Year: 2026
Type: preprint
Abstract: Given a smooth domain $D\subset\mathbb S^{N-1}$, $N\geq 3$, and a positive constant $H$, we study radial graphs over $D$ with constant mean curvature $H$ which meet orthogonally the boundary of the cone spanned by $D$. The corresponding problem is given by a quasilinear elliptic equation on $D$ with homogeneous Neumann boundary condition. We investigate the structure of the solution set under smooth perturbations of the domain which are parametrized by an open set of $C^{3,α}$-diffeomorphisms of the sphere. After pulling the equation back to a fixed reference domain, we apply a classical transversality theorem and prove that, in the class of domains for which $N-1$ does not belong to the Neumann spectrum of the Laplace-Beltrami operator, there is a residual set for which every solution is nondegenerate. A consequence of our result is that, for every such domain and every $R>0$, the set of solutions whose $C^{2,α}$-norm is bounded by $R$ is finite.
Source: arXiv (Cornell University)
Authors: Iacopetti, Alessandro, Pacella, Filomena
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221775872