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Fumihiko Onoue

Publications and source records attributed to Fumihiko Onoue.

9 recordsLinked to original sources
arXiv2026

Optimal sets for a geometric oscillation energy

We investigate the nonlocal energy corresponding to the $p$-oscillation of the unit normal vector for hypersurfaces, or the unit tangent vector for curves. The energy satisfies geometric inequalities with optimal constants $c(n,p)$ and $C(n,p)$ which are determined by a variational problem over the probability measures on the sphere. The extremal measures for such problem depend critically on the value of $p$. We prove existence of optimal sets for this energy under perimeter and volume constraint, and characterize their shape.

math.AP↗
arXiv2024

A notion of $s$-fractional mass for $1$-currents in higher codimension

In this paper we propose a notion of $s$-fractional mass for $1$-currents in $\R^d$. Such a notion generalizes the notion of $s$-fractional perimeters for sets in the plane to higher codimension one-dimensional singularities. Remarkably, the limit as $s\to 1$ of the $s$-fractional mass gives back the classical notion of length for regular enough curves in $\R^d$. We prove a lower semi-continuity and compactness result for sequences of $1$-currents with uniformly bounded fractional mass and support. Moreover, we prove the density of weighted polygonal, closed and compact oriented curves in the class of divergence-free 1-currents with compact support and finite fractional mass. Finally, we discuss some possible applications of our notion of fractional mass to build up purely geometrical approaches to the variational modeling of dislocation lines in crystals and to vortex filaments in superconductivity.

math.FA↗
arXiv2023

On the shape of hypersurfaces with boundary which have zero fractional mean curvature

We consider hypersurfaces with boundary in $\mathbb{R}^N$ that are the critical points of the fractional area introduced by Paroni, Podio-Guidugli, and Seguin in [R. Paroni, P. Podio-Guidugli, B. Seguin, 2018]. In particular, we study the shape of such hypersurfaces in several simple settings. First, we show that the critical points whose boundary is an $(N-2)$-sphere coincide with $(N-1)$-balls. Second, we show that the critical points whose boundary is the union of two parallel $(N-2)$-spheres do not coincide with two parallel $(N-1)$-balls. Moreover, the interior of the critical points does not intersect the boundary of the convex hull of the two $(N-2)$-spheres, while it can happen in the situation considered by Dipierro, Onoue, and Valdinoci in [S. Dipierro, F. Onoue, E. Valdinoci, 2022]. We also obtain a quantitative bound which may tell us how different the critical points are from the two $(N-1)$-balls. Finally, in the same setting as in the second case, we show that, if the two parallel boundaries are far away from each other, then the critical points are disconnected and, if the two parallel boundaries are close to each other, then the boundaries are in the same connected component of the critical points when $N \geq 3$. Moreover, by computing the fractional mean curvature of a cone with the same boundaries as those of the critical points, we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.

math.AP↗
arXiv2022

Local Hölder regularity of minimizers for nonlocal variational problems

We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) $β$-Hölder continuous for some $β\in (1-s,\,1]$, where $s \in (0,1)$ is a parameter related to the nonlocal operator, we prove that the solution is also $β$-Hölder continuous.

math.AP↗
arXiv2021

Existence of minimizers for a generalized liquid drop model with fractional perimeter

We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. Secondly, we show the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of minimizers converges to the Euclidean ball up to translations if the kernel of the Riesz potential decays sufficiently fast.

math.AP↗
arXiv2020

A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and the boundedness of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

math.AP↗
arXiv2020

(Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property

We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension $2$ we provide a quantitative bound on the stickiness property exhibited by the minimizers. Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.

math.AP↗
arXiv2019

Nonexistence of Minimizers for a Nonlocal Perimeter Functional with a Riesz and a Background Potential

We consider the nonexistence of minimizers for the energy containing a nonlocal perimeter with a general kernel $K$, a Riesz potential, and a background potential in $\mathbb{R}^N$ with $N\geq2$ under the volume constraint. We show that the energy has no minimizer for a sufficiently large mass under suitable assumptions on $K$. The proof is based on the partition of a minimizer and the comparison of the sum of the energy for each part with the energy for the original configuration. This strategy is shown in [R.A. Frank, R. Killip, P.T. Nam, 2016] and [D.A. La Manna, 2018]

math.AP↗