The \(n\)-dimensional analogue of a variational problem of Euler (Q6583561)

Let \(M\subset \mathbb{R}^{n+1}\) be a smooth hypersurface, and \(f\) a \(C^2\) function from \(\mathbb{R}^{n+1}\) into \(\mathbb{R}\). The authors consider the weighted area-functional \[\mathcal{E}_f(M):=\int_M f(x) d\mathcal{H}_n(x)\]\Nand derive first and second variation formulae for the integral \(\mathcal{E}_f(M)\). Note that \(M\) is stationary for \(f\) if the relation \N\[\Nf H =\sum_{i=1}^{n+1} f_{x_i} \nu_i\N\]\Nholds, where \(\nu_i\) are the components of the normal vector and \(H\) is the mean curvature. A stationary hypersurface is said to be stable if the second derivative of the functional \(\mathcal{E}_f(M)\) is non-negative. \N\NThe authors study the case \(f(x)=|x]^\alpha\) and prove that a stationary surface\N\N-- cannot be compact if \(\alpha > -n\), \N\N-- is necessarily a sphere centered at the origin if \(\alpha=-n\) (if we suppose \(M\) to be compact),\N\N-- and in the case \(\alpha < -n\) the closure of \(M\) contains the origin.\N\NThey also study the case of minimal cones over products of spheres with vertex at the origin: they are stationary for \(\mathcal{E}_f\) and the authors investigate their stability. In addition, complementing non-existence results for nontrivial stable cones are formulated and proved.\N\NThe authors conclude by proving minimizing properties of some symmetric cones under suitable conditions on \(\alpha\) and the dimension \(n\). Also planes through the origin and spheres centered at the origin are minimizers in certain cases. All results are proved by using some kind of calibration theory.