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Yoshihiro Tonegawa

Publications and source records attributed to Yoshihiro Tonegawa.

At least 19 recordsLinked to original sources
arXiv2026

Geometric flows of branched transportation networks

The branched transport problem is a nonconvex and nonsmooth variational optimization problem on normal $1$-currents in $\mathbb{R}^n$ with prescribed boundary. The optimality is with respect to some non-decreasing, lower semicontinuous, and subadditive function $τ:\mathbb{R}_+\to\mathbb{R}_+$ with $τ(0)=0$ describing the cost $τ(m)$ to move an amount of mass $m$ per unit distance. The subadditivity leads to complicated, hierarchically ramified patterns in the support of (suboptimal) solutions. These network-like sets appear to have regularity properties similar to those of the singular surfaces arising in the so-called Brakke flow, a weak generalization of the mean curvature flow. We construct a geometric flow of transportation networks, which correspond to normal real $1$-rectifiable currents, by modifying Brakke's variational approximation scheme. We prove the existence of a limit that is Hölder continuous with respect to the flat norm and whose branched transport cost decreases along the geometric evolution. We further analyze a closely related geometric flow approximated by $1$-varifolds, whose weight measures model the branched transport cost, and establish its $1$-rectifiability together with a motion law analogous to Brakke's inequality.

math.AP↗
arXiv2026

Existence of curvature flow with forcing in a critical Sobolev space

Suppose that a closed 1-rectifiable set $Γ_0\subset\mathbb R^2$ of finite 1-dimensional Hausdorff measure and a vector field $u$ in a dimensionally critical Sobolev space are given. It is proved that, starting from $Γ_0$, there exists a non-trivial flow of curves with the normal velocity given by the sum of the curvature and the given vector field $u$. The motion law is satisfied in the sense of Brakke and the flow exists through singularities.

math.AP↗
arXiv2026

Ferronematics: integrality of the limiting interface in the strong coupling regime

We consider a vectorial energy functional proposed in the physical literature as a simplified model for thin films of ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a nematic liquid crystals host. The model features two order parameters: a reduced $\Q$-tensor field $\Q$, describing the nematic liquid crystal component, and a vector field $\M$, accounting for the average polarisation vector field generated by the included particles. The energy contains a coupling term promoting alignment between $\Q$ and $\M$. It has been shown in~\cite{CDS1,CDS2} that, as a small parameter $\eps$ tends to zero, the energy of critical pairs $(\Q_\eps,\,\M_\eps)$ concentrates on \emph{distinct} singular sets: a finite set of points for the $\Q_\eps$-component and the support of a $\H^1$-rectifiable varifold for the $\M_\eps$-component, with first variation supported on the singular set for the $\Q_\eps$-component. We show in this paper that, if the coupling constant is above a certain (explicit) threshold, then, up to rescaling its density by a material constant, such a limiting varifold has integer multiplicity.

math.AP↗
arXiv2025

The epsilon-regularity theorem for Brakke flows near triple junctions

We establish the $\varepsilon$-regularity theorem for $k$-dimensional, possibly forced, Brakke flows near a static, multiplicity-one triple junction. This result provides the parabolic analogue to L. Simon's foundational work on the singular set of stationary varifolds and confirms that the regular structure of triple junctions persists under weak mean curvature flow. The regularity holds provided the flow satisfies a mild structural assumption on its 1-dimensional slices taken orthogonal to the junction's $(k-1)$-dimensional spine, which prohibits certain topological degeneracies. We prove that this assumption is automatically satisfied by two fundamental classes of flows where such singularities are expected: codimension-one multi-phase flows, such as the canonical $\mathrm{BV}$-Brakke flows constructed by the authors, and flows of arbitrary codimension with the structure of a mod 3 integral current, which arise from Ilmanen's elliptic regularization. For such flows, therefore, the Simon type regularity holds unconditionally.

math.AP↗
arXiv2025

Gradient flow of phase transitions with fixed contact angle

We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary non-concentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary.

math.AP↗
arXiv2024

Dynamical instability of minimal surfaces at flat singular points

Suppose that a countably $n$-rectifiable set $Γ_0$ is the support of a multiplicity-one stationary varifold in $\mathbb{R}^{n+1}$ with a point admitting a flat tangent plane $T$ of density $Q \geq 2$. We prove that, under a suitable assumption on the decay rate of the blow-ups of $Γ_0$ towards $T$, there exists a non-constant Brakke flow starting with $Γ_0$. This shows non-uniqueness of Brakke flow under these conditions, and suggests that the stability of a stationary varifold with respect to mean curvature flow may be used to exclude the presence of flat singularities.

math.AP↗
arXiv2024

End-time regularity theorem for Brakke flows

For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to $1$, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White's local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class.

math.AP↗
arXiv2022

On the existence of canonical multi-phase Brakke flows

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. As a consequence of the results recently established by Fischer et al. in arXiv:2003.05478, under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows.

math.AP↗
arXiv2022

Brakke's formulation of velocity and the second order regularity property

Suppose that a family of $k$-dimensional surfaces in $\mathbb R^n$ evolves by the motion law of $v=h+u^\perp$ in the sense of Brakke's formulation of velocity, where $v$ is the normal velocity vector, $h$ is the generalized mean curvature vector and $u^\perp$ is the normal projection of a given vector field $u$ in a dimensionally sharp integrability class. When the flow is locally close to a time-independent $k$-dimensional plane in a weak sense of measure in space-time, it is represented as a graph of a $C^{1,α}$ function over the plane. On the other hand, it is not known if the graph satisfies the PDE of $v=h+u^\perp$ pointwise in general. For this problem, when $k=n-1$ and under the additional assumption that the distributional time derivative of the graph is a signed Radon measure, it is proved that the graph satisfies the PDE pointwise. An application to a short-time existence theorem for a surface evolution problem is given.

math.AP↗
arXiv2021

An existence theorem for Brakke flow with fixed boundary conditions

Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-dimensional domain, and suppose that the set has finite $n$-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As $t \uparrow \infty$, the flow sequentially converges to non-trivial solutions of Plateau's problem in the setting of stationary varifolds.

math.AP↗
arXiv2020

Existence and regularity theorems of one-dimensional Brakke flows

Given a closed countably $1$-rectifiable set in $\mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class $W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.

math.DG↗
arXiv2020

A diffused interface with the advection term in a Sobolev space

We study the asymptotic limit of diffused surface energy in the van der Waals--Cahn--Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min-max method.

math.AP↗
arXiv2016

A fixed contact angle condition for varifolds

We define a generalized fixed contact angle condition for $n$-varifold and establish a boundary monotonicity formula. The results are natural generalizations of those for the Neumann boundary condition considered by Grüter-Jost.

math.AP↗
arXiv2016

Convergence of the Allen-Cahn equation with Neumann boundary conditions

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.

math.AP↗
arXiv2016

On the mean curvature flow of grain boundaries

Suppose that $Γ_0\subset\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\mathbb R^{n+1}\setminus Γ_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $Γ_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from $Γ_0$. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curvature flow.

math.DG↗
arXiv2016

The blow up method for Brakke flows: networks near triple junctions

We introduce a parabolic blow-up method to study the asymptotic behavior of an integral Brakke flow of planar networks (i.e. a 1-dimensional integral Brakke flow in a two dimensional region) weakly close in a space-time region to a static multiplicity 1 triple junction $J$. We show that such a network flow is regular in a smaller space-time region, in the sense that it consists of three curves coming smoothly together at a single point at 120 degree angles, staying smoothly close to $J$ and moving smoothly. Using this result and White's stratification theorem, we deduce that whenever an integral Brakke flow of networks in a space-time region ${\mathcal R}$ has no static tangent flow with density $\geq2$, there exists a closed subset $Σ\subset {\mathcal R}$ of parabolic Hausdorff dimension at most 1 such that the flow is classical in ${\mathcal R} \setminus Σ$, i.e. near every point in ${\mathcal R} \setminus Σ$, the flow, if non-empty, consists of either an embedded curve moving smoothly or three embedded curves meeting smoothly at a single point at 120 degree angles and moving smoothly. In particular, such a flow is classical at all times except for a closed set of times of ordinary Hausdorff dimension at most $\frac{1}{2}$.

math.AP↗
arXiv2014

Existence and regularity of mean curvature flow with transport term in higher dimensions

Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are $C^1$ for a short time and, even after some singularities occur, almost everywhere $C^1$ away from higher multiplicity region.

math.DG↗