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Tatsuya Miura

Publications and source records attributed to Tatsuya Miura.

32 records · Page 2Linked to original sources
arXiv2024

Delta-convex structure of the singular set of distance functions

For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta-convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

math.AP↗
arXiv2023

Li-Yau type inequality for curves in any codimension

For immersed curves in Euclidean space of any codimension we establish a Li--Yau type inequality that gives a lower bound of the (normalized) bending energy in terms of multiplicity. The obtained inequality is optimal for any codimension and any multiplicity except for the case of planar closed curves with odd multiplicity; in this remaining case we discover a hidden algebraic obstruction and indeed prove an exhaustive non-optimality result. The proof is mainly variational and involves Langer--Singer's classification of elasticae and André's algebraic-independence theorem for certain hypergeometric functions. We also discuss applications to elastic flows, networks, and knots.

math.DG↗
arXiv2023

Pinned planar p-elasticae

Building on our previous work, we classify all planar $p$-elasticae under the pinned boundary condition, and then obtain uniqueness and geometric properties of global minimizers. As an application we establish a Li--Yau type inequality for the $p$-bending energy, and in particular discover a unique exponent $p \simeq 1.5728$ for full optimality. We also prove existence of minimal $p$-elastic networks, extending a recent result of Dall'Acqua--Novaga--Pluda.

math.DG↗
arXiv2023

Optimal thresholds for preserving embeddedness of elastic flows

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational characterization that the thresholding shape is a minimizer of the bending energy (normalized by length) among all nonembedded planar closed curves of unit rotation number. It turns out that a minimizer is uniquely given by a nonclassical shape, which we call ``elastic two-teardrop''.

math.AP↗
arXiv2023

A diameter bound for compact surfaces and the Plateau-Douglas problem

In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The obtained estimate is expected to be optimal for minimal surfaces in the sense that optimality follows if the Topping conjecture holds true. Our result directly implies an explicit nonexistence criterion in the classical Plateau-Douglas problem. We exhibit examples of boundary contours to ensure that our criterion is of different type from classical criteria based on the maximum principle and White's criterion based on a density estimate.

math.DG↗
arXiv2021

Geometric inequalities involving mean curvature for closed surfaces

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we also address Topping's conjecture relating diameter and mean curvature for connected closed surfaces. We prove this conjecture in the class of simply-connected axisymmetric surfaces, and moreover obtain a sharp remainder term which ensures the first evidence that optimal shapes are necessarily straight even without convexity.

math.DG↗
arXiv2021

Sharp boundary $\varepsilon$-regularity of optimal transport maps

In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,α}$-boundary. Our main result asserts sharp $C^{1,α}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,α}$. Our method is completely variational and builds on the recently developed interior regularity theory.

math.AP↗
arXiv2020

Polar tangential angles and free elasticae

In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae.

math.DG↗
arXiv2019

On the isoperimetric inequality and surface diffusion flow for multiply winding curves

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $H^2$-close to a multiply covered circle and sufficiently rotationally symmetric.

math.DG↗
arXiv2018

Elastic curves and phase transitions

This paper is devoted to classical variational problems for planar elastic curves of clamped endpoints, so-called Euler's elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain several new results concerning properties of least energy solutions. In particular we reach a first uniqueness result that assumes no symmetry. As a key ingredient we develop a foundational singular perturbation theory for the modified total squared curvature energy. It turns out that our energy has almost the same variational structure as a phase transition energy of Modica-Mortola type at the level of a first order singular limit.

math.CA↗
arXiv2017

Overhanging of membranes and filaments adhering to periodic graph substrates

This paper mathematically studies membranes and filaments adhering to periodic patterned substrates in a one-dimensional model. The problem is formulated by the minimizing problem of an elastic energy with a contact potential on graph substrates. Global minimizers (ground states) are mainly considered in view of their graph representations. Our main results exhibit sufficient conditions for the graph representation and examples of situations where any global minimizer must overhang.

math-ph↗
arXiv2016

A characterization of cut locus for $C^1$ hypersurfaces

Let $Ω$ be an open set in $\mathbb{R}^n$ with $C^1$-boundary and $Σ$ be the skeleton of $Ω$, which consists of points where the distance function to $\partialΩ$ is not differentiable. This paper characterizes the cut locus (ridge) $\overlineΣ$, which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of $\overlineΣ$.

math.AP↗
arXiv2016

Singular perturbation by bending for an adhesive obstacle problem

A free boundary problem arising from materials science is studied in one-dimensional case. The problem studied here is an obstacle problem for the non-convex energy consisting of a bending energy, tension and an adhesion energy. If the bending energy, which is a higher order term, is deleted then "edge" singularities of the solutions (surfaces) may occur at the free boundary as Alt-Caffarelli type variational problems. The main result of this paper is to give a singular limit of the energy utilizing the notion of $Γ$-convergence, when the bending energy can be regarded as a perturbation. This singular limit energy only depends on the state of surfaces at the free boundary as seen in singular perturbations for phase transition models.

math.AP↗