LOCAL BIFURCATION STRUCTURE AND STABILITY OF THE MEAN CURVATURE EQUATION IN THE STATIC SPACETIME

. We consider the curvature equation in the static spacetime,


Introduction and main results
We consider a domain Ω ⊆ R N , where N is greater than or equal to 1. Let f be a smooth positive function on Ω.Consider the N + 1-dimensional product manifold M = I × Ω equipped with the Lorentzian metric g = −f 2 (x) dt 2 + dx 2 .
In [22,Lemma 12.37], it was established that M is static with respect to ∂ t /f .For each u ∈ C 2 (Ω), let M = {(x, u) : x ∈ Ω, u ∈ C 2 (Ω)}.A spacetime M is termed static in relation to an observer field Q if Q is irrotational and if there exists a smooth positive function such that f Q is a Killing vector field.Then, (M, g) = U represents an N -dimensional hypersurface in M at time t, which can be depicted by the graph of t = u.U is referred to as spacelike if |∇u| < 1/f in Ω (see [20]).We define U as being weakly spacelike if |∇u| ≤ 1/f , i.e., if it is in Ω.Given the mean curvature H for a spacelike graph U , Problem (1.1) has implications for classical relativity [4] and cosmology research [5,19,21].
For the case in which f is constantly equal to 1, Calabi [9] explored the properties of maximal surfaces and demonstrated that when N ≤ 4, equation (1.2) allows only linear solutions.Cheng and Yau [10] further investigated maximal surfaces, extending Calabi's findings to all dimensions, and proposed the Bernstein theorem.For cases in which f is constantly equal to 1, Treibergs [24] provided significant results for entire surfaces with a constant mean curvature.For cases in which f equals 1, Bartnik and Simon [4] considered the Dirichlet problem for equation (1.2) with surfaces of bounded mean curvature.
The authors of [6,11] used critical point theory and topological degree arguments to explore the nonexistence, existence, and multiplicity of positive solutions for f ≡ 1 in bounded domains.In [13], the authors investigated the nonexistence, existence, and multiplicity of positive radial solutions of equation (1.2) with N H = −λf (x, s) on the unit ball via the bifurcation method.This work was later extended to general domains in [14,16].The author in [18] studied the existence and uniqueness of classical solutions, the multiplicity of strong solutions, and the symmetry of positive solutions.The global structure of the positive solutions for this problem was also delineated.For more research results on the mean curvature equation, see references [7,8,17,15] and their cited literature.
In [1], the stability of hypersurfaces with a constant mean curvature was studied through the calculus of variations.The relationship between stability and constant mean curvature was presented under the condition that the hypersurface is compact.In [2], the stability of hypersurfaces with constant mean curvature in Riemannian manifolds was studied.Barros,Brasil,and Caminha [3] investigated stability issues concerning the generalized Robertson-Walker spacetime.In this work, we investigate the local bifurcation structure and stability of the mean curvature equation in static the spacetime.
We consider the following 0-Dirichlet problem involving the mean curvature operator in Minkowski space: Here, λ is a nonnegative parameter representing the strength of the mean curvature function, the real-valued function H gives the mean curvature, Ω is a C 2,α bounded domain in R N with N ≥ 1 for some α > 0, and , where d is the diameter of Ω.
Using the equation we can derive that From [18], we have div This equation is equivalent to (1.2).Next, we present the main theorem.
Here, z : ) is exactly zero on Ω, then we obtain , where H uuu (x, 0) denotes the third derivative of H with respect to its second variable at 0.
By Theorem 1.1, we can deduce the following stability result.
This article is organized as follows.Section 2 discusses the local bifurcation structure of the solution set of equation (1.1).Section 3 presents the stability results near the bifurcation point.

Local bifurcation structure
In this section, we provide the proof of Theorem 1.1.Consider the set X defined as X = {u ∈ C 1 (Ω) : u = 0 on ∂Ω} with the norm ∥u∥ := ∥f (x)∇u∥ ∞ .Let φ 1 be a positive eigenfunction corresponding to λ 1 with ∥φ 1 ∥ = 1.Let X 0 be a closed subspace of X such that where X 1 = span{φ 1 }.By applying the Hahn-Banach theorem, we can find a linear continuous functional l ∈ X * satisfying Consider the function defined by Since H 0 ∈ (0, +∞) and H(x, 0) = 0 holds for any x ∈ Ω, we have According to [18], we have lim This holds because H is a function with third-order continuous derivatives with respect to its second variable, and F is C 3 with respect to u in some small neighborhood V ⊂ X of 0. By calculation, we obtain where φ ∈ X .The function φ 1 is a positive eigenfunction corresponding to the principal eigenvalue λ 1 of the linearized problem associated with equation (1.1).Specifically, φ 1 is a solution of Since φ 1 is a nontrivial solution, it follows that φ 1 ̸ = 0 and φ 2 1 ≥ 0. Hence, the integral Ω φ 2 1 dx > 0. Thus, the kernel space is The codimension of the image space is Clearly, F is C 1 with respect to λ, and F λu exists and remains continuous in a small neighborhood of (λ 1 /H 0 , 0).From the calculations, we obtain and Combining this with H 0 > 0, we obtain Thus, Ω f H 0 φ 2 1 dx ̸ = 0.This leads to the conclusion that (2.3) By applying [16], we deduce that all the solutions near (λ 1 /H 0 , 0) for problem (1.1) can be expressed as (λ(s), sφ 1 + sz(s)), where s belongs to the interval (−δ, δ) for some positive value of δ, and that they satisfy the conditions λ(0) = λ 1 /H 0 and z(0) = 0.
We rescale φ 1 so that Then, we define the linear functional and Furthermore, by employing formula (4.5) derived in [23], we can deduce that (2.7) From (2.6) and (2.7), we obtain .

Stability properties
In this section, we provide the formal stability results near the bifurcation point.The stability properties are obtained via the exchange of stability theorem presented in [12], which is our fundamental tool.Theorem 3.1 (Crandall-Rabinowitz).Let X and Y be real Banach spaces, and let K : X → Y be two bounded linear operators.Assume that simple eigenvalue of operator T and a K-simple eigenvalue of T , then there locally exists a curve (λ(s), Moreover, if F (λ, u) = 0 with u ̸ = 0 and (λ, u) near (λ * , 0), then (λ, u) = (λ(s), u(s)) for some s ̸ = 0.
By Theorem 3.1, we obtain the following formula, which is convenient to be used.
Proposition 3.2.Under the assumption of Theorem 3.1, we have that where l ∈ X * satisfies N (l) = R(F u (λ 1 /H 0 , 0)), with X * being the dual space of X.In particular, if and Since β = 0 is an F λu (λ 1 /H 0 , 0)-simple eigenvalue of operator F u (λ 1 /H 0 , 0), we have By taking l on both sides of equation (3.1) and using the fact that we obtain that It can be deduced that l(Kφ 1 ) ̸ = 0; consequently, , which yields the desired formula.□ Before providing the stability result (in the linearized sense) for (1.1) near the bifurcation point, we review the concept of stability.The operator equation F (λ, x) = 0 represents the equilibrium form of the evolution equation dx dt = F (λ, x).
Suppose that F (λ 0 , x 0 ) = 0.If all the eigenvalues of F x (λ 0 , x 0 ) are negative, then x 0 is called an asymptotically linearly stable solution of (3.2).On the other hand, if a positive eigenvalue of F x (λ 0 , x 0 ) exists, then x 0 is called an unstable solution of (3.2).