NONSPURIOUS SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE PRESCRIBED MEAN CURVATURE SPACELIKE EQUATION IN A FRIEDMANN–LEMAÎTRE–ROBERTSON–WALKER SPACETIME

Abstract

We consider the differential and difference problems associated with the discrete approximation of radially symmetric spacelike solutions of the nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime

where $\mathcal{ℬ}$ is the unit ball in ${ℝ}^N$, $\text{div}$ denotes the divergence operator of ${ℝ}^N$, $\text{grad}u$ is the gradient of $u$, $|\cdot |$ denotes the Euclidean norm in ${ℝ}^N$, $f\in C^{\infty }(I)$, , $I$ is an open interval in $ℝ$, $f^{\prime }(u)≔f^{\prime }\circ u$ and $H:I\times [0,+\infty )\to ℝ$ is the prescribed mean curvature function. By using lower and upper solutions, we prove the existence of solutions of the corresponding differential and difference problems, and based on the ideas of a prior bound show the solutions of the discrete problem converge to the solutions of the continuous problem.