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

Takemura Kazuo, Kametaka Yoshinori, Watanabe Kohtaro, Nagai Atsushi, Yamagishi, Yamagishi Hiroyuki

Description

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.

Sujets

• Hurwitz polynomial
• Sobolev inequality
• Green function

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English

Extrait

Takemura et al.Boundary Value Problems2012,2012:95 http://www.boundaryvalueproblems.com/content/2012/1/95

R E S E A R C HOpen Access Sobolev type inequalities of time-periodic boundary value problems for Heaviside and Thomson cables 1* 23 14 Kazuo Takemura, Yoshinori Kametaka, Kohtaro Watanabe, Atsushi Nagaiand Hiroyuki Yamagishi

* Correspondence: [e-mail] 1 Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shinei, Narashino, 275-8576, Japan Full list of author information is available at the end of the article

Abstract We consider a time-periodic boundary value problem ofnth 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 Keywords:Hurwitz polynomial; Sobolev inequality; best constant; Green function; n-cascaded LRCG circuits

1 Introduction Fornwe consider the following boundary value problem for an, , . . . ,= ,nth order ordinary differential operatorP(d/dt)

BVP(n)  P(d/dt)u=f(t)   (i) (i) u() –u() =    (i)  u∈L(, )

( <t< ), (≤i≤n– ), (≤i≤n).

The characteristic polynomial with real coefficients

n–n   n–j P(z() =z+a j ) =p j z(p  = ) j=j=

(.)

is assumed to be a Hurwitz polynomial with real characteristic roots –a  , –a  , . . . , –a n– . For the sake of simplicity, we impose the following assumption.

Assumption

a  ≤a  ≤ ∙ ∙ ∙ ≤a n– ,

a j =  (j= , , . . . ,n– ).

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