Compact hypersurfaces in an ellipsoid with prescribed mean curvature (Q1192397)

Given a function \(H\), the author considers the problem of constructing a closed hypersurface of a given topological type whose mean curvature equals \(H\). More precisely, he treats the following special case. For \(a,b > 0\) let \[ M_{n+1}:=\left\{(x_ 1,x_ 2,\dots,x_{n+1},x_{n+2})\in\mathbb{R}^{n+2} \left|\;{x^ 2_ 1+x^ 2_ 2+\dots+x^ 2_{n+1}\over a^ 2}+{x^ 2_{n+2}\over b^ 2}=1\right\}\right. \] be an \((n+1)\)-dimensional ellipsoid in \(\mathbb{R}^{n+2}\), and let \(H\) be a smooth function on \(M_{n+1}\sim\{(0,\dots,0,\pm b)\}\). Assuming that \(H\) satisfies certain conditions the author proves that there is a compact hypersurface in \(M_{n+1}\), homeomorphic to \(S^ n\), whose mean curvature is given by \(H\). He also considers the three-dimensional ellipsoid \[ Y=\left\{(x_ 1,x_ 2,x_ 3,x_ 4)\in\mathbb{R}^ 4\;\left|\;{x^ 2_ 1+x^ 2_ 2\over a^ 2} + {x^ 2_ 3+x^ 2_ 4\over b^ 2}=1\right\}\right. \] and proves the existence of a torus of prescribed mean curvature \(H\) in \(Y\) provided \(H\) satisfies certain conditions. The method consists in deriving suitable a priori estimates for the \(C^ 1\)-norm of solutions to a family of quasilinear elliptic equations so that the continuity method can be applied and yields existence.