Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains
CCritical points of solutions for mean curvature equation in strictlyconvex and nonconvex domains ∗ 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 mainly investigate the set of critical points associated to solutions of meancurvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvexdomain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegeneratecritical point in a smooth, bounded and strictly convex domain of R n ( n ≥ K of solutions u for the constant mean curvature equation in aconcentric (respectively an eccentric) spherical annulus domain of R n ( n ≥ K exists(respectively does not exist) a rotationally symmetric critical closed surface S . In fact, in an eccentricspherical annulus domain, K is made up of finitely many isolated critical points ( p , p , · · · , p l ) on anaxis and finitely many rotationally symmetric critical Jordan curves ( C , C , · · · , C k ) with respect to anaxis. Key Words: mean curvature equation, critical point, nodal set.
In this paper we consider the following mathematical models with zero Dirichlet boundary condition (cid:26) Lu = f ( u ) in Ω ,u = 0 on ∂ Ω , (1.1)where f is a real value function, Ω is a smooth, bounded, strictly convex and nonconvex domain in R n ( n ≥
2) respectively, and L is a mean curvature operator Lu = div ( ∇ u √ |∇ u | ) = n (cid:80) i,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j , n ≥ , (1.2)where a ij = √ |∇ u | ( δ ij − u xi u xj |∇ u | ).Equation (1.2) is a special case of the following A -Laplacian equation (see [15]) (cid:26) div( A ( |∇ u | ) ∇ u ) = f ( u ) in Ω ,u = 0 on ∂ Ω , (1.3)which satisfies A ( h ) + hA (cid:48) ( h ) > , h > . For example, if A ( h ) ≡ , the A -Laplacian equation is thewell-known semilinear elliptic equation (cid:52) u = f ( u ) . On the other hand, if A ( h ) = √ h , we obtain the ∗ The work is supported by National Natural Science Foundation of China (No.11401307, No.11401310), High level talentresearch fund of Nanjing Forestry University (G2014022) and Postgraduate Research & Practice Innovation Program ofJiangsu Province (KYCX17 0321). The second author is sponsored by Qing Lan Project of Jiangsu Province. † Corresponding author E-mail: [email protected], Tel.: +86 15877935256. a r X i v : . [ m a t h . A P ] D ec ell-known mean curvature equation. In this paper, we mainly consider the critical points of solutionsfor mean curvature equation.Critical set of solutions for elliptic problems is a subject of important research. The investigationsabout critical points of solutions for elliptic equations have many results. However, the critical set K has not been fully investigated, except for some few special domains, which still is an open problem forgeneral domains, especially for higher dimension spaces. Now let us review some known results. In 1971Makar-Limanov [20] solved the Poisson equation with constant inhomogeneous term in a convex domain,and proved that u has one unique critical point. In 1985 Kawohl [18] extended Makar-Limanov’s resultunder some hypothesis on the second derivative of f. In 1998 Cabr´e and Chanillo [6] proved that thePoisson equation −(cid:52) u = f ( u ) in smooth, bounded and convex domains of R n ( n ≥
2) has exactly onenondegenerate critical point under the assumption of semi-stable solutions. Under the similar assumptionsabout the domains, Arango [2] showed Poisson equation has exactly one nondegenerate critical point,provided that f is a smooth and increasing function satisfying f (0) > . In 2008 Finn [13] provedthe uniqueness and nondegeneracy of critical points, under the same hypothesis of [6], and the weakerassumptions that Ω is a strictly convex C ,α domain. Moreover, other authors have solved this problemand some other related problems in convex domains (see [7, 8, 9, 12, 16, 23]). For instance, in 2017 Deng,Liu and Tian [12] proved that the solution of constant mean curvature equation with Neumann or Robinboundary condition has exactly one nondegenerate critical point in R n ( n ≥ K in nonconvex domains (see [1, 3, 11]).For nonconvex domains, the critical set K of solutions for elliptic problems seems to be less considered.In 1992 Alessandrini and Magnanini [1] studied the geometric structure of the critical set of solutions tosemilinear elliptic equations in a planar nonconvex domain, whose boundary is composed of finite simpleclosed curves. They deduced that the critical set is made up of finitely many isolated critical points.In 2012 Arango and G´omez [3] considered the geometric distribution of critical points of the solutionsto a quasilinear elliptic equation with Dirichlet boundary condition in strictly convex and nonconvexplanar domains respectively. In 2017 Deng and Liu [11] investigate the geometric stucture of interiorcritical points of solutions u to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundaryconditions in a simply connected or multiply connected domain Ω in R . They develop a new methodto prove Σ ki =1 m i + 1 = N or the different result Σ ki =1 m i = N, where m , · · · , m k are the respectivemultiplicities of interior critical points x , · · · , x k of u and N is the number of global maximum points of u on ∂ Ω.However, so far, the result about the critical set of solutions for quasilinear elliptic problems in higherdimension spaces is still an open problem. The goal of this paper is to obtain some results about thecritical set of solutions for mean curvature equation in smooth, bounded, strictly convex and nonconvexdomains of R n ( n ≥
2) respectively. Moreover, the domains Ω are some domains of revolution formedby taking a strictly convex and nonconvex planar domain about one axis respectively. Owing to thedomains are symmetric with respect to some axis, therefore, we consider that the solutions of meancurvature equation should be symmetric about some axis, and the detail conclusion about symmetricsolution of mean curvature equation can been seen in [4, 19, 24, 25, 27].Throughout this paper, we shall suppose that Ω is a smooth, bounded, strictly convex and nonconvexdomain respectively, and that f is a real analytic, nondecreasing function. As we know that the coefficients a ij of mean curvature operator L are analytic in R n , and L is uniformly elliptic. Within this assumptionsand conditions, the existence of solutions cannot be taken for granted, but if there has a positive solution,then it is unique and analytic (see [22, 26]). The key idea is that we turn quasilinear elliptic equationassociated to u into a linear elliptic equation associated to v = u θ . The results of this paper can be shownas the following three main theorems:
Theorem 1.
Let Ω be a smooth, bounded and strictly convex domain of rotational symmetry with respectto an axis in R n ( n ≥ . Suppose that f is a real analytic, nondecreasing function in R and that u is apositive solution of equation (1.1). Then u has one unique nondegenerate critical point in Ω . heorem 2. Let Ω be a symmetric concentric spherical annulus domain with external boundary S n − E and internal boundary S n − I in R n ( n ≥ , where the spherical surfaces S n − E and S n − I centered at theorigin. Let u be a solution of the constant mean curvature equation (1.1) for the case H = 0 . Then thecritical set K of u exists exactly one closed surface S , and S is a spherical surface centered at the origin. Theorem 3.
Let Ω be a rotationally symmetric eccentric spherical annulus domain with respect to anaxis in R n ( n ≥ , which has external boundary S n − E and internal boundary S n − I . Let u be a solutionof the constant mean curvature equation (1.1) for the case H = 0 . Then the critical set K of u does notexist a rotationally symmetric closed surface S with respect to an axis. In fact, K is made up of finitelymany isolated critical points ( p , p , · · · , p l ) on an axis and finitely many rotationally symmetric criticalJordan curves ( C , C , · · · , C k ) with respect to an axis. The rest of this paper is written as follows. In Section 2, we describe the nodal set N θ and thecritical set M θ of u θ , prove that N θ cannot enclose any subdomain of Ω and M θ = ∅ , i.e., the solution u of equation (1.1) is a Morse function. In Section 3, we give some descriptions about the geometricdistribution of critical points in planar domain Ω , where Ω is a strictly convex domain and annulusdomain respectively. In Section 4, our difficulty is to prove the rationality of projection. Firstly, we givesome known results about symmetric solution of mean curvature equation in strictly convex domain Ωof R n ( n ≥
3) and the detailed proof of Theorem 4.2. Furthermore, we study the geometric distributionabout the critical set K of solutions u for the constant mean curvature equation (1.1) for the case H = 0in a concentric (respectively an eccentric) spherical annulus domain Ω of R n ( n ≥ K exists (respectively does not exist) a rotationally symmetric critical closed surface S respectively. Infact, in an eccentric spherical annulus domain Ω, K is made up of finitely many isolated critical points( p , p , · · · , p l ) on an axis and finitely many rotationally symmetric critical Jordan curves ( C , C , · · · , C k )with respect to an axis. N θ and critical set M θ of u θ In order to conveniently describe the critical set K = { x ∈ Ω |∇ u ( x ) = 0 } , we need introduce somenotations and auxiliary terms. Now, for any direction θ = ( θ , θ , · · · , θ n ) ∈ S n − and any function u, we define the nodal set of directional derivative N θ = { x ∈ Ω | u θ = ∇ u ( x ) · θ = 0 } . We know that K = N θ ∩ N α , if θ and α are two noncollinear directions of S n − , and we consider the following criticalset of directional derivative u θ M θ = { x ∈ N θ |∇ u θ ( x ) = D u ( x ) · θ = 0 } ⊂ N θ , (2.1)where D u ( x ) denotes the Hessian matrix of u at x. Near the regular points of u θ , N θ can be locally parametrized as a solution of the following Hamiltoniansystem associated to u θ ˙ x ( t ) = B ∇ u θ ( x ) , (2.2)where B is the Poisson matrix.In particular when n = 2 , then B = (cid:18) − (cid:19) , we assume x ( t ) = x ( t ; θ, p ) be the solution of equation (2.2) satisfying x (0) = p. Since f is a real analyticfunction, we know that u is also analytic in Ω , so the solution of equation (2.2) is analytic too.Next, we will give a key idea for studying the critical set of solution u for mean curvature equation(1.1). Assume u is a positive solution of equation (1.1), for any θ ∈ S n − , we turn quasilinear elliptic3quation associated to u into a linear elliptic equation associated to v = u θ . Firstly, we differentiate theequation (1.1), then take inner product with θ, hence we can get the following equation L u v + h ( x ) ∂v∂x + h ( x ) ∂v∂x + · · · + h n ( x ) ∂v∂x n = f (cid:48) ( u ) v, (2.3)where L u v = n (cid:88) i,j =1 a ij ( ∇ u ) ∂ v∂x i ∂x j , n ≥ h k ( x ) = n (cid:88) i,j =1 u x i x j ∂a ij ∂u x k , ≤ k ≤ n. The following we will give the crucial lemma of the proof of Theorem 3.1 for dimension n = 2. Lemma 2.1.
Let Ω be a strictly convex planar domain. Suppose that f is a real analytic, nondecreasingfunction in R and that u is a positive solution of equation (1.1). Then for any θ ∈ S , N θ cannot encloseany subdomain of Ω , i.e., N θ without self-intersection.Proof. By contradiction. Assume N θ enclose some subdomain of Ω for some θ, let C ∈ N θ be a Jordancurve and Ω C be the intersection of the interior of C with Ω . Since f is an nondecreasing function and v satisfies the zero boundary condition. By the maximum principle, the only solution of equation (2.3)is v ≡ C . On the other hand, if C is the only boundary of Ω C , and u θ is analytic, then we have u θ = 0 in Ω , it means that u is a constant in the θ direction. Since u = 0 on ∂ Ω , we get u = 0 in Ω , which contradicts with the fact u > . Therefore C cannot be the only boundary of Ω C , and Ω C isnot simply connected. It is contradictory with the convexity of domain Ω , hence N θ cannot enclose anysubdomain of Ω . Lemma 2.2. (see[10, Theorem 2.5]) If v is a nonzero solution of equation (2.3) for n = 2 . Then thecritical points of v on its nodal set are isolated and the nodal set N θ is a regular analytic curve at regularpoints. Moreover, at any critical point, the nodal set is locally an equiangular system of at least four rayssplitting Ω into a finite number of connected subregions. Remark 2.3.
According to Lemma 2.1 and Lemma 2.2, we easily know that M θ = ∅ and N θ ishomeomorphic to the interval [0 , , since M θ = ∅ , we get ∇ u θ ( x ) = D u ( x ) · θ (cid:54) = 0 for any θ, so rank ( D u ( x )) = 2 . Furthermore, owing to K ⊂ N θ for any θ , we deduce that the solution u of equation(1.1) is a Morse function. The descriptions about the Morse and semi-Morse function have been alreadystudied(see [5]). In this section, we investigate the geometric distribution of critical points in planar domain Ω , whereΩ is a smooth, bounded, strictly convex domain and nonconvex domain respectively. Theorem 3.1.
Let Ω be a smooth, bounded and strictly convex domain in R , whose boundary has positivecurvature. Suppose that f is a real analytic, nondecreasing function in R and that u is a positive solutionof equation (1.1). Then u has one unique nondegenerate critical point in Ω . Proof.
We divide the proof into two steps.Step 1, By Lemma 2.1 and Remark 2.3, since M θ = ∅ for any θ ∈ S , we deduce that the solution u of equation (1.1) is a Morse function, so all critical points of u are nondegenerate.Step 2, we will prove the uniqueness of critical points. Indeed, if x ∈ Ω \ K, we note ∇ u ( x ) = |∇ u ( x ) | (cos λ ( x ) , sin λ ( x )) , (3.1)4hich λ ( x ) ∈ R and cos λ ( x ) = u x |∇ u | , sin λ ( x ) = u x |∇ u | . This expression defines a smooth function λ locally in x, for any x ∈ Ω \ K, we have x ∈ N θ if and only if θ ⊥ ∇ u ( x ) , since ∇ u · (cos( λ ( x ) ± π/ , sin( λ ( x ) ± π/ , hence we deduce x ∈ N λ ( x ) ± π/ . Next we need compute, locally, an expression for ∇ λ. By formula (3.1), since sin λ ( x ) = u x |∇ u | , then wehave λ x cos λ ( x ) = u x x u x − u x x u x u x |∇ u | ,λ x cos λ ( x ) = u x x u x − u x x u x u x |∇ u | , in case that cos λ ( x ) (cid:54) = 0 , we deduce ∇ λ = |∇ u | ( u x x u x − u x x u x , u x x u x − u x x u x )= |∇ u | D u · ( − u x , u x ) . Even though λ be defined only locally, the above expression allows us to define the vector field X = |∇ u | D u · ( − u x , u x ) in Ω \ K, (3.2)which accords with X ( x ) = ∇ λ ( x ) . On one hand, we can know that X ( x ) (cid:54) = 0 for any x ∈ Ω \ K. Indeed, since rank ( D u ( x )) = 2 , we get X = |∇ u | D u · ( − u x , u x ) (cid:54) = 0 . On the other hand, since rank ( D u ( x )) = 2 , at all points in some neighborhood of K, for someconstant C >
0, we obtain | X | = 1 |∇ u | | D u · ( − sin λ ( x ) , cos λ ( x )) ≥ C |∇ u | in a neighborhood of K. Since X ( x ) (cid:54) = 0 in x ∈ Ω \ K, then we deduce | X | ≥ C |∇ u | in x ∈ Ω \ K. (3.3)By Hopf boundary lemma, we get (cid:104) X, t (cid:105) = |∇ u | − u tt = κ, where (cid:104)· , ·(cid:105) denotes interior product, κ isthe curvature of ∂ Ω and t is the positive oriented tangent vector to ∂ Ω. Hence we have (cid:104)
X, t (cid:105) > ∂ Ω . Next we revise X in a neighborhood of Ω to agree with t on ∂ Ω . Further there exists a smooth vectorfield Y on Ω \ K such that Y = (cid:26) X in Ω \ K,t on ∂ Ω , (3.4)so (cid:104) Y, X (cid:105) > \ K. Then we define another vector field Z associated to Y, which is smooth in x ∈ Ω \ K, and tangent to ∂ Ω . Z = (cid:40) Y (cid:104) Y,X (cid:105) in Ω \ K, K, (3.5)we claim that field vector Z can be extended to be Lipschitz in Ω , that is | Z ( x ) − Z ( x ) | ≤ C | x − x | ∀ x , x ∈ Ω (3.6)for some constant
C > . Indeed, since Z = X | X | in Ω \ K, by (3.3) we have | Z | ≤ C |∇ u | in Ω (3.7)5or another positive constant C > , hence Z is continuous in Ω . Firstly, if either x or x belong to K, without loss of generality, let x ∈ K, then | Z ( x ) − Z ( x ) | ≤ | Z ( x ) | + | Z ( x ) | = | Z ( x ) |≤ C |∇ u ( x ) | = C |∇ u ( x ) − ∇ u ( x ) |≤ C | x − x | . (3.8)Secondly, if a segment l joining x and x intersects with K, note that l ⊂ Ω , since we select Ω to beconvex. According to (3.8), for any p ∈ K, we have | Z ( x ) − Z ( x ) | ≤ | Z ( x ) − Z ( p ) | + | Z ( x ) − Z ( p ) | = | Z ( x ) | + | Z ( x ) |≤ C | x − x | . Finally, suppose that this segment l is included in Ω \ K. Hence we can differentiate Z along l, where D denotes the full differential, by (3.3), we have | DZ | = | D ( X | X | ) | ≤ C | DX || X | ≤ C |∇ u | | DX | in Ω \ K. Note that differentiating X = |∇ u | D u · ( − u x , u x ) in Ω \ K, we get | DX | ≤ C |∇ u | − . So we deducethat | DZ | ≤ C along l, and complete the proof of (3.6).Since each Lipschitz vector field locally can generate a one-parameter transformation group(i.e., flow).Now, we assume that there is an open convex neighborhood E of K, provides K ⊂ E ⊂ E ⊂ Ω . Next,we will consider the one-parameter transformation group ϕ t at time t associated to the Lipschitz vectorfield Z, which satisfies (cid:40) ϕ ( p ) = p ∀ p ∈ E,ϕ s ◦ ϕ t = ϕ s + t ∀ s, t ∈ R , (3.9)and Z ( p ) = dϕ t ( p ) dt | t =0 ∀ p ∈ E. (3.10)Since Z is parallel to ∂ Ω , hence we know that ϕ t is a continuous flow that keep Ω invariant. Moreover,the flow ϕ t keep any critical point of u fixed. Then, we can easily to deduce that ϕ t ( N θ ) ⊂ N θ + t for any nodal set N θ (i.e., path associated with the flow ϕ t ) and any time t . Indeed, by reversing time, weknow that ϕ t is an homeomorphism from N θ onto N θ + t , i.e., ϕ t ( N θ ) = N θ + t . Then we choose the time t = π, we get ϕ π ( N ) = N π = N , (3.11)where ϕ π is a homeomorphism of N that interchanges the two end-points of N . According to Remark2.3, since N is homeomorphic to the interval [0 , , so ϕ π keep that N has one unique fixed point.Moreover, because the flow ϕ π keep any critical point of u fixed, thus the solution u of equation (1.1) hasexactly one critical point. Remark 3.2.
The convexity of all the nodal set of any solution u in Theorem 3.1 is an open problem,even for semilinear case in dimension 2. Note that the strict convexity of nodal set is a stronger propertythan the uniqueness of critical points of the solution u . The related research results can be found in [17]. Next we study the geometric structure about critical set K of solutions u for the constant meancurvature equation in planar nonconvex domain Ω , where Ω is a concentric and an eccentric circle annulusdomain respectively. 6 emma 3.3. Let Ω be a symmetric planar concentric circle annulus domain with internal boundary γ I and external boundary γ E , where the circles γ I and γ E centered at the origin. Let u be a solution of theconstant mean curvature equation (1.1). Then u has exactly one critical Jordan curve C in Ω , and C isa circle centered at the origin.Proof. Due to the results of Theorem 11 in [3], we know that the critical set K of solution u is eitherfinitely many isolated critical points, or is made up of exactly one critical Jordan curve, that is, thecritical points and critical Jordan curve can’t exist at the same time. And according to the results of[4], the constant mean curvature equation (1.1) has a unique radial symmetric solution u in concentriccircle annulus domain Ω, so we deduce that the critical set K of solution u reduces to exactly one criticalJordan curve C , and C is a circle centered at the origin. Remark 3.4.
For the case of dimension n = 2 , if the domain Ω is a symmetric planar eccentric annuluswith respect to one axis, with internal boundary γ I and external boundary γ E , moreover, γ I and γ E arecircles, then the constant mean curvature equation (1.1) exists only a finite number of critical points, theresult from [3]. At the same time, if critical set K exists a Jordan curve C in an annulus domain Ω , thenthe geometry structure of N θ as follows:(1) For any θ ∈ S , N θ contains the curve C .(2) There exists exactly two corresponding branching points of N θ in critical Jordan curve C, denotesby points p and p ∗ , and θ is tangent to C at this two points.(3) The nodal set N θ exists exactly four branches departing from p (respectively p ∗ ), where two branchesof them are included in C, and the other two branches end respectively at the internal boundary γ I andexternal boundary γ E . (4) The points on external boundary γ E (respectively γ I ), the branches starting at q and q ∗ end, arecorresponding points, and θ is tangent to ∂ Ω at this two points.(5) N θ contains no any other points except for the above (1) ∼ (4). For a given θ ∈ S , next we will give the geometric structure of N θ . Figure 1
The geometric distribution of N θ . This section is aimed to deduce that the geometric structure of critical set K for mean curvatureequation (1.1) in higher dimension spaces. Next we will give some results about radial symmetric solutionof mean curvature equation (1.1) in strictly convex domains, as follows: Lemma 4.1. (see[22, Theorem 8.2.2]) Let Ω be an open ball in R n , n ≥ . Assume u ∈ C (Ω) is adistribution solution of the problem (1.1), while the function f ( u ) is locally Lipschitz continuous in R +0 . Then every solution u ∈ C (Ω) is radially symmetric and satisfies u r < . According to the above results about symmetric solution of problem (1.1), next we investigate thegeometric distribution of critical set for higher dimension spaces. Our difficulty is to prove the rationalityof projection of higher dimensional space onto two dimension plane.7 heorem 4.2.
Let Ω be a smooth, bounded and strictly convex domain of rotational symmetry withrespect to an axis in R n ( n ≥ . Suppose that f is a real analytic, nondecreasing function in R and that u is a positive solution of equation (1.1). Then u has one unique nondegenerate critical point in Ω . Proof.
Without loss generality, let Ω be a domain of revolution formed by taking a strictly convex planardomain 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 − . Due to the results of Pucci and Serrin [22, 25], we deduce that the solution u satisfies u ( x (cid:48) , x n ) = u ( | x (cid:48) | , x n ) (cid:44) v ( r, x n ) (4.1)and ∂v∂r ( r, x n ) < r (cid:54) = 0 . (4.2)From (4.2), we can know that the critical points of u lie on x n axis. Next, according to (4.1) we havethat u x n ( x (cid:48) , x n ) = v x n ( r, x n ) . (4.3)Moreover, we can deduce that u x n satisfies the following equation( Lu ) x n = 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 = f (cid:48) ( u ) u x n , 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 − f (cid:48) ( u ) u x n = 0 , (4.4)where a ij = √ |∇ u | ( δ ij − u xi u xj |∇ u | ) and ∂a ij ( ∇ u ) ∂x n as the first derivative of u x n .By (1.1), the strict convexity of Ω and the Hopf boundary point lemma, we can know that u x n vanishesprecisely on the n − 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 (4.3), N is rotationallyinvariant about the x n axis.So we turn the mean curvature equation (1.1) for dimension nLu = div ( ∇ u √ |∇ u | ) = f ( u ) (4.5)into the following similar mean curvature equation for dimension 2 Lv = div ( ∇ v (cid:112) |∇ v | ) + 1 (cid:112) |∇ v | n − r v r = f ( v ) , that is Lv = (cid:80) i,j =1 a ij ( ∇ v ) v ij + √ |∇ v | n − r v r = f ( v ) , (4.6)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 . Next, the proof is essentially same as the proof of Theorem 2 in [6]. Their work is based on the results ofGidas, Ni and Nirenberg [14], the ideas of Payne [21] and Sperb [28]. To prove that, on one hand, whenevercritical set has exactly one point, since all critical points of u are contained in N ∩ { x = · · · = x n − = 0 } 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 in the x , x n x n axis, by (4.6),where N can be seen as the projection of N in the x , x n N cannot enclose anysubdomain of Ω (By Lemma 2.1, N cannot enclose any planar subdomain of Ω ∩ { x = · · · = x n − = 0 } , where N looks like the nodal set of some homogeneous polynomial in x , x n . ). Because N is symmetricwith respect to the x n axis and intersects the x n axis at exactly one point, hence we prove the uniquenessof critical points.On the other hand, how to show that critical point p is nondegenerate, we restatement that u isrotationally symmetric with respect to x n axis and critical point p lies on this axis. From (4.1) and(4.2), we have that { u x k = 0 } = { x k = 0 } ∩ Ω for all 1 ≤ k ≤ n − . Hence u x i x j ( p ) = 0 for anyindex 1 ≤ i < j ≤ n, that is, D u ( p ) is diagonal. By (4.2), we can know that u x k < D k = { x k > } ∩ Ω for 1 ≤ k ≤ n − . What’s more, in domain 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 − f (cid:48) ( u ) u x k = 0 , (4.7)where ∂a ij ( ∇ u ) ∂x k as the first derivative of u x k . According to the Hopf boundary point lemma, we deducethat u x k x k ( p ) < ≤ k ≤ n − , where critical point p ∈ ∂ D k . Finally, we recall that the function u x n satisfies (4.4). By the definition of N , u x n < N . Then applying the Hopf boundary point lemma to u x n at p ∈ N , we have that u x n x n ( p ) < . So weprove that the Hessian matrix D u ( x ) of u is diagonal and negative definite at critical point p , hence p is a unique nondegenerate critical point.Next we will study the geometric structure about critical set K of solutions u for the constant meancurvature equation (i.e. f ( u ) = H, H = constant ) in a symmetric concentric spherical annulus domainwith external boundary S n − E and internal boundary S n − I of R n ( n ≥ , where the spherical surfaces S n − E and S n − I centered at the origin. It is known that for H small enough, there exists a uniquesymmetric solution u for the constant mean curvature equation (1.1), where u satisfies that ∂u∂r > H = 0 (see [4]). Theorem 4.3.
Let Ω be a symmetric concentric spherical annulus domain with external boundary S n − E and internal boundary S n − I in R n ( n ≥ , where the spherical surfaces S n − E and S n − I centered at theorigin. Let u be a solution of the constant mean curvature equation (1.1) for the case H = 0 . Then thecritical set K of u exists exactly one closed surface S , and S is a spherical surface centered at the origin.Proof. Without loss of generality, we assume that the domain Ω is a domain of revolution formed bytaking a symmetric planar concentric circle annulus domain Ω (cid:48) in the x , x n plane about x n axis, wherethe domain Ω (cid:48) centered at the origin.By the results of Bergner [4], so we deduce that the solution u satisfies u ( x (cid:48) , x n ) = u ( | x (cid:48) | , x n ) (cid:44) v ( x n , r ) (4.8)and ∂v∂r ( x n , r ) > , (4.9)where x (cid:48) = ( x , · · · , x n − ) and r = (cid:113) x + · · · + x n − . So we turn the constant mean curvature equation (1.1) for dimension nLu = div ( ∇ u √ |∇ u | ) = 0 (4.10)into the following similar mean curvature equation for dimension 2 Lv = div ( ∇ v (cid:112) |∇ v | ) + 1 (cid:112) |∇ v | n − r v r = 0 , Lv = (cid:80) i,j =1 a ij ( ∇ v ) v ij + √ |∇ v | n − r v r = 0 , (4.11)where ∇ v = ( ∂v∂x n , ∂v∂r ) , a ij = √ |∇ v | ( δ ij − v i v j |∇ v | ) and v = ∂v∂x n , v = ∂v∂r . For any θ = ( θ , θ ) = (cos α, sin α ) ∈ S , where α ∈ [0 , π ) . We turn quasilinear elliptic equationassociated to v into a linear elliptic equation associated to w = v θ = ∇ v · θ. Firstly, we differentiate theequation (4.11), then take inner product with θ. In order to conveniently, we denote 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 [(1 + v y ) ∂w∂y − v y v y ∂w∂y ] = √ |∇ v | n − r v r θ , (4.12)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 ≤ . By (4.9) and (4.12), we deduce that Lw = L v w + h ( y ) ∂w∂y + h ( y ) ∂w∂y + |∇ v | ) n − r [(1 + v y ) ∂w∂y − v y v y ∂w∂y ] ≥ . (4.13)By (4.13), so we can consider the result that graphic projected onto x , x n plane. Due to Lemma 3.3, wehave known that the constant mean curvature equation (1.1) exists only one unique critical Jordan curve C in symmetric planar concentric circle annulus domain. In turn, we rotate the geometry distributionof critical Jordan curve C in symmetric planar concentric circle annulus domain with respect to x n axis.Therefore we get the geometry distribution of critical set K in a symmetric concentric spherical annulusdomain Ω as shown in the following Figure 2. Figure 2
The geometric distribution of critical set K in a symmetric concentric spherical annulus domain. Figure 2 shows that the critical set K of u exists exactly one closed surface S , and S is a sphericalsurface centered at the origin.The rest of this section is aimed to prove that, in the case of the constant mean curvature equationin a rotationally symmetric eccentric spherical annulus domain Ω with respect to an axis, the critical set K does not exist a critical closed surface S , where S is rotationally symmetric with respect to an axis. Theorem 4.4.
Let Ω be a rotationally symmetric eccentric spherical annulus domain with respect to anaxis in R n ( n ≥ , which has external boundary S n − E and internal boundary S n − I . Let u be a solutionof the constant mean curvature equation (1.1) for the case H = 0 . Then the critical set K of u does notexist a closed surface S , where S is rotationally symmetric with respect to an axis. roof. The proof is based on the idea of Theorem 4.3, and we prove the theorem for two cases. Withoutloss of generality, we assume that the center of spherical surfaces S n − E and S n − I both on x n axis, so wededuce that the solution u satisfies u ( x (cid:48) , x n ) = u ( | x (cid:48) | , x n )where x (cid:48) = ( x , · · · , x n − ) . Case 1, if the critical set K enclose a subdomain of Ω , denotes by closed surface S , where S isrotationally symmetric with respect to x n axis. According to the assumptions, we can know that criticalset K is the closed surface S and the center of S on x n axis. Because the domain Ω and the solution u are rotationally symmetric with respect to x n axis, in the same way, so we can consider the result thatgraphic projected onto a two-dimensional plane which pass through x n axis, without loss of generality,denotes by x , x n plane. Therefore we get the geometry distribution of critical Jordan curve C in planarnonconvex domain as shown in the following Figure 3. Figure 3
The geometric distribution of critical set K for case 1. Because the interior of critical Jordan curve C is simply connected, it is contradictory with Lemma2.1, so the critical set K cannot enclose a subdomain of Ω . Case 2, if the critical set K enclose the internal boundary S n − I , denotes by closed surface S , where S is rotationally symmetric with respect to x n axis. we can consider the result that graphic projected onto x , x n plane, hence we can get the following geometry distribution of critical Jordan curve C in planarnonconvex domain. Figure 4
The geometric distribution of critical set K for case 2. Due to Remark 3.4, we have known that constant mean curvature equation (1.1) exists only a finitenumber of isolated critical points in a symmetric planar eccentric circle annulus domain Ω with respectto one axis. Figure 4 shows that it is contradictory with Remark 3.4, Therefore K cannot enclose theinternal boundary S n − I . As an incidental consequence of Theorem 4.4 we can fully describe the geometric distribution of criticalset K to the constant mean curvature equation (1.1) in a rotationally symmetric eccentric sphericalannulus domain Ω with respect to an axis in R n ( n ≥ , as stated in the following corollary. Corollary 4.5.
We have known that the constant mean curvature equation (1.1) exists only a finitenumber of isolated critical points in a planar eccentric circle annulus domain Ω , where Ω is symmetric ith respect to one axis. Hence we can deduce that the geometric distribution of critical set K of theconstant mean curvature equation (1.1) for the case H = 0 in a rotationally symmetric eccentric sphericalannulus domain Ω with respect to an axis in R n ( n ≥ , without loss of generality, denotes by x n axis.Then critical set K is made up of finitely many isolated critical points ( p , p , · · · , p l ) on x n axis andfinitely many rotationally symmetric critical Jordan curves ( C , C , · · · , C k ) with respect to x n axis, andthe geometric distribution of critical set K as shown in the following Figure 5. Figure 5
The geometric distribution of critical set K in an eccentric spherical annulus. Remark 4.6.
If the general quasilinear elliptic equation Lu = n (cid:80) i,j =1 a ij ( ∇ u ) ∂ u∂x i ∂x j = f ( u ) satisfies theabove corresponding assumptions, conditions and exists the corresponding symmetric solution as in Theo-rem 4.2, Theorem 4.3 and Theorem 4.4, then the general quasilinear elliptic equation has the same resultsabout the geometric distribution of critical set K as in Theorem 4.2, Theorem 4.3 and Theorem 4.4. Acknowledgement.
The first author is very grateful to his advisor Professor Xiaoping Yang for hisexpert guidances and useful conversations.
References [1] G. Alessandrini, R. Magnanini, The index of isolated critical points and solutions of elliptic equations in the plane.Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19(4) (1992) 567-589.[2] J. Arango, Uniqueness of critical points for semi-linear elliptic problems in convex domains. Electron. J. DifferentialEquations, 2005 (2005) 1-5.[3] J. Arango, A. G´omez, Critical points of solutions to quasilinear elliptic problems. Nonlinear Anal. 75 (2012) 4375-4381.[4] M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains. DifferentialGeom. Appl. 27 (2009) 335-343.[5] R. Bott, Lectures on Morse theory, old and new. Bull. Amer. Math. Soc. (N.S.) 7 (1982) 331-358.[6] X. Cabr´e, S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains. Selecta Math. (N.S.) 4 (1998)1-10.[7] L. A. Caffarelli, A. Friedman, Convexity of solutions of semilinear elliptic equations. Duke Math. J. 52 (1985) 431-456.[8] L. A. Caffarelli, J. Sprock, Convexity properties of solutions to some classical variational problems. Comm. PartialDifferential Equations 7 (1982) 1337-1379.[9] J. T. Chen, W. H. Huang, Convexity of capillary surfaces in the outer space. Invent. Math. 67 (1982) 253-259.[10] S. Cheng, Eigenfunctions and nodal sets. Comment. Math. Helv. 51 (1976) 43-55.[11] H. Y. Deng, H. R. Liu, Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichletboundary conditions. preprint.[12] H. Y. Deng, H. R. Liu, L. Tian, Uniqueness of critical points of solutions to the mean curvature equation withNeumann and Robin boundary conditions. preprint.[13] D. Finn, Convexity of level curves for solutions to semilinear elliptic equations. Commun. Pure Appl. Anal. 7 (2008)1335-1343.[14] B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle. Commun. Math.Phys. 68 (1979) 209-243.[15] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order. New York: Springer-Verlag, 1983.