Yoshinori Kametaka

The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem

Greens function of the clamped boundary value problem for the differential operator on the interval is obtained. The best constant of corresponding Sobolev inequality is given by . In addition, it is shown that a reverse of the Sobolev best constant is the one which appears in the generalized Lyapunov inequality by Das and Vatsala (1975).

Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations

This note generalizes the well known Lyapunov-type inequalities for second-order linear differential equations to certain 2M-th order linear differential equations with five types of boundary conditions. The usage of the best constant of some Sobolev-type inequalities clarify the process for obtaining such inequality and sharpen the result of Cakmak [2].

Positivity and hierarchical structure of Green’s functions of 2-point boundary value problems for bending of a beam

Green’s functions to 2-point simple type self-adjoint boundary value problems for bending of a beam under relatively strong tension on an elastic foundation are studied. We have 9 different Green’s functions. All are positive-valued and have a suitable hierarchical structure.

A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph

This paper clarifies the hierarchical structure of the sharp constants for the discrete Sobolev inequality on a weighted complete graph. To this end, we introduce a generalized-graph Laplacian A = I − B on the graph, and investigate two types of discrete Sobolev inequalities. The sharp constants C 0 ( N ; a ) and C 0 ( N ) were calculated through the Green matrix G ( a ) = ( A + a I ) − 1 ( 0 < a < ∞ ) and the pseudo-Green matrix G ∗ = A † . The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of A. Based on this new discovery, we provide the first proof that each set of the sharp constants { C 0 ( n ; a ) } n = 2 N and { C 0 ( n ) } n = 2 N satisfies a certain hierarchical structure.

Sobolev type inequalities of time-periodic boundary value problems for Heaviside and Thomson cables

We consider a time-periodic boundary value problem of n th order ordinary differential operator which appears typically in Heaviside cable and Thomson cable theory. We calculate the best constant and a family of the best functions for a Sobolev type inequality by using the Green function and apply its results to the cable theory. Physical meaning of a Sobolev type inequality is that we can estimate the square of maximum of the absolute value of AC output voltage from above by the power of input voltage.MSC:46E35, 41A44, 34B27.

Symmetrization of Functions and the Best Constant of 1-DIM Sobolev Inequality

The best constants of Sobolev embedding of into for and are obtained. A lemma concerning the symmetrization of functions plays an important role in the proof.

The Best Constant of Discrete Sobolev Inequality on the C60 Fullerene Buckyball

The best constants of two kinds of discrete Sobolev inequalities on the C60 fullerene buckyball are obtained. All the eigenvalues of discrete Laplacian

The best constant of discrete Sobolev inequality

A

Lyapunov-type inequalities for 2M th order equations under clamped-free boundary conditions

corresponding to the buckyball are found. They are roots of algebraic equation at most degree

Solutions to Some Fractional Differential Equations and Their Integrable Discretizations

4