arXiv subjects

Reinaldo Resende

Publications and source records attributed to Reinaldo Resende.

8 recordsLinked to original sources

arXiv2025

Rigidity of critical points of hydrophobic capillary functionals

We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^\circ$ and $120^\circ$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^\circ$ and $180^\circ$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.

math.AP↗

arXiv2024

On the regularity of area minimizing currents at boundaries with arbitrary multiplicity

In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, taking a boundary $\Gamma$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $\Gamma$ and $\Sigma$ are $C^{3, \kappa}$. We prove a sharp generalization of Allard's boundary regularity theorem to a higher multiplicity setting. Precisely, we prove that the set of density $Q/2$ singular boundary points of $T$ is $\mathcal{H}^{m-3}$-rectifiable. As a consequence, we show that the entire boundary regular set, without any assumptions on the density, is open and dense in $\Gamma$ which is also dimensionally sharp. Moreover, we prove that if $p \in \Gamma$ admits an open neighborhood in $\Gamma$ consisting of density $Q/2$ points with a tangent cone supported in a half $m$-plane, then $p$ is regular. Furthermore, we show that if the convex barrier condition is satisfied-namely, if $\Gamma$ is a closed manifold that lies at the boundary of a uniformly convex set and $\Sigma = \mathbb{R}^{m+n}$-then the entire boundary singular set is $\mathcal{H}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $\Gamma$ that enable us to provide further information about the singular boundary set.

math.AP↗

arXiv2024

Interior regularity of area minimizing currents within a $C^{2,\alpha}$-submanifold

Given an area-minimizing integral $m$-current in $\Sigma$, we prove that the Hausdorff dimension of the interior singular set of $T$ cannot exceed $m-2$, provided that $\Sigma$ is an embedded $(m+\bar{n})$-submanifold of $\mathbb{R}^{m+n}$ of class $C^{2,\alpha}$, where $\alpha>0$. This result establishes the complete counterpart, in the arbitrary codimension setting, of the interior regularity theory for area-minimizing integral hypercurrents within a Riemannian manifold of class $C^{2,\alpha}$.

math.AP↗

arXiv2023

Lipschitz approximation for general almost area minimizing currents

We prove a Lipschitz approximation with superlinear error terms for integral currents $\omega$-minimizing the area functional, where $\omega$ is a modulus of continuity satisfying a Dini condition. We also present an almost monotonicity result for the density of these general almost area minimzing currents.

math.AP↗

arXiv2023

Boundary regularity for anisotropic minimal Lipschitz graphs

We prove that $m$-dimensional Lipschitz graphs in any codimension with $C^{1,\alpha}$ boundary and anisotropic mean curvature bounded in $L^p$, $p > m$, are regular at every boundary point with density bounded above by $1/2 +\sigma$, provided the anisotropic energy satisfies the uniform scalar atomic condition.

math.AP↗

arXiv2022

Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity

In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of $(C_0, \alpha_0, r_0)$-almost area minimizing currents of arbitrary dimension $m$ and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current $T$ is an open dense set in the boundary.

math.AP↗

arXiv2022

Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds

We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation (with two equations) involving a triple-well (triphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $\Gamma$-converges to the weighted multi-perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to the ACH system are nondegenerate.

math.AP↗

arXiv2020

On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry

For a complete Riemannian manifold with bounded geometry, we prove the existence of isoperimetric clusters and also the compactness theorem for sequence of clusters in a larger space obtained by adding finitely many limit manifolds at infinity. Moreover, we show that isoperimetric clusters are bounded. We introduce and prove the Holder continuity of the multi-isoperimetric profile which has been explored by Emanuel Milman and Joe Neeman with a Gaussian-weighted notion of perimeter. We yield a proof of classical existence theorem, e.g. in space forms, for isoperimetric cluster using the results presented here. The results in this work generalize previous works of Stefano Nardulli, Andrea Mondino, Frank Morgan, Matteo Galli and Manuel Ritor\'e from the classical Riemannian and sub-Riemannian isoperimetric problem to the context of Riemannian isoperimetric clusters and also Frank Morgan and Francesco Maggi works on the clusters theory in the Euclidean setting.

math.DG↗