ( p ) = 0 for some point p ∈ Ω . Thenthe Gaussian curvature K ( p ) > for any critical point p . roof. We set up the usual contradiction argument. Suppose that there exists interior criticalpoint p such that K ( p ) ≤
0. Without loss of generality, by using a suitable parallel translationand a rotation of coordinates, we may suppose that p = (0 , , [ D ij v ( p )] = diag [ λ , λ ] , where λ + λ = H > , λ > , λ ≤ . By Lemma 2.1, then the difference v − q around ( x , x ) = (0 ,
0) is given by v ( x , x ) − q ( x , x ) = P k ( x , x ) + o (( x + x ) k ) , where P k ( x , x ) is a homogeneous harmonic polynomial of degree k in Ω, and q ( x , x ) = v (0 ,
0) + 12 λ x + 12 λ x . Firstly, we study the case of Neumann boundary condition in (4.1). Next, we consider (cid:98) I + = (cid:8) x ∈ Ω; v ( x ) − q ( x ) > (cid:9) , and (cid:98) I − = (cid:8) x ∈ Ω; v ( x ) − q ( x ) < (cid:9) . Since v ( x , x ) − q ( x , x ) vanishes up to second order derivatives at (0 ,
0) and P k ( x , x ) is realanalytic. Then it follows from Lemma 2.1 and Remark 2.3 that k ≥ (cid:98) I + and (cid:98) I − have at least three componentsand each of them meets the boundary ∂ Ω . (4.3)Now we set (cid:98) γ + = (cid:110) x ∈ ∂ Ω; ∂ ( v − q ) ∂ −→ n ( x ) > (cid:111) , (4.4)and (cid:98) γ − = (cid:110) x ∈ ∂ Ω; ∂ ( v − q ) ∂ −→ n ( x ) < (cid:111) . (4.5)Since Ω is convex and q ( x , x ) = v (0 ,
0) + λ x + λ x with λ + λ = H > , λ > , λ ≤ , then we know that (cid:98) γ + and (cid:98) γ − has at most two components on ∂ Ω . The rest of the proof is sameas the proof of Lemma 3.3. This contradicts with (4.3).Secondly, we consider the case of Robin boundary condition in (4.1). We can use the samemethod in the case of Neumann boundary condition. This completes the proof.Next we will show the uniqueness of the interior minimal points of u in Ω by using the continuityargument. Lemma 4.4.
For any t ∈ [0 , . Then v t has exactly one minimal point in Ω . Proof.
We set M = [0 ,
1] and divide M into two sets M and M as follows: M = { t ∈ M ; v t has only one minimal point in Ω } , (4.6)and M = { t ∈ M ; v t has more than two minimal points in Ω } . (4.7)10hen M = M + M and M ∩ M = ∅ . Lemma 4.2 and Lemma 4.3 imply that 0 ∈ M , i.e., M (cid:54) = ∅ . Now we show that M is open in M. That is, for any t (cid:63) ∈ M , there exists a constant ε > t (cid:63) − ε, t (cid:63) + ε ) ⊂ M . In fact, the follows from Lemma 4.1 and Lemma 4.2 and inverse functiontheorem that v t has as many critical points as v t (cid:63) when t is near t (cid:63) . Suppose by contradiction thatthere exists a sequence { t k } ∈ M such that { v t k } has only one minimal point and t k ∈ ( t (cid:63) − k , t (cid:63) + k )for some positive t (cid:63) ∈ M . Then it follows from Lemma 4.1 and Lemma 4.2 that v t k does not hasthe saddle points, i.e., v t k has exactly one critical point. By Lemma 4.1 and continuity, we maytake a subsequence { v t kj } of { v t k } such that p k j → p, ∇ v t kj ( p k j ) = 0 , K t kj ( p k j ) > , ∇ v t (cid:63) ( p ) = 0 , K t (cid:63) ( p ) > . (4.8)Since t (cid:63) ∈ M , then there exists another point q ∈ U ( p ) ⊂ Ω and a sequence of point { q k j } suchthat q k j → q, ∇ v t kj ( q k j ) → ∇ v t (cid:63) ( q ) = 0 . According to the uniqueness of the critical point of v t k , we can take a subsequence { v t kj } of { v t k } such that v t kj are all monotone in the line γ ( p k j , q k j ). Therefore there exists a sequence of points { z k j ; z k j ∈ γ ( p k j , q k j ) } which satisfy |∇ v t kj ( z k j ) | ≤ |∇ v t kj ( q k j ) | → , | K t kj ( z k j ) | = |∇ v tkj ( q kj ) || p kj − q kj | → . (4.9)By (4.9) and continuity, then there should be a point z ∈ γ ( p, q ) such that ∇ v t (cid:63) ( z ) = 0 , K t (cid:63) ( z ) = 0 , this contradicts with Lemma 4.1, then we complete the proof which M is open set in M. On the other hand, we show that M is closed in M. In fact, let { t i } be a sequence of points in M such that t i → t as i → ∞ . Then Lemma 4.2 and the continuity argument imply that thereexists a subsequence { t j } of { t i } , a sequence { p j } and a point p ∈ Ω such that p j → p as j → ∞ , ∇ v t j ( p j ) = 0 , and K t j ( p j ) < . (4.10)By (4.10) and continuity, we have ∇ v t ( p ) = 0 , and K t ( p ) ≤ . (4.11)Since ∇ v t (cid:54) = 0 on ∂ Ω , then we have p ∈ Ω . Hence it follows from Lemma 4.1, Lemma 4.2, Lemma4.3 and (4.11) that t ∈ M . This shows that M is closed in M. Then M must be M or ∅ . Since M (cid:54) = ∅ , so M = ∅ and M = M. This completes the proof.
Proof of Theorem 1.1 and Theorem 1.2.
By Lemma 3.3 and Lemma 3.5, we know that the Gaus-sian curvature K ( p ) (cid:54) = 0 for any critical point p and solution u does not have maximum points inΩ. In addition, Lemma 3.6 shows thatif ∃ p ∈ Ω such that ∇ u ( p ) = 0 and K ( p ) < ⇔ (cid:93) { minimal points of u } ≥ . (4.12)On the other hand, by Lemma 4.4, we know that u has exactly one minimal point in Ω . Therefore, u does not have saddle points in Ω , this implies that u has exactly one critical point p in Ω and p is a non-degenerate interior minimal point of u .11 The proof of Theorem 1.3
In this section, we investigate the geometric structure of critical point set K of solutionsto prescribed constant mean curvature equation with Neumann and Robin boundary conditionsrespectively in higher dimensional spaces. Proof Theorem 1.3.
We divide the proof into three steps.Step 1, we turn the mean curvature equation (1.1) for n -dimension into the similar meancurvature equation for 2-dimension. Without loss generality, let Ω be a domain of revolutionformed by taking a strictly convex planar domain in the x , x n plane with respect to the x n axis.In the sequel, x = ( x (cid:48) , x n ) , x (cid:48) = ( x , · · · , x n − ) and r = (cid:113) x + · · · + x n − . By the assumptions, we have that the solution u satisfies u ( x (cid:48) , x n ) = u ( | x (cid:48) | , x n ) (cid:44) v ( r, x n ) (5.1)and ∂v∂r ( r, x n ) > r (cid:54) = 0 . (5.2)From (5.2), we can know that the critical points of u lie on x n axis. Next, according to (5.1),we have that u x n ( x (cid:48) , x n ) = v x n ( r, x n ) . (5.3)Moreover, we can deduce that u x n satisfies the following equation n (cid:88) i,j =1 a ij ( ∇ u ) ∂ u x n ∂x i ∂x j + n (cid:88) i,j =1 ∂a ij ( ∇ u ) ∂x n ∂ u∂x i ∂x j = 0 n ≥ . That is L u x n := n (cid:80) i,j =1 a ij ( ∇ u ) ∂ u xn ∂x i ∂x j + n (cid:80) i,j =1 ∂ u∂x i ∂x j ∂a ij ( ∇ u ) ∂x n = 0 , (5.4)where a ij ( ∇ u ) = √ |∇ u | ( δ ij − u xi u xj |∇ u | ) , ∂a ij ( ∇ u ) ∂x n = |∇ u | ) / (cid:2) ( u xi u xj |∇ u | − δ ij )( ∇ u · ∇ u x n ) − ( u x i u x n x j + u x j u x n x i ) (cid:3) is the first derivative term of u x n .By the assumptions, the strict convexity of Ω and the Hopf lemma, we can know that u x n vanishes precisely on the ( n −
2) dimensional sphere given by S = { x n = a } ∩ ∂ Ω , for some a ∈ R . For convenience, we define the nodal set N = { x ∈ Ω | u x n ( x ) = 0 } . It is clear that all critical points of solution u are contained in N . Also from (5.3), N is rotationallyinvariant about the x n axis.Now we turn the mean curvature equation (1.1) for n -dimension div ( ∇ u √ |∇ u | ) = H into the following similar mean curvature equation on 2-dimension div ( ∇ v (cid:112) |∇ v | ) + 1 (cid:112) |∇ v | n − r v r = H, (cid:80) i,j =1 a ij ( ∇ v ) v ij + √ |∇ v | n − r v r = H, (5.5)where ∇ v = ( ∂v∂r , ∂v∂x n ) , a ij = √ |∇ v | ( δ ij − v i v j |∇ v | ) and v = ∂v∂r , v = ∂v∂x n . For any θ = ( θ , θ ) = (cos α, sin α ) ∈ S , where α ∈ [0 , π ) . We turn the quasilinear ellipticequation associated to v into a linear elliptic equation associated to w = v θ = ∇ v · θ. Firstly,we differentiate the equation (5.5), then take inner product with θ. For convenience, we set y =( y , y ) = ( x n , r ) , hence we can get the following equation L v w + h ( y ) ∂w∂y + h ( y ) ∂w∂y + |∇ v | ) n − r (cid:104) (1 + v y ) ∂w∂y − v y v y ∂w∂y (cid:105) = √ |∇ v | n − r v r θ , (5.6)where L v w := (cid:88) i,j =1 a ij ( ∇ v ) ∂ w∂y i ∂y j and h k ( y ) = (cid:88) i,j =1 v y i y j ∂a ij ∂v y k , k = 1 , . By (5.2) and (5.6), we deduce that L v w + h ( y ) ∂w∂y + h ( y ) ∂w∂y + |∇ v | ) n − r (cid:104) (1 + v y ) ∂w∂y − v y v y ∂w∂y (cid:105) ≥ . (5.7)By (5.7), so we can consider the result of projecting a graph onto x , x n plane (see Fig. 4). Fig. 4.
The graphic projection of higher dimensional space onto two dimensional plane.
Step 2, we show the uniqueness of critical points. This subsection is based on the resultsof Gidas, Ni and Nirenberg [18] and Caffarelli and Friedman [10], the ideas of Payne [25] andSperb [28]. To prove that whenever critical set has exactly one point, since all critical pointsof u are contained in N ∩ { x = · · · = x n − = 0 } and lie on the x n axis. The nodal set N = { x ∈ Ω | u x n ( x ) = 0 } is rotationally invariant about the x n axis, formed by a set N contained inthe x , x n x n axis, by (5.5), where N can be seen as theprojection of N in the x , x n N cannot enclose any subdomain of Ω (ByLemma 2.1 in [14], N cannot enclose any planar subdomain of Ω ∩ { x = · · · = x n − = 0 } , wherelocally N looks like the nodal set of some homogeneous polynomial in x , x n . ). Because N issymmetric with respect to the x n axis and intersects the x n axis at exactly one point, hence weprove the uniqueness of critical points.Step 3, we show the non-degeneracy of critical point. How to show that critical point p is non-degenerate, we restatement that u is rotationally symmetric with respect to x n axis and critical13oint p lies on this axis. From (5.1) and (5.2), we have that { u x k = 0 } = { x k = 0 } ∩ Ω for all1 ≤ k ≤ n − . Hence u x i x j ( p ) = 0 for any index 1 ≤ i < j ≤ n, that is, D u ( p ) is diagonal. By(5.2), we can know that u x k > D k = { x k > } ∩ Ω for 1 ≤ k ≤ n − . Furthermore, indomain D k , u x k satisfies L u x k = n (cid:80) i,j =1 a ij ( ∇ u ) ∂ u xk ∂x i ∂x j + n (cid:80) i,j =1 ∂ u∂x i ∂x j ∂a ij ( ∇ u ) ∂x k = 0 , (5.8)where ∂a ij ( ∇ u ) ∂x k = |∇ u | ) / (cid:2) ( u xi u xj |∇ u | − δ ij )( ∇ u ·∇ u x k ) − ( u x i u x k x j + u x j u x k x i ) (cid:3) is the first derivativeterm of u x k . According to the Hopf lemma, we deduce that u x k x k ( p ) > ≤ k ≤ n − , where critical point p ∈ ∂ D k . Finally, we recall that the function u x n satisfies (5.4). By the definition of N , u x n > N . By applying the Hopf lemma to u x n at p ∈ N , we have that u x n x n ( p ) > . So we provethat the Hessian matrix D u ( x ) of u is diagonal and positive definite at critical point p , hence p is the unique critical point and p is a non-degenerate interior minimal point of u . This completesthe proof of Theorem 1.3. References [1] G. Alessandrini, R. Magnanini, The index of isolated critical points and solutions of elliptic equations in theplane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (4) (1992) 567-589.[2] G. Alessandrini, R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, andStekloff eigenfunctions, SIAM J. Math. Anal. 25 (1994) 1259-1268.[3] G. Alessandrini, D. Lupo, E. Rosset, Local behavior and geometric properties of solutions to degeneratequasilinear elliptic equations in the plane, Appl. Anal. 50 (1993), no. 3-4, 191-215.[4] G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. PisaCl. Sci. (4) 14 (1987) 229-256.[5] G. Alessandrini, M. Sigalotti, Geometric properties of solutions to the anisotropic p-Laplace equation in di-mension two, Ann. Acad. Sci. Fenn. Math. 26 (2001) 249-266.[6] J. Arango, A. G´omez, Critical points of solutions to quasilinear elliptic problems, Nonlinear Anal. 75 (2012)4375-4381.[7] L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math. 8 (1955)473-496.[8] R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N.S.) 7 (1982) 331-358.[9] X. Cabr´e, S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.)4 (1998) 1-10.[10] L.A. Caffarelli, A. Friedman, Partial regularity of the zero-set of solutions of linear and superlinear ellipticequations, J. Differential Equations 60 (1985), no. 3, 420-433.[11] A. Cavicchioli, Covering numbers of manifolds and critical points of a Morse function, Israel J. Math. 70 (1990)279-304.[12] S. Cecchini, R. Magnanini, Critical points of solutions of degenerate elliptic equations in the plane, Calc. Var.Partial Differential Equations 39 (2010) 121-138.[13] J.T. Chen, W.H. Huang, Convexity of capillary surfaces in the outer space, Invent. Math. 67 (1982) 253-259.[14] H.Y. Deng, H.R. Liu, L. Tian, Critical points of solutions for mean curvature equation in strictly convex andnonconvex domains, arXiv preprint arXiv:1712.08431 (2017).[15] H.Y. Deng, H.R. Liu, L. Tian, Critical points of solutions to a quasilinear elliptic equation with nonhomogeneousDirichlet boundary conditions, J. Differential Equations (2018), https://doi.org/10.1016/j.jde.2018.05.031[16] A. Enciso, D. Peralta-Salas, Critical points of Green’s functions on complete manifolds, J. Differential Geom.92 (2012) 1-29.