arXiv subjects

Shota Fukushima

Publications and source records attributed to Shota Fukushima.

11 recordsLinked to original sources

arXiv2026

Ellipsoidal characterization of neutral inclusions for imperfect bonding of high-conductivity type

This paper concerns neutral inclusions for imperfect bonding of high-conductivity type. An inclusion, which is a bounded domain, is said to be of imperfect bonding of high-conductivity type if the flux is discontinuous along its boundary while the potential is continuous. The inclusion is neutral to a uniform field if the presence of the inclusion does not perturb the field outside the inclusion. It is known that ellipses and ellipsoids can be neutral to all uniform fields by introducing a proper imperfect bonding coefficient on boundaries. The purpose of this paper is to prove the converse. We prove that if an inclusion of imperfect bonding of high-conductivity type is neutral to all uniform fields, then it is an ellipse or an ellipsoid. The neutrality condition is given by existence of the solution to a certain differential equation on the boundary surface and the main result is proved by converting the neutrality condition into an algebraic boundary identity characterizing ellipses and ellipsoids.

math.AP↗

arXiv2026

The transmission problem with imperfect interfaces of small resistance

We consider the transmission problem in presence of interfaces with imperfect bonding. The imperfect bonding condition is characterized by the positive resistance along the interface, which causes discontinuity of the potential across the interface while the flux is continuous. If the interface resistance is zero, then the interface is of perfect bonding, where both the potential and the flux of the solution are continuous across the interface. In this paper, we first construct using layer potentials the solution to the transmission problem with imperfect interfaces. We then prove that the solutions converge in various Sobolev spaces to the solution to the transmission problem with perfect interfaces as the interface resistance tends to zero. In particular, it is shown that the gradient of the solution converges in the uniform norm if the boundary is sufficiently regular.

math.AP↗

arXiv2024

Finiteness of the stress in presence of closely located inclusions with imperfect bonding

If two conducting or insulating inclusions are closely located, the gradient of the solution may become arbitrarily large as the distance between inclusions tends to zero, resulting in high concentration of stress in between two inclusions. This happens if the bonding of the inclusions and the matrix is perfect, meaning that the potential and flux are continuous across the interface. In this paper, we consider the case when the bonding is imperfect. We consider the case when there are two circular inclusions of the same radii with the imperfect bonding interfaces and prove that the gradient of the solution is bounded regardless of the distance between inclusions if the bonding parameter is finite. This result is of particular importance since the imperfect bonding interface condition is an approximation of the membrane structure of biological inclusions such as biological cells.

math.AP↗

arXiv2024

Spectral structure of the Neumann-Poincaré operator on axially symmetric functions

We consider the Neumann-Poincaré operator on a three-dimensional axially symmetric domain which is generated by rotating a planar domain around an axis which does not intersect the planar domain. We investigate its spectral structure when it is restricted to axially symmetric functions. If the boundary of the domain is smooth, we show that there are infinitely many axially symmetric eigenfunctions and derive Weyl-type asymptotics of the corresponding eigenvalues. We also derive the leading order terms of the asymptotic limits of positive and negative eigenvalues. The coefficients of the leading order terms are related to the convexity and concavity of the domain. If the boundary of the domain is less regular, we derive decay estimates of the eigenvalues. The decay rate depends on the regularity of the boundary. We also consider the domains with corners and prove that the essential spectrum of the Neumann-Poincaré operator on the axially symmetric three-dimensional domain is non-trivial and contains that of the planar domain.

math.SP↗

arXiv2023

Decompositions of surface vector fields and topological characterizations of the codimensions

We prove that the space of vector fields on the boundary of a bounded domain in three dimensions is decomposed into three subspaces orthogonal to each other: elements of the first one extend to the inside of the domain as gradient fields of harmonic functions, the second one to the outside of the domain as gradient fields of harmonic functions, and the third one to both the inside and the outside as divergence-free harmonic vector fields. We also characterize by the first Betti numbers the codimensions of inclusions of various subspaces related to the decomposition.

math.AP↗

arXiv2023

A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincaré operator in elasticity

We prove that the space of vector fields on the boundary of a bounded domain with the Lipschitz boundary in three dimensions is decomposed into three subspaces: elements of the first one extend to the inside the domain as divergence-free and rotation-free vector fields, the second one to the outside as divergence-free and rotation-free vector fields, and the third one to both the inside and the outside as divergence-free harmonic vector fields. We then show that each subspace in the decomposition is infinite-dimensional. We also prove under a mild regularity assumption on the boundary that the decomposition is almost direct in the sense that any intersection of two subspaces is finite-dimensional. We actually prove that the dimension of intersection is bounded by the first Betti number of the boundary. In particular, if the boundary is simply connected, then the decomposition is direct. We apply this decomposition theorem to investigate spectral properties of the Neumann-Poincaré operator in elasticity, whose cubic polynomial is known to be compact. We prove that each linear factor of the cubic polynomial is compact on each subspace of decomposition separately and those subspaces characterize eigenspaces of the Neumann-Poincaré operator. We then prove all the results for three dimensions, decomposition of surface vector fields and spectral structure, are extended to higher dimensions. We also prove analogous but different results in two dimensions.

math.AP↗

arXiv2023

Decay rate of the eigenvalues of the Neumann-Poincaré operator

If the boundary of a domain in three dimensions is smooth enough, then the decay rate of the eigenvalues of the Neumann-Poincaré operator is known and it is optimal. In this paper, we deal with domains with less regular boundaries and derive quantitative estimates for the decay rates of the Neumann-Poincaré eigenvalues in terms of the Hölder exponent of the boundary. Estimates in particular show that the less the regularity of the boundary is, the slower is the decay of the eigenvalues. We also prove that the similar estimates in two dimensions. The estimates are not only for less regular boundaries for which the decay rate was unknown, but also for regular ones for which the result of this paper makes a significant improvement over known results.

math.SP↗

arXiv2022

Semiclassical pseudodifferential operators and resolvent parametrices on manifolds with ends

We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.

math.DG↗

arXiv2022

Propagation of singularities under Schrödinger equations on manifolds with ends

We prove a microlocal smoothing effect of Schrödinger equations on manifolds. We employ radially homogeneous wavefront sets introduced by Ito and Nakamura (Amer. J. Math., 2009). In terms of radially homogeneous wavefront sets, we can apply our theory to both of asymptotically conical and hyperbolic manifolds. We relate wavefront sets in initial states to radially homogeneous wavefront sets in states after a time development. We also prove a relation between radially homogeneous wavefront sets and homogeneous wavefront sets and prove a special case of Nakamura (2005).

math.AP↗

arXiv2021

Time-slicing approximation of Feynman path integrals on compact manifolds

We construct fundamental solutions to the time-dependent Schrödinger equations on compact manifolds by the time-slicing approximation of the Feynman path integral. We show that the iteration of short-time approximate solutions converges to the fundamental solutions to the Schrödinger equations modified by the scalar curvature in the uniform operator topology from the Sobolev space to the space of square integrable functions. In order to construct the time-slicing approximation by our method, we only need to consider broken paths consisting of sufficiently short classical paths. We prove the convergence to fundamental solutions by proving two important properties of the short-time approximate solution, the stability and the consistency.

math-ph↗

arXiv2020

$L^2$ boundedness of pseudodifferential operators on manifolds with ends

We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which naturally appears in the quantum mechanics on curved spaces. We prove a Calderón-Vaillancourt type theorem for our pseudodifferential operators and discuss a construction of parametrix of elliptic differential operators on manifolds with ends.

math.AP↗