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

- Kazuo Takemura
^{1}Email author, - Yoshinori Kametaka
^{2}, - Kohtaro Watanabe
^{3}, - Atsushi Nagai
^{1}and - Hiroyuki Yamagishi
^{4}

**2012**:95

https://doi.org/10.1186/1687-2770-2012-95

© Takemura et al.; licensee Springer 2012

**Received: **9 May 2012

**Accepted: **16 August 2012

**Published: **31 August 2012

## 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.

### Keywords

Hurwitz polynomial Sobolev inequality best constant Green function*n*-cascaded LRCG circuits

## 1 Introduction

*n*th order ordinary differential operator $P(d/dt)$

is assumed to be a Hurwitz polynomial with real characteristic roots $-{a}_{0},-{a}_{1},\dots ,-{a}_{n-1}$. For the sake of simplicity, we impose the following assumption.

**Assumption**

It is shown later in Section 3 that the function $G(t-s)$ is the Green function of $BVP(n)$.

The conclusion of this paper is as follows.

**Theorem 1.1**

*For any function*$u(t)$

*satisfying*${u}^{(i)}(t)\in {L}^{2}(0,1)$ ($0\le i\le n$)

*and*${u}^{(i)}(1)-{u}^{(i)}(0)=0$ ($0\le i\le n-1$),

*there exists a positive constant*

*C*

*which is independent of*$u(t)$

*such that the following Sobolev type inequality holds*:

*Among such*

*C*,

*the best constant*$C(n;a)$

*is equal to the square of*${L}^{2}$-

*norm of the Green function*,

*and it can be expressed as*

*Let*$u(t)=U(t)$

*be a solution of*$BVP(n)$

*for*$f(t)=G(-t)$ ($0<t<1$).

*Then*,

*if we replace*

*C*

*by*$C(n;a)$

*in*(1.2),

*the equality holds for*

*where*${t}_{0}$

*is a real number satisfying*$0\le {t}_{0}\le 1$,

*c*

*is an arbitrary complex number and*$U(t)$

*is given by*

The results [4] and [5] are prior works related to the subject of the present paper. In [5] in particular, we considered the boundary value problem of a similar *n* th 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. Moreover, the result [4] relates the best constant of the Sobolev inequality to a 2*n* th order Hurwitz differential operator. In this paper, we consider a problem similar to that of [5] with different boundary condition (time-periodic boundary condition).

This paper is organized as follows. First, we construct the Green function using Fourier series expansion in Sections 2 and 3 and derive a Sobolev type inequality from a solution formula to the time-periodic boundary value problem in Section 4. Using the solution formula, we compute the best constant and best function of the Sobolev type inequality as in [5]. The best constant is expressed as a function of ${a}_{0},{a}_{1},\dots ,{a}_{n-1}$. Several concrete forms of the best constant and best function are presented in Section 5. Section 6 presents an interesting application of results for an analog electric circuit. At this point, the physical meaning of a Sobolev type inequality becomes that the square of maximum of the absolute value of AC output voltage is estimated above by the constant multiple of the power of input voltage.

## 2 Fourier series expansion

Let us first introduce the trigger function $\{t\}=t-[t]$, where $[t]$ is the integral part of *t*. Note that $\{t\}$ is a periodic function of *t* with period 1. We enumerate important properties of the Fourier transform on ${L}^{2}(0,1)$.

*Proof of Proposition 2.1*(1) is obtained as follows by a straightforward calculation.

This completes the proof of Proposition 2.1. □

## 3 Green function

In this section, we will obtain a concrete expression of the Green function $G(t)$. Concerning the uniqueness and existence of the solution to $BVP(n)$, we have the following theorem.

**Theorem 3.1**

*For any function*$f(t)\in {L}^{2}(0,1)$, $BVP(n)$

*has a unique solution*$u(t)$

*expressed as*

*By using the functions*

*the Green function*$G(t)$

*can be expressed as*

*In the above determinants*, *the exponent* *i* *and the index* *j* *are such that* $0\le i\le n-2$ *and* $0\le j\le n-1$ *in the numerator*, *and* $0\le i$, $j\le n-1$ *in the denominator*.

**Remark 3.1**In the case of multiple roots, we can also have the Green function by taking a suitable limit in (3.3). For example, in the case $n=2$, we have

by taking the limit ${a}_{1}\to {a}_{0}$ in (3.3).

The above method is a well-known technique of Heaviside calculus (see reference [1] A.§5, 22.2, for example).

*Proof of Theorem 3.1*Taking the Fourier series expansion on both sides of (1.1), we have

which completes the proof of Theorem 3.1. □

## 4 Sobolev type inequality

In this section, we give a proof of Theorem 1.1.

*Proof of Theorem 1.1*For any function $u(t)$ satisfying ${u}^{(i)}(t)\in {L}^{2}(0,1)$ ($0\le i\le n$) and ${u}^{(i)}(1)-{u}^{(i)}(0)=0$ ($0\le i\le n-1$), we define $f(t)\in {L}^{2}(0,1)$ by the following relation:

*s*, so we have the following Sobolev type inequality:

which completes the proof of Theorem 1.1. □

## 5 The best constant and best function

The concrete form of the best function $U(t)$ which appeared in Theorem 1.1, is given by the following lemma.

**Lemma 5.1**

*The best function*

*is expressed as*

*If all*${a}_{j}$

*’s are distinct*, $U(t)$

*is also rewritten as*

*Here*${H}_{j}(t)$

*is defined as*

*In the above determinants*, *the exponent* *i* *and the index* *j* *are such that* $0\le i\le n-2$ *and* $0\le j\le n-1$ *in the numerator*, *and* $0\le i,j\le n-1$ *in the denominator*.

*Proof of Lemma 5.1*We first prove (5.2). Taking the Fourier series expansion on both sides to the best function, we have

we have (5.2). This completes the proof of Lemma 5.1. □

**Theorem 5.1**

*For*$n=1,2,3,\dots $,

*the best constant*$C(n;a)$

*is expressed by the following concrete forms*:

*For*$n\ge 2$,

*we have the following expression*:

*In the above determinants*, *the exponent* *i* *and the index* *j* *are such that* $0\le i\le n-2$ *and* $0\le j\le n-1$ *in the numerator*, *and* $0\le i,j\le n-1$ *in the denominator*.

*Proof of Theorem 5.1*Putting $t=0$ into (5.1), we have

□

## 6 Heaviside cable and Thomson cable

*n*-cascaded LRCG units (Figure 1).

${L}_{i}$, ${R}_{i}$, ${C}_{i}$, ${G}_{i}$ are inductance, resistance, capacitance and conductance respectively. They are nonnegative constants and not all of them are zero. ${u}_{i-1}={u}_{i-1}(t)$ and ${u}_{i}={u}_{i}(t)$ are input and output voltage respectively. ${v}_{i}={v}_{i}(t)$ is current. Output end is open, ${v}_{n+1}=0$. Input voltage ${u}_{0}(t)$ is a given function of *t*. We investigate the relation between output voltage $u(t)={u}_{n}(t)$ and input voltage ${u}_{0}(t)$.

Heaviside cable is a discrete model of the transmission line treated by Oliver Heaviside (see references [2, 3] for example).

In our previous work [5], we found that these polynomials (HC) and (TC) are Hurwitz polynomials, in which all roots have positive real parts.

## 7 Examples

Physical meaning of the 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. The authors believe that this is a new point of view compared with classical theory which mainly considers input-output gain.

In this section, let us calculate the best constants of the Sobolev type inequality for specific cases.

(1 unit Heaviside cable)

## Declarations

### Acknowledgements

The authors KT and AN are supported by J. S. P. S. Grant-in-Aid for Scientific Research (C) No. 21540148 and No. 20540138.

## Authors’ Affiliations

## References

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*Differentialgleichungen, Lösungsmethoden und Lösungen I*. 3rd edition. Chelsea, New York; 1971.Google Scholar - Heaviside O:
*Electromagnetic Induction and Its Propagation - XLVII*. The Electrician, London; 1887:189-191.Google Scholar - Nahin PJ:
*Oliver Heaviside, Sage in Solitude: The Life, Work, and Times of an Electrical Genius of the Victorian Age*. IEEE Press, New York; 1988:230-232.Google Scholar - Kametaka Y, Nagai A, Watanabe K, Takemura K, Yamagishi H: Giambelli’s formula and the best constant of Sobolev inequality in one dimensional Euclidean space.
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