Symmetric solutions of the singular minimal surface equation

We classify all rotational symmetric solutions of the singular minimal surface equation in both cases α<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <0$$\end{document} and α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document}. In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.


Introduction
The singular (or symmetric) minimal surface equation (in short: s.m.s.e.) is the equation where u ∶ → ℝ , ⊂ ℝ n open, denotes some function and ∈ ℝ stands for some real number. Observe that for any ∈ ℝ equation (1) is the Euler equation of the variational integral which is the nonparametric counterpart of the energy functional where M ⊂ ℝ n × ℝ + denotes some C 2 -hypersurface and H n stands for the n-dimensional Hausdorff measure. M is stationary for the energy (3) There are several key motivations to considering equation (1) and the associated variational integral (2) in both cases > 0 and < 0 , respectively. On the one hand, the variational problem connected to the integral (2) is a singular problem which admits in general only 1 2 -Hölder continuous solutions, cp. Bemelmans, Dierkes [1], Dierkes [3], Tennstädt [13], and on the other hand (1) with = 1 appears as a model problem for the multi-dimensional analogue of the catenary, while, for = m ∈ ℕ , equation (1) describes symmetric minimal hypersurfaces in ℝ n × ℝ m × ℝ . Finally, for = −n equation (1) is the minimal surface equation in the hyperbolic space ℝ n × ℝ + of curvature K = −1 . A further application in architecture lends special interest to the problem. We refer the reader to Dierkes [7,8] for more detailed information and further references to the literature.
Clearly, the case = 0 corresponds to classical minimal surfaces and will not be considered here.
(8) tan = y � x � or x � , y � = (cos , sin ), Further insights are obtained if we also consider the stationarity condition in the "phase space" ( , ) , where ∈ C 1 (I) stands for the polar angle Then we have which in turn leads to the planar ordinary system Symmetric solutions of (1) and (4) resp., i.e., solutions of (6), (7), (9), (11) and (12), respectively, have been considered in the papers by Keiper [10] (preprint), Lopez [11] and Dierkes [3,4,7]. The reason why we start a new discussion is twofold: Firstly, we add some new solutions to (6) or (9) by carefully analyzing system (12) in the phase space (cp. Theorem 1, (iii) and Theorem 7). Secondly, we accomplish Keiper's [10] paper by thoroughly examining, extending and proving his assertions and indications of proofs (cp. Theorem 14). In addition to that we completely classify the stability and (non-) minimizing properties of the solutions (cp. Theorems 17 and 20 ) and prove a Bernstein-type result (cp. Theorem 19) (Fig. 1). Finally, we think that our arguments are slightly cleaner than the original proofs in [10] and [11].

Solutions for ˛< 0
For negative , symmetric solutions of the s.m.s.e (1) are classified as follows, cp. also Lopez [11].
Theorem 1 Let = (x(s), y(s)) ⊂ ℝ × ℝ + , s ∈ I , denote a maximal solution of (7), < 0 , then can be described by one of the following three cases where (iii) only occurs if −1 < < 0 : is the graph of a strictly concave symmetric function on a bounded interval of the x-axis which attains its maximum at x = 0 and intersects the x-axis orthogonally.
(ii) stays on one side of the y-axis (on x > 0 , say) and intersects the x-axis orthogonally in both end points. x = x(s) attains exactly one interior minimum in the interval I and no maximum. has a horizontal tangent at the unique maximum of y = y(s) . Furthermore, has no self-intersections.
stays on one side of the y-axis (on x > 0 , say) and is the graph of a strictly concave function, which is defined over some compact interval of the x-axis. At both end points intersects the x-axis orthogonally.
We split the proof of Theorem 1 in five Lemmata and start with some simple properties of solutions of system (9): is a maximal solution that intersects the y-axis, then can be written as the graph of a strictly concave function on a bounded interval of the x-axis that is symmetric about the y-axis. The function has a unique maximum at x = 0 . At the boundary points a, b of I, intersects the x-axis.
Assume there exists a smallest s 1 ∈ (0, b) with � s 1 = 0 . Then ′′ s 1 ≥ 0 and, by taking the derivative in (13), Plugging in s 1 , we obtain 0 ≤ �� s 1 = sin s 1 cos s 1 , and it follows that s 1 ∈ 0, 2 + ℤ. Independently of the existence of s 1 , now assume there is a smallest s 2 ∈ (0, b) with The existence of either s i implies the existence of the other s j in 0, s i , leading to a contradiction. Hence, we must have � (s) < 0 and (s) ∈ − 2 , 0 for all s ∈ (0, b) and is the graph of a strictly concave function. It remains to determine the behavior of (s) near s = b.
Since x and y are monotone and bounded and s is the arc length, it follows that b < ∞ . Because b is maximal, (x, y, ) has to leave every compact subset of ℝ + × ℝ + × ℝ near b. The only way this is possible is that y(b) ∶= lim s→b y(s) = 0 . Furthermore, the limits For negative s close to zero, we have strictly decreasing, (s) ∈ 0, 2 and x(s) > 0. Assume the existence of a maximal s 1 ∈ (a, 0) with s 1 ∈ 0, 2 and � s 1 = 0 . Then ′′ s 1 ≤ 0 , but also If (s) < 2 for all s ∈ (a, 0) , the behavior of is described by case (b). Otherwise, there must exist a maximal s 2 ∈ (a, 0) with s 2 = 2 . There we have � s 2 = 1−n x(s 2 ) < 0 , so is still decreasing. Because of x � s 2 = 0 and x �� s 2 = − � s 2 > 0 , x has a local minimum at s 2 .
Assume now that there is a maximal s 4 ∈ a, s 2 with s 4 = 2 . Then ′ s 4 ≥ 0 , but on the other hand � s 4 = 1−n x(s 4 ) < 0 , another contradiction.
for all s ∈ a, s 2 . Next, we will show that ′ does not stay negative on all of I. Assuming otherwise that � (s) < 0 for all s ∈ a, s 2 , consider the following cases: exist by monotonicity and boundedness. Because is parametrized by arc length, we must also have (a) = and lim s→a x(s) = ∞ . Using (9), we obtain lim s→a This yields the existence of a point s 5 ∈ a, s 2 with � s 5 = 0 . Since the second derivative �� s 5 = sin s 5 cos s 5 dle point of (12) with stable manifold in (1, 0)-direction and unstable manifold in 1, − n -direction. For ( , ) = − 2 , 0 , we have the linearized system with matrix Hence, (− 2 , 0) is a stable node. The eigenvectors corresponding to −1 and are (1, 0) and 1, − +1 n−1 , respectively. Therefore, the behavior depends on the size of : The trajectories spiral toward the -axis in negative direction.

Lemma 4
Every maximal solution of (9) in the quadrant ∈ 0, 2 intersects either the xand y-axis or the x-axis in the respective endpoints at a right angle. In particular this also holds in case (b) of Lemma 3.
Proof We look at the region ( , ) ∈ ℝ × 0, 2 in the phase plane of system 12 and show that all trajectories start and end in a singular (or equilibrium) point. Clearly, it suffices to consider the asymptotic behavior as t → ∞.
Assume on the contrary, there is a trajectory not approaching a singular point as t → ∞ . Then, by the Poincaré-Bendixson theorem (see Hartman [9, Section VII.4]) it must approach a periodic orbit or is periodic itself. This orbit must contain a critical point on the inside (see Corollary 2 in Perko [12, Section 3.12]). However, there is no critical point in ( , ) ∈ ℝ × 0, 2 ; hence, we obtain a contradiction. In the particular case of the left branch of a solution of type (b) in Lemma 3, the only possible end point is of type ( , ) = 2 , 0 because of the monotonicity of . Also x cannot converge to zero as y → 0 , since for small s → −∞ the trajectory has small values of > 0 and is close to 2 . Finally, every trajectory that is not of type as in Lemma 2 starts in an unstable and ends in a stable node (Fig. 2), so there always exists a (finite) point with ∈ ℤ . ◻ It still remains to show that both cases in Lemma 3 are possible: Lemma 5 There exists a value 0 = 0 (n, ) ∈ 0, 2 such that for every maximal solution (x, y, ) of (9) in the region ∈ 0, 2 with (0) = 0 , we have: Lemma 3.
Case − 1 < < 0 ∶ There is exactly one trajectory that leaves 2 , 0 in direction −1, +1 n−1 . Since we then have (t) < 2 for small t > −∞ and the value = 2 cannot be assumed later, the solutions belonging to must be of type (b). Furthermore, the value 0 ∈ 0, 2 is uniquely determined by the requirement that passes through the point ( , ) = 0, 0 . Every trajectory with (t) < 0 at the point t where = 0 has to leave 2 , 0 in direction (−1, 0) and therefore corresponds to (b). Otherwise, if > 0 at = 0 , it leaves the starting point with direction (1, 0) , and the solutions belong to case (a). ◻ Finally, also solutions of type (a) have no self-intersections.

Lemma 6 For every solution
Proof Assume this is not the case and choose a minimal s 2 such that there is a (maximal) Recall that the trajectories of solutions that touch the y-axis divide the phase plane in regions of -width equal to , and these trajectories meet every value of ∈ 0, 2 exactly once. In other words, the existence of a solution and points s 1 , s 2 ∈ I = (a, b) as above is impossible.
We still have to consider the case s 1 = a and s 2 = b . Since orthogonally meets the x-axis at both points, we must have we can pick the s i such that s 1 = s 2 . Again, this leads to a contradiction. ◻ The proof of Theorem 1 now easily follows from Lemma 2-6. The solutions of type (i) can be used in conjunction with a maximum principle to show that (1) has no solutions defined on all of ℝ n for < 0 . Also type (i) solutions minimize locally the energy E (⋅) in suitable classes of BV-functions (see Dierkes [7]).

Solutions for ˛> 0
The first part of this section is based on the unpublished work by Keiper [10] who investigated the case = 1. In addition to carefully scrutinizing his results, we give more detailed arguments for the proofs and generalize to arbitrary > 0 . Also we adopt Keiper's notation to denote smooth entire solutions of the system (15) as the n--tectum (the Latin word for "roof").

Classification and geometry
As before, we consider the system as well as system (12). In analogy to Keiper, we refer to any solution which intersects the y-axis orthogonally as the n--tectum (which is unique up to homotheties) (Fig. 3, left).
All solutions of (15) are classified in the following Theorem (Figs. 3, 4):  We start our analysis with the n--tectum.

Lemma 8
The n--tectum is the graph of a symmetric function on some interval of the x-axis which assumes a global minimum at x = 0 and is strictly increasing for x > 0 . (s) is defined for every s ∈ ℝ and is unbounded.
Proof Without loss of generality, let x(0) = 0 and (0) = 0 . By symmetry, we only have to consider s ≥ 0. At s = 0 , we get d ds = ny > 0 from (15) and therefore y has a local minimum there and ∈ 0, 2 for small s > 0. With the same line of arguments as in the previous section, it is easy to see that stays in the interval 0, 2 for all s > 0 . In particular, dx ds > 0 and dy ds > 0 for all s > 0. Since (x, y, ) is a maximal solution to (15) and is parametrized by arc length, must be defined for all s ∈ ℝ and is unbounded. ◻ For > 0 , the system has the following four types of singular points, where k 1 , k 2 ∈ ℤ , and n ∶= arctan √ n−1 .

Lemma 9
The system (12) has no nontrivial periodic solutions. Every trajectory in the --plane is bounded.
Proof It is sufficient to consider ∈ 0, 2 . For every point in ( , ) ∈ 2 , × 0, 2 , we have which means that the trajectories in this rectangle cannot be periodic and have to leave it at = 2 . Once a trajectory is in ( , ) ∈ 0, 2 × 0, 2 , it cannot leave this rectangle (Fig. 5). It remains to show that no periodic solutions are contained in this interval. Defining a function B( , ) ∶= sin −1 , we calculate Using the Bendixson-Dulac theorem (see Perko [12, Section 3.9]) concludes the proof. ◻ Lemma 10 Every trajectory of (12) within ∈ 0, 2 approaches singular points as t → ∞ and t → −∞ and -for each trajectory -at least one end point is of the form n + k , n for some k ∈ ℤ. In particular, every solution is of one of the four types (i)-(iv) in Theorem 7, and every case occurs.
Proof By the Poincaré-Bendixson theorem (see Hartman [9, Section VII.4]) and Lemma 9, every trajectory approaches singular points at both end points. By virtue of the anti-periodicity of (12), it is sufficient to analyze the points n , n , 0, 2 and 2 , 0 : 0, 2 ∶ The matrix of the linearized system is  The point is a saddle point, as in the case for < 0 . The only trajectory in ∈ 0, 2 starting at this point corresponds to the n--tectum (type (i)). It leaves the point with direction ( , −n) , entering ( , ) ∈ 0, 2 × 0, 2 which was already shown in Lemma 8. It is impossible to leave this rectangle and as t → ∞ , the trajectory of the n--tectum must approach n , n .
2 , 0 ∶ Here we have the linearized system given by the matrix and hence another saddle point. The only trajectory starting here leaves in the direction (1 − n, + 1) . Arguing as before we find that the trajectory has to approach the singular point n , n and the solution curve is described by (iii). 8 . In any case both real parts are negative and we therefore have a stable node, if n + ≥ 4 + √ 8 , or a stable focus, if n + < 4 + √ 8. All other trajectories in the phase space have to connect singular points of types n + (2l − 1) , n and n + 2k , n for some l, k ∈ ℤ . Additionally, since the phase plane is divided into regions of -width by the three other types of trajectories (i), (ii) and (iii), we must have l ∈ {k, k + 1} . Hence, these trajectories correspond to solutions of system (9) which are described in case (iv) of Theorem 7.  Since 1 belongs to the eigenvector e 1 = (1, 0, 0) , we can estimate with suitable constants c 2 ,c 3 ∈ ℝ and as s, t → ∞ . This yields again as s, t → ∞ . In other words, the n--cone is indeed an asymptotic line. ◻ Theorem 7 follows from Lemmata 8-11.

Intersections of the n-˛-tectum and the n-˛-cone
We show here that there are no points of intersection of the tectum and the cone, if for example (n + ) > 4 + √ 8 and n ≥ 6 , cp. Theorem 14 and, for a more general result, see Theorem 15. In these cases the n--tectum turns out to be stable for symmetric perturbation, see Theorem 20. Recall that a point of intersection is characterized by the condition tan = √ n−1 and that the unstable manifold starting at the saddle point ( , ) = (0, 2 ) has direction (1, − n ) . Our aim is to construct a region R ⊂ (0, 2 ) × (0, 2 ) in the --plane such that R ⊂ { > n } and the vector field (12) points inward R whence solution trajectories of (12) remain trapped inside R and converge to the equilibrium ( n , n ) . To this end we let R denote the region enclosed by the three curves where a ≤ 0 (so R ⊂ { > n }) and b ∶= (1 − a) √ n−1 , such that (C3) and (C2) intersect at ( , ) = ( n , n ) (Fig. 6). Clearly, the vector field (12), when restricted to the curve (C1), points into R.
Finally, we write (C3) as a graph = h( ) ∶= arctan(a tan + b) and find h � ( ) = a cos 2 cos 2 and the condition h( ) ≤ g( ) requires that a = h � ( n ) ≥ g � ( n ) = −1 . By virtue of (11) any trajectory in R cannot leave R across curve (C3), if we can find some a ∈ [−1, 0] , such that for all , along (C3) we have In other words, defining the polynomial we have the following sufficient condition:

Lemma 12 Suppose there exists an a ∈ [−1, 0] such that
Then the trajectory of the n--tectum stays inside the region R. In particular, it does neither intersect nor touch the n--cone.
Remark For a < 0 , p n (a, z) is a polynomial of degree four in z. Using p n a, √ n−1 = 0 , it can be reduced to a polynomial of degree three.

Proof
Step 1: The choice of a.
Proof In order to minimize p n � √ n−1 , we put a ∶= − 1 2 ⋅ n+ −2 n+ −1 . Then set We have to show that r n (z) < 0 for all z ∈ (0, 1) in order to be able to apply Lemma 12. From our choice of a and n + > 4 + √ 8 , we already know this is fulfilled near the boundary.
Without loss of generality, let us assume n = 6 . Then r 6 (z) becomes We observe r 6 (0) < −1 < 0 , and the roots of the derivative of r these are real and distinct.
where q z ( ) is defined as the following function in : We have q �� z ( ) < 0 , so In the same way, it follows that by p and p replaced by , starts at ( 2 , 0) and ends at the equilibrium point ( n , n ) where n ∶= arctan √ p , p = n − 1.
Here we obtain Jacobi's equation simply as follows We have the following  The analogue result for not necessarily symmetric, stable entire C 2 -solutions of equation (1) was shown to hold under the stronger condition (n + ) < 4 + √ 2 n+ , i.e., (n + ) < 5.23... , cp. Dierkes [6]. We conjecture that Theorem 19 holds for all stable, entire solutions of equation (1). The assertion of Theorem 19 also holds for Lipschitz regular cones with vertex at the origin, see the paper [5].
We conclude with the following result on stability and minimizing properties of the n--tecti, if n + > 4 + (B) Suppose that one of the following conditions holds: In all these cases the n--tectum minimizes the energy E in suitable classes of nonnegative functions with bounded variation. Proof ad A): By Theorem 14 we have tan > √ n−1 > tan , so Theorem 17 implies stability of the n--tectum; hence, (i) follows. Furthermore, from Lemma 12 and

3
Theorem 14 we obtain the existence of a field (or "calibration") of n--tecti lying completely above the n--cone. This in turn leads to a function of least -gradient which is defined in the open set A ∶= {x ∈ ℝ n × ℝ + ;x n+1 > √ n−1 (x 2 1 + ... + x 2 n ) 1 2 } ; since the procedure is analogue to the construction device in [4] and [3] we skip over the details, referring to the papers [3,4]. It follows that every n--tectum minimizes (in a very general sense) the energy E with respect to variations inside the set A. For a more detailed discussion we refer to the papers [3,4] ad (iii): If the 6-1-tectum were minimizing, another minimizing solution v could be constructed which were not of class C 1 in the set {x ∈ ℝ n ;v(x) > 0} , which, however, contradicted known regularity results proved in [1]. Since the argument is similar to the proof of Theorem 2 in [4], we omit the details and refer the reader to [4]. ad B): Under the conditions (i),(ii) or (iii) the minimizing property of the n--tecti was implicitly established in the proof of Theorem 1 of [4], c.p. also [3].
The proof follows from the construction of the function f with least -gradient in the upper half space ℝ n × ℝ + which has the n--cones and the n--tecti as level surfaces. Lemmata 1 & 2 of [4] guarantee the existence of the field (or "calibration") which leads to the function f, while in [3] the minimizing properties of the corresponding level sets are proved. Since the n--tecti appear as level surfaces of the function f in the respective cases, they are minimizers of the energy integral E (⋅) . ◻ Funding Open Access funding enabled and organized by Projekt DEAL. The authors did not receive support from any organization for the submitted work.
Availability of Data All data generated or analyzed during this study are included in this published article [and its supplementary information files].

Conflict of interest
The authors declare that they have no conflict of interest.
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