On the completeness and the basis property of the modified Frankl problem with a nonlocal oddness condition in the Sobolev space $(W^{1}_{p}(0,\pi))$

Abstract

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal oddness condition of the first kind in the Sobolev space $(W^{1}_{p}(0,\pi))$. We analyze the completeness, the basis property, and the minimality of the eigenfunctions in the space $(W^{1}_{p}(0,\pi))$.

1 Introduction

The classical Frankl problem was considered in [1]. The problem was further developed in [2], pp.339-345, [3], pp.235-252. The modified Frankl problem with a nonlocal boundary condition of the first kind was studied in [4, 5]. The basis property of eigenfunctions of the Frankl problem with nonlocal parity conditions in the Sobolev space was studied in [4]. The coefficients β are found by Theorem 1 in [6], using the results of [6], pp.177-179. In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal oddness condition of the first kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions in the space $({W}^{1}_{p}(0,\pi))$, where $({W}^{1}_{p}(0,\pi))$ is the space of absolutely continuous functions on $[0,\pi]$. So we can obtain new results by the expansion into cosines that are related to new coefficients which we calculated. This analysis and results may be of interest in itself.

2 Statement of the modified Frankl problem

Definition 2.1

In the domain $D=(D_{+}\cup D_{-1}\cup D_{-2})$, we seek a solution of the modified generalized Frankl problem

$$ u_{xx}+\operatorname{sgn}(y) u_{yy}+ \mu^{2}\operatorname{sgn}(x+y)u=0 \quad\mbox{in } (D_{+}\cup D_{-1}\cup D_{-2}), $$

(1)

with the boundary conditions

$$\begin{aligned}& u(1,\theta)=0,\quad \theta\in\biggl[0,\frac{\pi}{2}\biggr], \end{aligned}$$

(2)

$$\begin{aligned}& \frac{\partial u}{\partial x}(0,y)=0,\quad y\in(-1,0)\cup(0,1), \end{aligned}$$

(3)

$$\begin{aligned}& u(0,y)=u(0,-y),\quad y\in[0,1], \end{aligned}$$

(4)

where $u(x,y)$ is a regular solution in the class

$$u\in C^{0} (\overline{D_{+}\cup D_{-1}\cup D_{-2}})\cap C^{2}(\overline{D_{-1}})\cap C^{2}(\overline{D_{-2}}), $$

and where

$$ \begin{aligned} &D_{+} =\biggl\{ (r,\theta): 0< r< 1, 0< \theta< \frac{\pi}{2}\biggr\} , \\ &D_{-1} =\biggl\{ (x,y): -y< x< y+1, \frac{-1}{2}< y< 0\biggr\} , \\ &D_{-2} =\biggl\{ (x,y): x-1< y< -x, 0< x< \frac{1}{2}\biggr\} , \\ &\kappa\frac{\partial u}{\partial y}(x,+0)=\frac{\partial u}{\partial y}(x,-0),\quad -\infty< \kappa< \infty, 0< x< 1. \end{aligned} $$

(5)

Theorem 2.2

([7])

The eigenvalues and eigenfunctions of problem (1)-(5) can be written out in two series. In the first series, the eigenvalues $\lambda= \mu^{2}_{nk}$ are found from the equation

$$ J_{4n}(\mu_{nk})= 0, $$

(6)

where $\mu_{nk}$, $n =0, 1,2, \ldots$ , $k = 1,2,\ldots$ , are roots of the Bessel equation (6), $J_{\alpha}(z)$ is the Bessel function [8], and the eigenfunctions are given by the formula

$$ u_{nk}= \textstyle\begin{cases} A_{nk}J_{4n}(\mu_{nk}r)\cos(4n)(\frac{\pi}{2}-\theta) & \textit{in } D^{+}; \\ A_{nk}J_{4n}(\mu_{nk}\rho)\cosh(4n)\psi & \textit{in } D_{-1}; \\ A_{nk}J_{4n}(\mu_{nk}R)\cosh(4n)\varphi & \textit{in } D_{-2}, \end{cases} $$

(7)

where $x=r\cos\theta$, $y=r\sin\theta$ for $0\leq\theta\leq\frac{\pi }{2}$, $0< r<1$, and $r^{2}=x^{2}+y^{2}$ in $D_{+}$, $x=\rho\cosh\psi$, $y=\rho\sinh\psi$ for $0<\rho<1$, $-\infty<\psi<0$, $\rho^{2}= x^{2}-y^{2}$ in $D_{-1}$ and $x=R\sinh\varphi$, $y=-R\cosh\varphi$ for $0<\varphi<+\infty$, $R^{2}=y^{2}-x^{2}$ in $D_{-2}$.

In the second series, the eigenvalues $\tilde{\lambda}=\tilde{\mu }^{2}_{nk}$ are found from the equation

$$ J_{4(n-\Delta )}(\tilde{\mu}_{nk})= 0, $$

(8)

where $n =1,2, \ldots$ , and $k = 1,2,\ldots$ , and $(\tilde{\mu}_{nk})$ are the roots of the Bessel equation (8).

$$ \tilde{u}_{nk}= \textstyle\begin{cases} \tilde{A}_{nk} J_{4(n-\Delta )}(\tilde{\mu}_{nk}r)\cos 4(n-\Delta )(\frac{\pi}{2}-\theta) & \textit{in } D^{+}; \\ \tilde{A}_{nk}J_{4(n-\Delta )}(\tilde{\mu}_{nk}\rho)[\cosh 4(n-\Delta )\varphi \cos4(n-\Delta )\frac{\pi}{2} \\ \quad{}+\kappa\sinh4(n-\Delta )\psi\cos4(n-\Delta )] & \textit {in } D_{-1}; \\ \tilde{A}_{nk}J_{4(n-\Delta )}(\tilde{\mu}_{nk}R)\cosh 4(n-\Delta )\varphi[\cos4(n-\Delta )\frac{\pi}{2} \\ \quad{}-\kappa\sin4(n-\Delta )\frac{\pi}{2}] & \textit{in } D_{-2}, \end{cases} $$

(9)

where $\Delta=\frac{1}{\pi}\arcsin\frac{\kappa}{\sqrt{1+\kappa^{2}}}$, $\Delta \in(0,\frac{1}{2})$, and

$$\begin{aligned}& A^{2}_{nk}\int^{1}_{0}J^{2}_{4n}( \mu_{nk}r)r\,dr=1, \\& \tilde{A}^{2}_{nk}\int^{1}_{0}J^{2}_{4n-1}( \tilde{\mu}_{nk}r)r\,dr=1, \end{aligned}$$

$A_{nk}>0$ and $\tilde{A}_{nk}>0$.

Theorem 2.3

(see [5])

The function system

$$ \biggl\{ \cos(4n) \biggl(\frac{\pi}{2}-\theta\biggr)\biggr\} ^{\infty}_{n=0}, \qquad \biggl\{ \cos4(n-\Delta ) \biggl( \frac{\pi}{2}-\theta\biggr)\biggr\} ^{\infty}_{n=1} $$

(10)

is a Riesz basis in $L_{2}(0,\frac{\pi}{2})$ provided that $\Delta \in(0,\frac{3}{4})$.

3 The completeness, the basis property and minimality of the eigenfunctions

Theorem 3.1

The system of functions $\{\cos(n-\frac{\beta}{2})\theta\}^{\infty}_{n=0}$ is a Riesz basis in $(W^{1}_{p}(0,\pi))$ if and only if $\beta\in(-\frac{1}{p},2-\frac {1}{p})$, $\beta\neq1$.

Proof

Using the formula (20) of [9], we have the relation

$$ f(\theta)=\sum^{\infty}_{n=1}B_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta+B_{0}, $$

(11)

where

$$ B_{n}=-\int^{\pi}_{0}f'( \theta)h^{\beta}_{n}\,d\theta\biggl(n-\frac{\beta }{2} \biggr)^{-1}\quad (n=1,2,\ldots). $$

(12)

The coefficient $B_{0}$ depends on $B_{n}$ (see [9]). Consider the formally differentiated series (11)

$$ \sum^{\infty}_{n=1}{B}_{n} \biggl(n-\frac{\beta}{2}\biggr)\sin\biggl(n-\frac{\beta }{2}\biggr)\theta. $$

(13)

Since the coefficient $B_{n}$ is found by formula (12), using the results of [7], we obtain that series (11) converges to $f'(\theta)$ in the space $L_{p}(0,\pi)$. Integrating series (11) from 0 to θ, we obtain the relation

$$ f(\theta)-f(0)=\sum^{\infty}_{n=1}B_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta -\sum^{\infty}_{n=1}B_{n}, $$

(14)

which has a meaning if the following series converges

$$ \sum^{\infty}_{n=1}B_{n}. $$

(15)

By using the results of [9], we obtain that the numerical series (15) converges and relation (11) uniformly converges on $[0,\pi]$, and therefore it converges in the space $L_{p}(0,\pi)$. Now we assume that

$$B_{0}=f(0)-\sum^{\infty}_{n=1}B_{n}. $$

Then expression (14) coincides with expression (11), and therefore series (11) converges to a function in the space $(W^{1}_{p}(0,\pi))$.

Now let us show that the coefficients $B_{n}$ are uniquely found by using relation (11). Indeed, if series (11) converges in the space $(W^{1}_{p}(0,\pi))$, then series (15) converges in the space $L_{p}(0,\pi)$ (see [9]), this implies that $\lim_{n\to \infty}B_{n}=0$. For $\beta\in(-\frac{1}{p},2-\frac{1}{p})$. Now let us show that the system $\{\cos(n-\frac{\beta}{2})\theta,1\}^{\infty}_{n=1}$ does not compose a basis for $\beta\notin(-\frac{1}{p},2-\frac{1}{p})$. If $\beta\in(2-\frac {1}{p},4-\frac{1}{p})$, then, using the substitution $\beta-2=\beta'$ and removing the first cosine, we obtain the cosine system $\{\cos(n-\frac{\beta'}{2})\theta^{\infty }_{n=1},1\}$, which, as was proved above, composes a basis in $(W^{1}_{p}(0,\pi))$, and therefore the initial cosine system is not minimal in $(W^{1}_{p}(0,\pi ))$. Analogously, for $\beta\in(-2-\frac{1}{p},-\frac{1}{p})$, the substitution $\beta+2=\beta'$ reduces the initial cosine system to the system with $\beta'\in(-\frac{1}{p},2-\frac{1}{p})$, in which there is no function $(\cos(1-\frac{\beta'}{2})\theta)$, and therefore the initial cosine system is not complete. Other ranges of the parameter $\beta\in(-\frac{1}{p}+2k,2-\frac{1}{p}+2k)$, $k=\pm1,\pm2,\ldots$ , can be considered analogously. Furthermore, for $\beta=2-\frac{1}{p}$ in the space $(W^{1}_{p}(0,\pi))$, where $\hat{p}>p$, we have $-\frac {1}{\hat{p}}<\beta<2-\frac{1}{\hat{p}}$, and therefore the cosine system composes a basis in $W^{1}_{\hat{p}}(0,\pi)$, and hence it is complete in the space $(W^{1}_{p}(0,\pi))$.

For $\beta=-\frac{1}{p}$, the cosine system is minimal since, as was proved above, the coefficients $B_{n}$ are found by concrete formulas in the form of an integral. Let us show that for $\beta=2-\frac{1}{p}$, the cosine system is not minimal. By using the results of [7], we obtain that for $\beta=2-\frac{1}{p}$, the cosine system is complete but not minimal, and hence, for $\beta=-\frac{1}{p}$, the cosine system is complete (since it is minimal in this case). Now let us prove that for $\beta=-\frac{1}{p}$, the cosine system does not compose a basis. Let $f(\theta)=\theta$, then $f(\theta)\in(W^{1}_{p}(0,\pi))$, $f'(\theta)=1$, and the coefficients $B_{n}$ can be calculated by using formula (12) exactly in the same way as in [7], where it was shown that a series converges to a function not belonging to $L_{p}(0,\pi)$, thus Theorem 3.1 is proved. □

Theorem 3.2

The cosine system $\{\cos(n-\frac{\beta}{2})\theta\}^{\infty}_{n=0}$ forms a basis in the space $(W^{1}_{p}(0,\pi))$ if and only if $\beta \in(-\frac{1}{p},2-\frac{1}{p})$, $\beta\neq1$. The expansion into cosines has the form

$$f(\theta)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta, $$

where the coefficients $D_{n}$ are calculated according to the formulas

$$ D_{0}= \int^{\pi}_{0}f( \theta)H^{\beta}_{0}(\theta) \,d(\theta) $$

(16)

for $\beta<1$ and

$$\begin{aligned} D_{0}&=\frac{8(1-\beta)}{\pi\beta(2-\beta)} \int^{\pi }_{0} \frac{\sin(\theta)\sin(\frac{\beta\theta}{2})}{(2\cos\frac{\theta }{2})^{\beta}} \,d(\theta) \\ & = \int^{\pi}_{0}f(\theta)H^{\beta}_{0}( \theta) \,d(\theta) + \int^{\pi }_{0} \frac{f'(\theta)h^{\beta}_{1}}{1-\frac{\beta}{2}}\,d(\theta) \end{aligned}$$

(17)

for $\beta>1$ and for all $n\in N$, $D_{n}$ is given by

$$ D_{n}=-\int^{\pi}_{0} \biggl(f'+D_{0}\biggl(\frac{\beta}{2}\biggr)\sin\biggl( \frac{\beta\theta }{2}\biggr)\biggr)h^{\beta}_{n}\,d(\theta) \biggl(n-\frac{\beta}{2}\biggr)^{-1}, $$

(18)

where $H^{\beta}_{n}$ and $h^{\beta}_{n}(\theta)$ were studied in [10].

Proof

Analogously to the proof of relation (14), we obtain the relation

$$ f(\theta)-f(0)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta -\sum^{\infty}_{n=1}D_{n}. $$

(19)

The convergence of numerical series $\sum^{\infty}_{n=0}D_{n}$ is proved analogously to the proof of the convergence of series $\sum^{\infty}_{n=1}B_{n}$. This implies the uniform convergence of series (19).

First let $\beta<1$, then multiply series (19) by $H^{\beta }_{0}$. Integrating over the closed interval $[0,\pi]$ and taking into account relations (6) of [9] and (16) or (17), we have the relation

$$f(0)=\sum^{\infty}_{n=0}D_{n}. $$

Therefore, instead of the relation, we can write

$$ f(\theta)=\sum^{\infty}_{n=0}D_{n} \cos\biggl(n-\frac{\beta}{2}\biggr)\theta. $$

(20)

For $\beta>0$, we multiply series (19) by $H^{\beta-2}_{0}(\theta )$ and integrate the obtained relation over the closed interval $[0,\pi ]$. Using relation (9) of [9], we obtain

$$\begin{aligned} \int^{\pi}_{0}f(\theta)H^{\beta-2}_{0}( \theta) \,d(\theta )={}&D_{0}\int^{\pi}_{0} \cos\frac{\beta\theta}{2}H^{\beta-2}_{0}(\theta) \,d( \theta)+D_{1} \\ &{} + \Biggl(f(0)-\sum^{\infty}_{n=0}D_{n} \Biggr) \int^{\pi}_{0}H^{\beta -2}_{0}( \theta)\, d(\theta). \end{aligned}$$

Substituting the expression for $D_{1}$ from (18) in the latter relation, we obtain

$$\begin{aligned} &\int^{\pi}_{0}f(\theta)H^{\beta-2}_{0}( \theta) \,d(\theta )-D_{0}\int^{\pi}_{0}\cos \frac{\beta\theta}{2}H^{\beta-2}_{0}(\theta) \,d(\theta) \\ &\qquad{} + \int^{\pi}_{0}f'( \theta)h^{\beta}_{1}(\theta) \,d(\theta )\frac{1}{1-\frac{\beta}{2}} \\ &\qquad{} +D_{0}\int^{\pi}_{0}\sin \frac{\beta\theta}{2}h^{\beta }_{1}(\theta) \,d(\theta) \frac{\beta}{2(1-\frac{\beta}{2})} \\ &\quad=\Biggl(f(0)-\sum^{\infty}_{n=0}D_{n} \Biggr) \int^{\pi}_{0}H^{\beta-2}_{0}( \theta )\, d(\theta). \end{aligned}$$

(21)

Now let us show that the left-hand side of relation (21) vanishes, this will imply

$$f(0)=\sum^{\infty}_{n=0}D_{n}. $$

Indeed, integrating relation (9) of [9] by parts, we obtain the relation

$$\frac{\beta}{2(1-\frac{\beta}{2})}\int^{\pi}_{0}H^{\beta-2}_{0}( \theta )\cos\frac{\beta\theta}{2}\, d(\theta)= \biggl(1-\frac{\beta}{2}\biggr) \frac{2}{\beta }\int^{\pi}_{0}\sin \frac{\beta\theta}{2}h^{\beta}_{1}(\theta) \,d(\theta). $$

Furthermore, substituting this formula in (21), we immediately see that

$$\begin{aligned} &\int^{\pi}_{0}\biggl(f(\theta)H^{\beta-2}_{0}( \theta)+ \frac{f'(\theta )h^{\beta}_{1}(\theta)}{1-\frac{\beta}{2}}\biggr) \,d(\theta) \\ &\qquad{}+D_{0}\int^{\pi }_{0}\sin \frac{\beta\theta}{2}h^{\beta}_{1}(\theta)\, d(\theta) \biggl( \frac {2}{2-\beta}-\frac{2-\beta}{\beta}\biggr) \\ &\quad = \int^{\pi}_{0}\biggl(f( \theta)H^{\beta-2}_{0}(\theta)+ \frac{f'(\theta )h^{\beta}_{1}(\theta)}{1-\frac{\beta}{2}}\biggr) \,d( \theta) \\ &\qquad{}+ \biggl(\frac{4D_{0}(\beta-1)}{\beta(2-\beta)}\biggr)\int^{\pi}_{0} \sin\frac {\beta\theta}{2}h^{\beta}_{1}(\theta) \,d(\theta). \end{aligned}$$

By using relations (16) (or (17)) and (9) of [9], we annihilate the latter relation, i.e., we obtain relation (20) for $\beta>1$. The remaining part of Theorem 3.2 is proved analogously to Theorem 3.1. □

Remark 3.3

In case $\kappa>0$. The system of functions (10) is a Riesz basis in $({W}^{1}_{p}(0,\pi))$ if $\Delta \in(\frac {-1}{4},0)\cup(0,\frac{3}{4})$.

If $\Delta \geq\frac{3}{4}$, $\Delta \neq1,2,3,\ldots$ , then system (10) is complete but is not minimal in $({W}^{1}_{p}(0,\pi))$.

If $\Delta =\frac{-1}{4}$, then system (10) is complete and minimal but is not basis in $({W}^{1}_{p}(0,\pi))$.

If $\Delta <\frac{-1}{4}$, $\Delta \neq1,2,3,\ldots$ , then system (10) is not complete but is minimal in $({W}^{1}_{p}(0,\pi))$.

Proof

The proof of Remark 3.3 reproduces that of Theorem 3.1 and Theorem 3.2. □