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In this paper, we consider the radially symmetric spacelike solutions of a nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime. By using a conformal change of variable, this problem can be translated an equivalent problem in the Minkowski spacetime. By using the lower and upper solution method, fixed point, a priori bounds and topological degree method, we obtain the existence, nonexistence and multiplicity of radially symmetric spacelike solutions.

Abstract Let I ⊂ R $I\subset\mathbb{R}$ be an open interval with 0 ∈ I $0\in I$ , and let g ∈ C 1 ( I , ( 0 , + ∞ ) ) $g\in C^{1}(I, (0,+\infty))$ . Let N ∈ N $N\in\mathbb{N}$ be an integer with N ≥ 4 $N\geq4$ , [ 2 , N − 1 ] Z : = { 2 , 3 , … , N − 1 } $[2, N-1]_{\mathbb{Z}}:=\{2, 3,\ldots,N-1\}$ . We are concerned with the existence of solutions for the discrete Neumann problem { ∇ ( k n − 1 △ v k 1 − ( △ v k ) 2 ) = n k n − 1 [ − g ′ ( ψ − 1 ( v k ) ) 1 − ( △ v k ) 2 + g ( ψ − 1 ( v k ) ) H ( ψ − 1 ( v k ) , k ) ] , k ∈ [ 2 , N − 1 ] Z , Δ v 1 = 0 = Δ v N − 1 $$\textstyle\begin{cases} \nabla(k^{n-1}\frac{\triangle v_{k}}{\sqrt{1-(\triangle v_{k})^{2}}} )=nk^{n-1}[-\frac{ g'(\psi^{-1}(v_{k}))}{\sqrt{1-(\triangle v_{k})^{2}}}+g(\psi^{-1}(v_{k}))H(\psi^{-1}(v_{k}),k)],\quad k\in[2, N-1]_{\mathbb{Z}},\\ \Delta v_{1}=0=\Delta v_{N-1} \end{cases} $$ which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where ψ ( s ) : = ∫ 0 s d t g ( t ) $\psi(s):=\int_{0}^{s}\frac{dt}{g(t)}$ , ψ − 1 $\psi ^{-1}$ is the inverse function of ψ, and H : R × [ 2 , N − 1 ] Z → R $H:\mathbb{R}\times[2, N-1]_{\mathbb{Z}}\to\mathbb{R}$ is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann-Lemaître-Robertson-Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space. [ABSTRACT FROM AUTHOR]

We identify a family of Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes such that the radially symmetric prescribed curvature problem with Neumann boundary condition is solvable on a ball of small radius. Such family contains some examples of interest in Cosmology. [ABSTRACT FROM AUTHOR]

The kinematical nature of the spectral shift in different types of spacetime, cosmological redshift and frequency shift in Schwarzschild spacetime being the most prominent representatives, has been a subject of recent interest. We demonstrate that the kinematical nature of the spectral shift is an inherent feature of the particular type of spacetimes. The general solution is found: necessary and sufficient conditions for the spectral shift arising from a kinematical origin are formulated. The status of the relative speed is discussed. [ABSTRACT FROM AUTHOR]