arXiv subjects
Publications and source records attributed to Érico Melo Silva.
We show that certain functions whose nodal sets lie near a fixed nondegenerate minimal hypersurface satisfy a strong min-max principle for the Allen--Cahn energy which is analogous to the strong min-max principle for non-degenerate minimal hypersurfaces first proved by Brian White.
We introduce an Allen-Cahn type functional, $\text{BE}_ε$, that defines an energy on separating hypersurfaces, $Y$, of closed Riemannian Manifolds. We establish $Γ$-convergence of $\text{BE}_ε$ to the area functional, and compute first and second variations of this functional under hypersurface pertrubations. We then compute an explicit expansion for the variational formula as $ε\to 0$. A key component of this proof is the invertibility of the linearized Allen-Cahn equation about a solution, on the space of functions vanishing on $Y$. We also relate the index and nullity of $\text{BE}_ε$ to the Allen-Cahn index and nullity of a corresponding solution vanishing on $Y$. We apply the second variation formula and index theorems to show that the family of $2p$-dihedrally symmetric solutions to Allen-Cahn on $S^1$ have index $2p - 1$ and nullity $1$.