The mean curvature equation on semidirect products $\mathbb{R}^2\rtimes_A\mathbb{R}$: Height estimates and Scherk-like graphs
On the ambient space of a Lie group with a left invariant metric that is isometric and isomorphic to a semidirect product $\mathbb{R}^2\rtimes_A\mathbb{R}$, we consider a domain $Ω\subseteq \mathbb{R}^2\rtimes_A\{0\}$ and vertical $π$-graphs over $Ω$ and study the partial differential equation a function $u:Ω\rightarrow \mathbb{R}$ must satisfy in order to have prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a $π-$graph has non-negative mean curvature, then it satisfy some uniform height estimates that depend on $Ω$ and on a parameter $α$, given a priori. When $\text{trace}(A) > 0$, these estimates imply that the oscillation of a minimal graph assuming the same constant value $n$ along the boundary tends to zero when $n\rightarrow + \infty$ and goes to $+ \infty$ if $n\rightarrow - \infty$. Furthermore, we use some of the estimates, allied with techniques from Killing graphs, to prove the existence of minimal $π-$graphs assuming the value $0$ along a piecewise smooth curve $γ$ with endpoints $p_1,\,p_2$ and having as boundary $γ\cup (\{p_1\}\times[0,\,+\infty))\cup(\{p_2\}\times[0,\,+\infty))$.