Variational mean curvatures for general ${\mathrm{C}}^{1,\alpha }$ domains

The notion of variational mean curvature (VMC) arises from Massari’s variational approach to hypersurfaces of prescribed mean curvature and expresses a minimizing property for Massari’s functional. Here, we complement the theory of VMCs with a new sharp relation between hypersurface regularity and VMC integrability. In fact, our main result asserts that every bounded open set in ${\mathbb{R}}^n$ with ${\mathrm{C}}^{1,\alpha }$ boundary, $0<\alpha <1$, has a global VMC in ${\mathrm{L}}^p({\mathbb{R}}^n)$ for every $p<\frac{1+\alpha }{1-\alpha }$. In addition, we underpin sharpness of this conclusion by constructing examples of bounded open sets in ${\mathbb{R}}^n$ with ${\mathrm{C}}^{1,\alpha }$ boundary which do not admit a VMC in ${\mathrm{L}}^p({\mathbb{R}}^n)$ for any $p>\frac{1+\alpha }{1-\alpha }$. Our proofs are based on Barozzi’s level-set construction of global VMCs and determine integrability of these VMCs by geometric investigation of their superlevel sets and of related unions of balls.