Abstract
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.
Highlights
For n =, . . . , we consider the following boundary value problem for an nth order ordinary differential operator P(d/dt) BVP(n)⎧ ⎪⎪⎨P(d/dt)u = f (t) ( < t < ),⎪⎪⎩uu((ii))(∈ )L– u(i)( ) (, ) =( ≤ i ≤ n – ), ( ≤ i ≤ n). ( . )The characteristic polynomial with real coefficients n– n
In [ ] in particular, we considered the boundary value problem of a similar nth linear ordinary differential operator P(d/dt) and computed the best constant and best function of a Sobolev type inequality by using the Green function
In our previous work [ ], we found that these polynomials (HC) and (TC) are Hurwitz polynomials, in which all roots have positive real parts
Summary
Kazuo Takemura1*, Yoshinori Kametaka[2], Kohtaro Watanabe[3], Atsushi Nagai[1] and Hiroyuki Yamagishi[4]
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