arXiv subjects

Oscar Ivan Agudelo Rico

Publications and source records attributed to Oscar Ivan Agudelo Rico.

3 recordsLinked to original sources

arXiv2026

The Allen-Cahn equation and general minimal cones

We construct solutions to the Allen-Cahn equation $Δu+u-u^3=0$ in $\mathbb{R}^{N+1}$, with $N\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.

math.AP↗

arXiv2025

Fredholm properties of the jacobi Operator of minimal conical hypersurfaces

In this paper we study non-degeneracy properties of $Σ$ via the Jacobi operator $J_Σ:=Δ_Σ+|A_Σ|^2$ of a given minimal hypersurface $Σ$ asymptotic to a cone $C\subset \mathbb{R}^{N+1}$ of co-dimension one. Here $Δ_Σ$ is the Laplace Beltrami operator of $Σ$ and $|A_Σ|$ is the norm of the second fundamental form of $Σ$. We also construct a right inverse of $J_Σ$, that is, we prove that the Jacobi equation $J_Σϕ=f$ is solvable in $Σ$, at least under some suitable non-degeneracy assumptions about $Σ$ and about the asymptotic behavior of $f$ at infinity. We also discuss some examples where our results can be applied.

math.DG↗

arXiv2021

k-ended $O(m) times O(n)$ invariant solutions to the Allen-Cahn equation with infinite Morse index

In this work we study existence, asymptotic behaviour and stability properties of $O(m) times O(n)$ invariant solutions of the Allen-Cahn equation $Delta u+u(1-u^2)=0$ in $R^m times R^n$ with $m,n ge 2$, $m+n ge 8$. We exhibit four families of solutions whose nodal sets are smooth logarithmic corrections of the Lawson cone and with infinite Morse index. This work complements the study started by Pacard and Wei (about stable solutions) and by Agudelo, Kowalczyk and Rizzi (about 2 ended solutions).

math.AP↗