1 Introduction. Formulation of the problem

The boundary value problems for elliptic equations have been studied extensively by many authors (see, e.g., [1, 2] and the references therein). In [3], the following problem for the homogeneous heat conduction equation in the domain (\(0< x<\infty\), \(t>0\)) was considered:

$${\sum_{k=1}^{m}{a_{k}}} \frac{\partial^{k}u(0,t)}{\partial x^{k}}=f(x,t), \qquad u(x,0)=0, $$

and the uniqueness and existence of the solution of this problem were proved. In [4], for the Laplace equation in an n-dimensional bounded domain D a problem with the boundary condition of the form

$$\frac{d^{m}u}{d\nu^{m}}=f(x), \quad x\in\partial D, $$

was investigated, and its Fredholm property was proved. For the Laplace, Poisson, and Helmholtz equations, the boundary value problems in the unit ball with higher-order derivatives in the boundary conditions were studied by Karachik [5–8], Sokolovskiĭ [9], and others. In the papers [4–8], the boundary conditions were given on the whole boundary. Therefore, the uniqueness of problems was proved within homogeneous polynomials of certain degree. In a rectangular domain for the heat conduction equation, the initial-boundary value problem with higher-order derivative in the initial condition was studied in [10]. Boundary value problems in the rectangular domains were studied by Sabitov (see, e.g., [11–15]).

In the present paper, we consider the equation

$$ \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}} =f ({x,y} ) $$
(1.1)

in the domain \(\Omega = \{ { ( {x,y} ):0 < x < p, 0 < y < q} \}\).

Problem

Find a function \(u(x,y)\in C^{2}(\overline{\Omega})\), \(\frac{\partial^{k}u}{\partial y^{k}}\in C(\overline{\Omega}) \) satisfying equation (1.1) in Ω and the following conditions:

$$\begin{aligned}& u ({0,y} )=0,\quad 0< y< q, \end{aligned}$$
(1.2)
$$\begin{aligned}& u ({p,y} )=0,\quad 0< y< q, \end{aligned}$$
(1.3)
$$\begin{aligned}& \frac{\partial^{k}u}{\partial y^{k}}(x,0)=\varphi ({x} ),\quad 0< x< p, \end{aligned}$$
(1.4)
$$\begin{aligned}& \frac{\partial^{k}u}{\partial y^{k}}(x,q)=\psi ({x} ),\quad 0< x< p, \end{aligned}$$
(1.5)

where k is a fixed nonnegative integer. If \(k=0 \), then it is necessary for the functions \(\varphi(x)\) and \(\psi(x)\) to satisfy the following conditions: \(\varphi(0)=\varphi(p)=0 \), \(\psi(0)=\psi (q)=0\). In case \(k=0 \) and \(f(x,y)=0 \) in Ω, problem (1.1)-(1.5) was studied in [2]. In the papers [16] and [17], the authors used similar procedures.

In this paper, our goal is to show the existence of a unique regular solution for this problem.

2 Uniqueness of the solution of problem (1.1)-(1.5)

Here, we prove the uniqueness of the solution of problem (1.1)-(1.5).

Theorem 2.1

The solution of problem (1.1)-(1.5) is unique if it exists.

Proof

Assume that

$$\varphi(x)=0, \qquad \psi(x)=0,\quad 0\leq x \leq p, \qquad f(x,y)=0,\quad (x,y) \in \overline {\Omega}. $$

We will prove that \(u(x,y)=0 \) in Ω̅. In order to show this, we refer to [18] and consider the integral

$$ \alpha_{n}(y)= \int_{0}^{p}u(x,y)X_{n}(x)\,dx, $$
(2.1)

where

$$X_{n}(x)=\sqrt{\frac{2}{p}}\sin\lambda_{n}x\, dx, \quad \lambda_{n}=\frac{n \pi }{p}, n=1,2,\ldots $$

is a complete orthonormal system in \(L_{2}[0,p] \) (see, e.g., [19]). Differentiating (2.1) twice with respect to y, we get

$$\alpha_{n}''(y)= \int_{0}^{p}\frac{\partial^{2}u}{\partial y^{2}}X_{n}(x) \,dx. $$

From the homogeneous equation (1.1) we have

$$\alpha_{n}''(y)=- \int_{0}^{p}u_{xx}(x,y)X_{n}(x) \,dx. $$

Applying integration by parts and using conditions (1.2) and (1.3), we get

$$ \alpha_{n}''(y)-\lambda_{n}^{2} \alpha_{n}(y)=0. $$
(2.2)

The general solution of equation (2.2) has the form

$$\alpha_{n}(y)=a_{n}e^{-\lambda_{n}y}+b_{n}e^{\lambda_{n}y}, $$

where \(a_{n}\) and \(b_{n} \) are unknown constant coefficients. In order to find \(a_{n} \) and \(b_{n} \), we use conditions (1.4) and (1.5), which imply

$$ \alpha_{n}^{(k)}(0)=0,\qquad \alpha_{n}^{(k)}(q)=0. $$
(2.3)

The derivative \(\alpha_{n}^{(k)}(y) \) has form

$$\alpha_{n}^{(k)}(y)=\lambda_{n}^{k} \bigl[ (-1)^{k}a_{n}e^{-\lambda _{n}y}+b_{n}e^{\lambda_{n}y} \bigr]. $$

Using (2.3), we have

$$\left \{ \textstyle\begin{array}{l} (-1)^{k}a_{n}+b_{n}=0, \\ (-1)^{k}a_{n}e^{-\lambda_{n}q}+b_{n}e^{\lambda_{n}q}=0. \end{array}\displaystyle \right . $$

The determinant of this system equals \((-1)^{k}2 \operatorname{sh} (\lambda _{n}q) \neq0\). Therefore, \(a_{n}=b_{n}=0\). Consequently, \(\alpha _{n}(y)=0\). Finally, from completeness of the functions \(X_{n}(x) \) in \(L_{2}(0,p)\) and from (2.1) we obtain \(u(x,y)=0 \) in Ω̅.

Theorem 2.1 is proved. □

3 Existence of the solution of problem (1.1)-(1.5)

In this section, we first construct a formal solution of problem (1.1)-(1.5). Then, we prove some lemmas on convergence of the series in the formal solution and its derivatives. Finally, we formulate the theorem on solvability of problem (1.1)-(1.5). We seek a formal solution of this problem in the form of Fourier series

$$ u(x,y)=\sum_{n=1}^{\infty}u_{n}(y)X_{n}(x) $$
(3.1)

expanded along system \(X_{n}(x)\). It is clear that \(u(x,y)\) satisfies conditions (1.2)-(1.3). Assume that

$$\begin{aligned}& f(x,y)\in C^{2}(\Omega), \qquad f(0,y)=f(p,y)=0,\qquad \varphi(x)\in C^{2}[0,p],\qquad \varphi(0)=\varphi(p)=0, \\& \psi(x)\in C^{2}[0,p],\qquad \psi(0)=\psi(p)=0. \end{aligned}$$

We expand the given functions \(f(x,y)\), \(\varphi(x)\), and \(\psi(x)\) in the Fourier series along the functions \(X_{n}(x)\):

$$\begin{aligned}& f(x,y)=\sum_{n=1}^{\infty}f_{n}(y)X_{n}(x), \end{aligned}$$
(3.2)
$$\begin{aligned}& \varphi(x)=\sum_{n=1}^{\infty} \varphi_{n}X_{n}(x), \end{aligned}$$
(3.3)
$$\begin{aligned}& \psi(x)=\sum_{n=1}^{\infty} \psi_{n}X_{n}(x), \end{aligned}$$
(3.4)

where

$$\begin{aligned}& f_{n}(y)= \int_{0}^{p}f(x,y)X_{n}(x)\,dx, \end{aligned}$$
(3.5)
$$\begin{aligned}& \varphi_{n}= \int_{0}^{p}\varphi(x)X_{n}(x)\,dx, \end{aligned}$$
(3.6)
$$\begin{aligned}& \psi_{n}= \int_{0}^{p}\psi(x)X_{n}(x)\,dx. \end{aligned}$$
(3.7)

Using Fourier’s method, we get a solution of problem (1.1)-(1.5) in the form

$$\begin{aligned} u(x,y) =&\sum_{n=1}^{\infty}X_{n}(x) \Biggl\{ \frac{e^{\lambda _{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{(-1)^{k}2\operatorname{sh}\lambda_{n}q}\Biggl[\frac {\varphi_{n}}{\lambda_{n}^{k}}-\sum _{s=0}^{[\frac{k-2}{2}]}\frac {f_{n}^{(k-2-2s)}(0)}{\lambda_{n}^{k-2s}} \\ &{}+\frac{1}{2\lambda_{n}} \int_{0}^{q}e^{-\lambda_{n}\eta}f_{n}(\eta) \,d\eta\Biggr]+ \frac{(-1)^{k}e^{\lambda_{n}y}y-e^{-\lambda_{n}y}}{(-1)^{k}2\operatorname{sh}\lambda _{n}q}\Biggl[\frac{\psi_{n}}{\lambda_{n}^{k}} \\ &{}-\sum_{s=0}^{[\frac{k-2}{2}]}\frac{f_{n}^{(k-2-2s)}(q)}{\lambda _{n}^{k-2s}}+ \frac{(-1)^{k}}{2\lambda_{n}} \int_{0}^{q}e^{-\lambda _{n}(q-\eta)}f_{n}(\eta) \,d\eta\Biggr] \\ &{}-\frac{1}{2\lambda_{n}} \int_{0}^{y}e^{-\lambda_{n}(y-\eta)}f_{n}(\eta ) \,d\eta-\frac{1}{2\lambda_{n}} \int_{y}^{q}e^{-\lambda_{n}(\eta -y)}f_{n}(\eta) \,d\eta\Biggr\} . \end{aligned}$$
(3.8)

Now, let us consider the derivatives

$$\begin{aligned}& \frac{\partial^{2}u}{\partial x^{2}}=-\sum_{n=1}^{\infty}X_{n}(x) \Biggl\{ \frac {e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{(-1)^{k}2\operatorname{sh}\lambda _{n}q}\Biggl[\frac{\varphi_{n}}{\lambda_{n}^{k-2}} \\& \hphantom{\frac{\partial^{2}u}{\partial x^{2}}={}}{}-\sum_{s=0}^{[\frac{k-2}{2}]} \frac{f_{n}^{(k-2-2s)}(0)}{\lambda _{n}^{k-2-2s}}+\frac{1}{2}\sum_{s=0}^{1} \frac{1}{\lambda _{n}^{s}}\bigl(f_{n}^{(s)}(0)-f_{n}^{(s)}(q)e^{-\lambda_{n}q} \bigr) \\& \hphantom{\frac{\partial^{2}u}{\partial x^{2}}={}}{}+\frac{1}{2\lambda_{n}} \int_{0}^{q}f_{n}''( \eta)e^{-\lambda_{n}\eta }\,d\eta\Biggr]+\frac{(-1)^{k}e^{\lambda_{n}y}-e^{-\lambda _{n}y}}{(-1)^{k}2\operatorname{sh}\lambda_{n}q}\Biggl[\frac{\psi_{n}}{\lambda_{n}^{k-2}} \\& \hphantom{\frac{\partial^{2}u}{\partial x^{2}}={}}{}-\sum_{s=0}^{[\frac{k-2}{2}]} \frac{f_{n}^{(k-2-2s)}(q)}{\lambda _{n}^{k-2-2s}}+\frac{(-1)^{k}}{2}\sum_{s=0}^{1} \frac{(-1)^{s}}{\lambda _{n}^{s}}\bigl[f_{n}^{(s)}(q)-f_{n}^{(s)}(0)e^{-\lambda_{n}q} \bigr] \\& \hphantom{\frac{\partial^{2}u}{\partial x^{2}}={}}{}+\frac{(-1)^{k}}{2\lambda_{n}} \int_{0}^{q}f_{n}''( \eta)e^{-\lambda _{n}(q-y)}\,d\eta\Biggr]-\frac{1}{2}\sum _{s=0}^{1}\frac{(-1)^{s}}{\lambda _{n}^{s}}\bigl[f_{n}^{(s)}(y)=f_{n}^{(s)}(0)e^{-\lambda_{n}y} \bigr] \\& \hphantom{\frac{\partial^{2}u}{\partial x^{2}}={}}{}-\frac{1}{2\lambda_{n}} \int_{0}^{y}f_{n}''( \eta)e^{-\lambda _{n}(y-\eta)}\,d\eta-\frac{1}{2}\sum_{s=0}^{1} \frac{1}{\lambda _{n}^{s}}\bigl[f_{n}^{(s)}(y)-f_{n}^{(s)}(q)e^{-\lambda_{n}(q-y)} \bigr] \\& \hphantom{\frac{\partial^{2}u}{\partial x^{2}}={}}{}-\frac{1}{2\lambda_{n}} \int_{y}^{q}f_{n}''( \eta)e^{-\lambda_{n}(\eta -y)}\,d\eta\Biggr\} , \end{aligned}$$
(3.9)
$$\begin{aligned}& \frac{\partial^{2}u}{\partial y^{2}}=\sum_{n=1}^{\infty}X_{n}(x) \Biggl[\frac {e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{(-1)^{k}2\operatorname{sh}\lambda _{n}q}\Biggl[\frac{\varphi_{n}}{\lambda_{n^{k-2}}}-\sum _{s=0}^{[\frac {k=2}{2}]}\frac{f_{n}^{(k=2-2s)}(0)}{\lambda_{n}^{k-2-2s}} \\& \hphantom{\frac{\partial^{2}u}{\partial y^{2}}={}}{}+\frac{1}{2}\sum_{s=0}^{1} \frac{1}{\lambda _{n}^{s}}\bigl(f_{n}^{(s)}(0)-f_{n}^{(s)}(q)e^{-\lambda_{n}q} \bigr)+\frac {1}{2\lambda_{n}} \int_{0}^{q}f_{n}''( \eta)e^{-\lambda_{n}\eta}\,d\eta\Biggr] \\& \hphantom{\frac{\partial^{2}u}{\partial y^{2}}={}}{}+\frac{(-1)^{k}e^{\lambda_{n}y}-e^{-\lambda_{n}y}}{(-1)^{k}2\operatorname{sh}\lambda _{n}q}\Biggl[\frac{\psi_{n}}{\lambda_{n}^{k-2}}-\sum _{s=0}^{[\frac {k-2}{2}]}\frac{f_{n}^{(k-2-2s)}(q)}{\lambda_{n}^{k-2-2s}}+\frac {(-1)^{k}}{2}\sum _{s=0}^{1}\frac{(-1)^{s}}{\lambda_{n}^{s}} \bigl(f_{n}^{(s)}(q) \\& \hphantom{\frac{\partial^{2}u}{\partial y^{2}}={}}{}-f_{n}^{(s)}(0)e^{-\lambda_{n}q}\bigr)+ \frac{(-1)^{k}}{2\lambda_{n}} \int _{0}^{q}f_{n}''( \eta)e^{-\lambda_{n}(q-\eta)}\,d\eta\Biggr]+f_{n}(y)-\frac {1}{2}\sum _{s=0}^{1}\frac{(-1)^{s}}{\lambda_{n}^{s}} \bigl[f_{n}^{(s)}(y) \\& \hphantom{\frac{\partial^{2}u}{\partial y^{2}}={}}{}-f_{n}^{(s)}(0)e^{-\lambda_{n}y}\bigr]- \frac{1}{2\lambda_{n}} \int _{0}^{y}f_{n}''( \eta)e^{-\lambda_{n}(y-\eta)}\,d\eta-\frac{1}{2}\sum_{s=0}^{1} \frac{1}{\lambda_{n}^{s}}\bigl[f_{n}^{(s)}(y) \\& \hphantom{\frac{\partial^{2}u}{\partial y^{2}}={}}{}-f_{n}^{(s)}(q)e^{-\lambda_{n}(q-y)}\bigr]- \frac{1}{2\lambda_{n}} \int _{y}^{q}f_{n}''( \eta)e^{-\lambda_{n}(\eta-y)}\,d\eta\Biggr], \end{aligned}$$
(3.10)

and

$$\begin{aligned} \frac{\partial^{k}u}{\partial y^{k}} =&\sum_{n=1}^{\infty}X_{n}(x) \Biggl\{ \frac {\operatorname{sh}\lambda_{n}(q-y)}{\operatorname{sh}\lambda_{n}q}\Biggl[\varphi_{n}-\sum _{s=0}^{[\frac {k-2}{2}]}\lambda_{n}^{2s}f_{n}^{(k-2-2s)}(0)+ \frac{1}{2}\sum_{s=0}^{k-1} \lambda_{n}^{k-2-s}\bigl[f_{n}^{(s)}(0) \\ &{}-f_{n}^{(s)}(q)e^{-\lambda_{n}q}\bigr]+\frac{1}{2\lambda_{n}} \int _{0}^{q}f_{n}^{(k)}( \eta)e^{-\lambda_{n}\eta}\,d\eta\Biggr]+\frac{\operatorname{sh}\lambda _{n}y}{\operatorname{sh}\lambda_{n}q}\Biggl[\psi_{n}- \sum_{s=0}^{[\frac{k-2}{2}]}\lambda _{n}^{2s}f_{n}^{(k-2-2s)}(q) \\ &{}+\frac{1}{2}\sum_{s=0}^{k-1} \frac{(-1)^{k+s}}{\lambda _{n}^{s+2-k}}\bigl(f_{n}^{(s)}(q)-f_{n}^{(s)}(0)e^{-\lambda_{n}q} \bigr)+\frac {1}{2\lambda_{n}} \int_{0}^{q}f_{n}^{(k)}( \eta)e^{-\lambda_{n}(q-\eta )}\,d\eta\Biggr] \\ &{}+\sum_{s=0}^{[\frac{k-2}{2}]}\lambda_{n}^{2s}f_{n}^{(k-2-2s)}(y)- \frac {(-1)^{k}}{2}\sum_{s=0}^{k-1}(-1)^{s} \lambda _{n}^{k-2-2s}\bigl[f_{n}^{(s)}(y)-f_{n}^{(s)}(0)e^{-\lambda_{n}y} \bigr] \\ &{}-\frac{1}{2\lambda_{n}} \int_{0}^{y}f_{n}^{(k)}( \eta)e^{-\lambda _{n}(y-\eta)}\,d\eta-\frac{1}{2}\sum_{s=0}^{k-1} \lambda _{n}^{k-2-s}\bigl[f_{n}^{(s)}(y)-f_{n}^{(s)}(q)e^{-\lambda_{n}(q-y)} \bigr] \\ &{}-\frac{1}{2\lambda_{n}} \int_{y}^{q}f_{n}^{(k)}( \eta)e^{-\lambda_{n}(\eta -y)}\,d\eta\Biggr\} . \end{aligned}$$
(3.11)

Denote by \(C_{x,y}^{1,0}(\overline{\Omega}) \) the class of the functions \(u(x,y) \) such that \(u(x,y), u_{x}(x,y)\in C(\overline {\Omega}) \).

We have the following lemmas.

Lemma 3.1

If

$$f(x,y)\in C_{x,y}^{1,0}(\overline{\Omega}),\qquad \frac{\partial f}{\partial x}\in \operatorname{Lip}_{\alpha}[0,p],\quad 0< \alpha< 1, $$

uniformly with respect to y, then the series in (3.2) absolutely and uniformly converges in Ω̅.

Proof

Applying integration by parts to (3.5), we obtain

$$ f_{n}(y)=\frac{1}{\lambda_{n}}f_{n}^{(1)}(y), $$
(3.12)

where

$$f_{n}^{(1)}(y)= \int_{0}^{p}\frac{\partial f}{\partial x}(x,y)\sqrt{ \frac {2}{p}}\cos\lambda_{n}x\, dx. $$

According to [20], \(|f_{n}^{(1)}(y)|\leq\frac{c}{n^{\alpha}} \), where \(c>0 \) is a constant. Then, \(|f_{n}(y)|<\frac{c p}{\pi}\frac {1}{n^{1+\alpha}} \), and the series \(\sum_{n=1}^{\infty}\frac {1}{n^{1+\alpha}} \) is convergent. Consequently, the series in (3.2) is absolutely and uniformly convergent in Ω̅.

Lemma 3.1 is proved. □

Lemma 3.2

If

$$\varphi(x)\in W_{2}^{1}(0,p),\qquad \varphi(0)=\varphi(p)=0 $$

and

$$\psi\in W_{2}^{1}(0,p),\qquad \psi(0)=\psi(p)=0, $$

then the series (3.3) and (3.4) are absolutely and uniformly convergent in \([0,p]\).

Proof

Integrating the integral in (3.6) by parts, we have

$$ \varphi_{n}=\frac{1}{\lambda_{n}}\varphi_{n}^{(1)}, $$
(3.13)

where

$$\varphi_{n}^{(1)}= \int_{0}^{p}\varphi'(x)\sqrt{ \frac{2}{p}}\cos \lambda_{n}x\, dx. $$

Taking into account equality (3.13) and applying the Hölder inequality to the sum (see, e.g., [21]) of the series

$$\frac{p}{\pi}\sum_{n=1}^{\infty} \frac{1}{n}\bigl\vert \varphi_{n}^{(1)}\bigr\vert , $$

we get

$$\frac{p}{\pi} \sum_{n=1}^{\infty} \frac{1}{n}\bigl\vert \varphi_{n}^{(1)}\bigr\vert \leq \frac{p}{\pi}\Biggl(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\Biggr)^{\frac{1}{2}}\Biggl(\sum_{n=1}^{\infty} \bigl\vert \varphi_{n}^{(1)}\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$

Using Bessel’s inequality (see, e.g., [21]), we find

$$\Biggl(\sum_{n=1}^{\infty}\bigl\vert \varphi_{n}^{(1)}\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}\leq\bigl\Vert \varphi '\bigr\Vert _{L_{2}(0,p)}. $$

Furthermore,

$$\frac{p}{\pi}\Biggl(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\Biggr)^{\frac{1}{2}}=\frac {p}{\pi}\biggl( \frac{\pi^{2}}{6}\biggr)^{\frac{1}{2}}=\frac{p}{\sqrt{6}}. $$

Hence, we have

$$\frac{p}{\pi}\sum_{n=1}^{\infty} \frac{1}{n}\bigl\vert \varphi_{n}^{(1)}\bigr\vert \leq\frac {p}{\sqrt{6}}\bigl\Vert \varphi'(x)\bigr\Vert _{L_{2}(0,p)}. $$

Consequently, the series (3.3) is absolutely and uniformly convergent in \([0,p]\). The proof of absolute and uniform convergence of the series (3.4) is analogous.

Lemma 3.2 is proved. □

Lemma 3.3

If

$$\varphi(x)\in W_{2}^{3}(0,p),\qquad \varphi(0)= \varphi(p)=0,\qquad \varphi ''(0)=\varphi''(p)=0 $$

and

$$\psi(x)\in W_{2}^{3}(0,p),\qquad \psi(0)=\psi(p)=0, \qquad \psi''(0)=\psi''(p)=0, $$

then, for any \(k\geq0 \), the series

$$ \sum_{n=1}^{\infty}\frac{e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda _{n}(q-y)}}{(-1)^{k}2 \operatorname{sh} \lambda_{n}q} \frac{\varphi_{n}}{\lambda _{n}^{k-2}} $$
(3.14)

and

$$ \sum_{n=1}^{\infty}\frac{(-1)^{k}e^{\lambda_{n}y}-e^{-\lambda _{n}y}}{(-1)^{k}2 \operatorname{sh} \lambda_{n}q} \frac{\psi_{n}}{\lambda _{n}^{k-2}} $$
(3.15)

are absolutely and uniformly convergent in Ω̅.

Proof

We show the inequalities

$$\begin{aligned}& 0\leq\frac{e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{2 \operatorname{sh} \lambda_{n}q}\leq C_{0}, \end{aligned}$$
(3.16)
$$\begin{aligned}& 0\leq\biggl\vert \frac{(-1)^{k}e^{\lambda_{n}y}-e^{-\lambda_{n}y}}{(-1)^{k}2 \operatorname{sh} \lambda_{n}q}\biggr\vert \leq C_{0}, \end{aligned}$$
(3.17)

where \(C_{0}=\frac{2}{1-e^{-2\frac{\pi q}{p}}}\).

Indeed,

$$\frac{e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{e^{\lambda _{n}q}-e^{-\lambda_{n}q}}=\frac{e^{\lambda_{n}(q-y)}}{e^{\lambda _{n}q}}\frac{1-(-1)^{k}e^{-2\lambda_{n}(q-y)}}{1-e^{-2\lambda_{n}q}}. $$

Since

$$\frac{e^{\lambda_{n}(q-y)}}{e^{\lambda_{n}q}}\leq1,\qquad 1-(-1)^{k}e^{-2\lambda_{n}(q-y)}\leq2, \qquad 1-e^{-2\lambda _{n}q}>1-e^{-2\frac{\pi q}{p}}, $$

we have

$$\frac{e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{2\operatorname{sh}\lambda _{n}q}\leq\frac{2}{1-e^{-2\frac{\pi q}{p}}}=C_{0}. $$

Inequality (3.16) is proved.

Similarly, one can verify inequality (3.17). In the case \(k=2\), the series (3.14) and (3.15) are absolutely and uniformly convergent in Ω̅ according to Lemma 3.2. When \(k>2 \), the series (3.14) and (3.15) evidently are absolutely and uniformly convergent in Ω̅. Let \(k=0\). We consider the absolute value of the series in (3.14)

$$\sum_{n=1}^{\infty}\frac{e^{\lambda_{n}(q-y)}-(-1)^{k}e^{-\lambda _{n}(q-y)}}{2\operatorname{sh}\lambda_{n}q} \lambda_{n}^{2}|\varphi_{n}|. $$

In order to prove the convergence of the last series, we apply integration by parts in (3.6). We have

$$ \varphi_{n}=-\frac{1}{\lambda_{n}^{3}}\varphi_{n}^{(3)}, $$
(3.18)

where

$$\varphi_{n}^{(3)}= \int_{0}^{p}\varphi'''(x) \sqrt{\frac{2}{p}}\cos \lambda_{n}x\, dx. $$

Using (3.16) and (3.18) in the last series, we get

$$C_{0}\sum_{n=1}^{\infty} \lambda_{n}^{2}\frac{1}{\lambda_{n}^{3}}\bigl\vert \varphi _{n}^{(3)}\bigr\vert =C_{0}\sum _{n=1}^{\infty}\frac{1}{\lambda_{n}}\bigl\vert \varphi_{n}^{(3)}\bigr\vert . $$

Applying Hölder’s inequality to the sum to the last series, we obtain

$$\begin{aligned} C_{0}\sum_{n=1}^{\infty} \frac{1}{\lambda_{n}}\bigl\vert \varphi_{n}^{(3)}\bigr\vert \leq& C_{0}\frac{p}{\pi}\Biggl(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\Biggr)^{\frac {1}{2}}\Biggl(\sum_{n=1}^{\infty} \bigl\vert \varphi_{n}^{(3)}\bigr\vert ^{2} \Biggr)^{\frac{1}{2}} \\ =&C_{0}\frac{p}{\pi}\frac{\pi}{\sqrt{6}}\Biggl(\sum _{n=1}^{\infty}\bigl\vert \varphi _{n}^{(3)} \bigr\vert ^{2}\Biggr)^{\frac{1}{2}} \\ =&\frac{C_{0}p}{\sqrt{6}}\Biggl(\sum _{n=1}^{\infty}\bigl\vert \varphi _{n}^{(3)}\bigr\vert ^{2}\Biggr)^{\frac{1}{2}}. \end{aligned}$$

Using Bessel’s inequality, we have

$$\Biggl(\sum_{n=1}^{\infty}\bigl\vert \varphi_{n}^{(3)}\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}\leq\bigl\Vert \varphi ''' \bigr\Vert _{L_{2}(0,p)}. $$

Further, we have

$$C_{0}\sum_{n=1}^{\infty} \frac{1}{\lambda_{n}}\bigl\vert \varphi_{n}^{(3)}\bigr\vert \leq \frac{C_{0}p}{\sqrt{6}}\bigl\Vert \varphi''' \bigr\Vert _{L_{2}(0,p)}. $$

Consequently, the series (3.14) converges. Analogously, the proof of convergence of the series (3.15) can be obtained, and, thus, we do not give it here.

Lemma 3.3 is proved. □

Lemma 3.4

If \(\frac{\partial^{k-2}f(x,y)}{\partial y^{k-2}}\in C(\overline{\Omega})\), then the series

$$ \sum_{n=1}^{\infty}X_{n}(x) \frac{e^{\lambda _{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{(-1)^{k}2\operatorname{sh}\lambda_{n}q}\sum_{s=0}^{[\frac{k-2}{2}]} \frac{f_{n}^{(k-2-2s)}(0)}{\lambda_{n}^{k-2s}} $$
(3.19)

and

$$ \sum_{n=1}^{\infty}X_{n}(x) \frac{(-1)^{k}e^{\lambda_{n}y}-e^{-\lambda _{n}y}}{(-1)^{k}2\operatorname{sh}\lambda_{n}q}\sum_{s=0}^{[\frac{k-2}{2}]} \frac {f_{n}^{(k-2-2s)}(q)}{\lambda_{n}^{k-2s}} $$
(3.20)

are absolutely and uniformly convergent in Ω̅.

Proof

By the condition of the lemma we have

$$ \bigl\vert f_{n}^{(k-2-2s)}(a)\bigr\vert \leq C_{1}, $$
(3.21)

where \(a=0 \) or \(a=q \), and \(C_{1}>0 \) is constant. Taking into account (3.16), (3.17), and the last inequality, we conclude that the series

$$ C_{0}C_{1}\sqrt{\frac{2}{p}}\sum _{n=1}^{\infty}\sum_{s=0}^{[\frac {k-2}{2}]} \frac{1}{\lambda_{n}^{k-2s}} $$
(3.22)

is a majorant for the series (3.19) and (3.20). If \(s=0 \) and \(k\geq 2 \), then the series (3.22) converges. At \(s=[\frac{k-2}{2}]\), we have

$$k-2s=k-2\biggl[\frac{k-2}{2}\biggr]= \left \{ \textstyle\begin{array}{l@{\quad}l} 3 & \mbox{if } k \mbox{ is odd}, \\ 2 &\mbox{if } k \mbox{ is even}. \end{array}\displaystyle \right . $$

Consequently, at \(s=[\frac{k-2}{2}] \), if k is an odd number, then the series (3.22) has the form \(\sum_{n=1}^{\infty}\frac{1}{\lambda _{n}^{3}}\); if k is an even number, then the series (3.22) has the form \(\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}^{2}}\). In both cases, the series (3.22) is convergent. Therefore, the series (3.19) and (3.20) are absolutely and uniformly convergent in Ω̅.

Lemma 3.4 is proved. □

Lemma 3.5

If

$$\frac{\partial^{m-1}f(x,y)}{\partial y^{m-1}}\in C( \overline{\Omega }),\qquad \frac{\partial^{m}f(x,y)}{\partial y^{m}}\in L_{2}(0,q), $$

then we have the estimate

$$ \frac{1}{2\lambda_{n}}\biggl\vert \int_{0}^{q}e^{-\lambda_{n}(q-\eta )}f_{n}^{(m)}( \eta)\,d\eta\biggr\vert \leq\frac{C}{n^{\frac{3}{2}}}\bigl\Vert f_{n}^{(m)} \bigr\Vert _{L_{2}(0,q)}, $$
(3.23)

where \(C=(\frac{p}{2\pi})^{\frac{3}{2}} \), and \(m=2 \) or \(m=k \).

Proof

Applying the Hölder inequality (see, e.g.,[21]) to the integral on the left-hand side of inequality (3.23), we obtain

$$\begin{aligned} \frac{1}{2\lambda_{n}}\biggl\vert \int_{0}^{q}e^{-\lambda_{n}(q-\eta )}f_{n}^{(m)}( \eta)\,d\eta\biggr\vert \leq&\frac{1}{2\lambda_{n}}\biggl( \int _{0}^{q}e^{-2\lambda_{n}(q-\eta)}\,d\eta \biggr)^{\frac{1}{2}}\biggl( \int _{0}^{q}\bigl(f_{n}^{(m)}( \eta)\bigr)^{2}\,d\eta\biggr)^{\frac{1}{2}} \\ =&\biggl[\frac{1}{2\lambda_{n}}\bigl(1-e^{-2\lambda_{n}q}\bigr)\biggr]^{\frac{1}{2}} \frac {1}{2\lambda_{n}}\bigl\Vert f_{n}^{(m)}\bigr\Vert _{L_{2}(0,q)} \\ \leq&\frac{1}{(2\lambda _{n})^{\frac{3}{2}}}\bigl\Vert f_{n}^{(m)} \bigr\Vert _{L_{2}(0,q)}=\frac{1}{(\frac{2n\pi}{p})^{\frac{3}{2}}}\bigl\Vert f_{n}^{(m)}\bigr\Vert _{L_{2}(0,q)} \\ =&\frac{C}{n^{\frac{3}{2}}}\bigl\Vert f_{n}^{(m)} \bigr\Vert _{L_{2}(0,q)}. \end{aligned}$$

Lemma 3.5 is proved. □

Lemma 3.6

If \(\frac{\partial^{k-2}f(x,y)}{\partial y^{k-2}}\in C(\overline{\Omega})\), then the series

$$ \sum_{n=1}^{\infty}X_{n}(x) \frac{e^{\lambda _{n}(q-y)}-(-1)^{k}e^{-\lambda_{n}(q-y)}}{(-1)^{k}2\operatorname{sh}\lambda_{n}q}\sum_{s=0}^{[\frac{k-2}{2}]} \frac{f_{n}^{(k-2-2s)}(0)}{\lambda _{n}^{k-2-2s}} $$
(3.24)

and

$$ \sum_{n=1}^{\infty}X_{n}(x) \frac{(-1)^{k}e^{\lambda_{n}y}-e^{-\lambda _{n}y}}{(-1)^{k}2\operatorname{sh}\lambda_{n}q}\sum_{s=0}^{[\frac{k-2}{2}]} \frac {f_{n}^{(k-2-2s)}(q)}{\lambda_{n}^{k-2-2s}} $$
(3.25)

absolutely and uniformly converge in Ω̅.

Proof

Taking into account (3.16) and (3.17), we conclude that the series

$$ C_{0}\sqrt{\frac{2}{p}}\sum_{n=1}^{\infty} \sum_{s=0}^{[\frac {k-2}{2}]}\frac{|f_{n}^{(k-2-2s)}(a)|}{\lambda_{n}^{k-2-2s}}, $$
(3.26)

where \(a=0 \) or \(a=q \), is majorant for the series (3.24) and (3.25). If \(s=[\frac{k-2}{2}]\), then we have

$$k-2-2s=k-2-2\biggl[\frac{k-2}{2}\biggr]= \left \{ \textstyle\begin{array}{l@{\quad}l} 1& \mbox{if } k \mbox{ is odd}, \\ 0& \mbox{if } k \mbox{ is even}. \end{array}\displaystyle \right . $$

Therefore, in the case where k is an odd number, the series (3.26) has the form

$$ \sum_{n=1}^{\infty}\frac{|f_{n}'(a)|}{\lambda_{n}}, $$
(3.27)

and if k is an even number, then the series (3.26) has the form

$$ \sum_{n=1}^{\infty}\bigl\vert f_{n}(a)\bigr\vert . $$
(3.28)

Applying the Hölder inequality to the series (3.27), we get

$$\sum_{n=1}^{\infty}\frac{\vert f_{n}'(a)\vert }{\lambda_{n}}\leq \Biggl(\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}^{2}} \Biggr)^{\frac{1}{2}}\Biggl(\sum_{n=1}^{\infty} \bigl\vert f_{n}'(a)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$

Using Bessel’s inequality, we find

$$\Biggl(\sum_{n=1}^{\infty}\bigl\vert f_{n}'(a)\bigr\vert ^{2}\Biggr)^{\frac{1}{2}} \leq\biggl\Vert \frac {\partial f(x,a)}{\partial y}\biggr\Vert _{L_{2}(0,q)}. $$

Since \((\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}^{2}})^{\frac {1}{2}}=\frac{p}{\sqrt{6}}\), the series (3.27) converges. In order to proof the convergence of the series (3.28), we integrate by parts the integral

$$f_{n}(a)= \int_{0}^{p}f(x,a)X_{n}(x)\,dx, $$

and we obtain

$$ f_{n}(a)=\frac{1}{\lambda_{n}}f_{n}^{(1)}(a), $$
(3.29)

where

$$f_{n}^{(1)}(a)= \int_{0}^{p}\frac{\partial f(x,a)}{\partial x}\sqrt{ \frac {2}{p}}\cos\lambda_{n}x\, dx. $$

Substituting (3.29) into the series (3.28), we find

$$\sum_{n=1}^{\infty} \bigl\vert f_{n}(a)\bigr\vert =\sum_{n=1}^{\infty} \frac{1}{\lambda _{n}}\bigl\vert f_{n}^{(1)}(a)\bigr\vert . $$

Further, as in the case of (3.27), we have

$$\sum_{n=1}^{\infty}\bigl\vert f_{n}(a)\bigr\vert \leq\frac{p}{\sqrt{6}}\biggl\Vert \frac{\partial f(x,a)}{\partial x}\biggr\Vert _{L_{2}(0,p)}. $$

If \(s\leq[\frac{k-2}{2}]\), then the series (3.26) converges for both odd and even k. Consequently, the series (3.24) and (3.25) are absolutely and uniformly convergent in Ω̅.

Lemma 3.6 is proved. □

Lemma 3.7

If

$$\frac{\partial^{k-1}f(x,y) }{\partial x^{i}\, \partial y^{j}}\in C(\overline{\Omega}),\quad i,j=0,1,\ldots,k-1, i+j=k-1, \qquad \frac{\partial ^{2l}f}{\partial x^{2l}}(0,y)=\frac{\partial^{2l}f }{\partial x^{2l}}(p,y)=0, $$

where

$$ l=0,1,\ldots, \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{k-2}{2} & \textit{if } k \textit{ is even}, \\ \frac{k-3}{2} & \textit{if } k \textit{ is odd}, \end{array}\displaystyle \right . $$
(3.30)

then the series

$$\begin{aligned} \sum_{n=1}^{\infty}\sum _{s=0}^{[\frac{k-2}{2}]}\lambda _{n}^{2s}\bigl\vert f_{n}^{(k-2-2s)}(a)\bigr\vert =&\sum _{n=1}^{\infty }\Biggl[\bigl\vert f_{n}^{(k-2)}(a) \bigr\vert +\lambda_{n}^{2}\bigl\vert f_{n}^{(k-4)}(a)\bigr\vert \\ &{}+\cdots+\left \{ \textstyle\begin{array}{l@{\quad}l} \lambda_{n}^{k-2}\vert f_{n}(a)\vert & \textit{if } k \textit{ is even},\\ \lambda_{n}^{k-3}\vert f_{n}'(a)\vert & \textit{if } k \textit{ is odd} \end{array}\displaystyle \right . \Biggr] \end{aligned}$$
(3.31)

converges, where \(a=0 \) or \(a=q \).

Proof

Let k be an even number. If \(s=[\frac{k-2}{2}] \), then \(2s=2[\frac{k-2}{2}]=k-2\). If the series

$$ \sum_{n=1}^{\infty}\lambda_{n}^{k-2} \bigl\vert f_{n}(a)\bigr\vert $$
(3.32)

converges, then the series (3.31) is also convergent, where

$$f_{n}(a)= \int_{0}^{p}f(x,a)\sqrt{\frac{2}{p}}\sin \lambda_{n}x\, dx. $$

Integrating the last integral by parts \(k-1 \) times, we have

$$ f_{n}(a)=\frac{1}{\lambda_{n}^{k-1}}f_{n}^{(k-1,0)}(a), $$
(3.33)

where

$$f_{n}^{(k-1,0)}(a)= \int_{0}^{p}\frac{\partial^{k-1}f(x,a)}{\partial x^{k-1}}\sqrt{ \frac{2}{p}}\sin\biggl[(k-1)\frac{\pi}{2}+x\biggr]\,dx. $$

Taking into account (3.33), the series (3.32) gives

$$\sum_{n=1}^{\infty}\lambda_{n}^{k-2} \bigl\vert f_{n}(a)\bigr\vert \leq\sum _{n=1}^{\infty }\frac{1}{\lambda_{n}}\bigl\vert f_{n}^{(k-1,0)}(a)\bigr\vert . $$

Applying the Hölder inequality to the right-hand side of the last inequality, we have

$$\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}}\bigl\vert f_{n}^{(k-1,0)}(a)\bigr\vert \leq\Biggl(\sum _{n=1}^{\infty}\frac{1}{\lambda_{n}^{2}}\Biggr)^{\frac{1}{2}} \Biggl(\sum_{n=1}^{\infty}\bigl\vert f_{n}^{(k-1,0)}(a)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}=\frac{p}{\sqrt {6}}\Biggl(\sum_{n=1}^{\infty} \bigl\vert f_{n}^{(k-1,0)}(a)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$

Using Bessel’s inequality, we find

$$\Biggl(\sum_{n=1}^{\infty}\bigl\vert f_{n}^{(k-1,0)}(a)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}\leq\biggl\Vert \frac {\partial^{k-1}f(x,a)}{\partial x^{k-1}}\biggr\Vert _{L_{2}(0,p)}. $$

Hence, we obtain

$$\sum_{n=1}^{\infty}\lambda_{n}^{k-2} \bigl\vert f_{n}(a)\bigr\vert \leq\frac{p}{\sqrt{6}}\biggl\Vert \frac{\partial^{k-1}f(x,a)}{\partial x^{k-1}}\biggr\Vert _{L_{2}(0,p)}. $$

The convergence of the series (3.32) is proved.

Let k be an odd number and \(s=[\frac{k-2}{2}]\). In this case, \(2s=2[\frac{k-2}{2}]=k-3\). If the series

$$ \sum_{n=1}^{\infty}\lambda_{n}^{k-3} \bigl\vert f_{n}'(a)\bigr\vert $$
(3.34)

is convergent, then the series (3.31) is also convergent for odd k. The proof of this assertion is analogous to the proof of convergence of the series (3.32).

Lemma 3.7 is proved. □

Lemma 3.8

Let the conditions of Lemma  3.7 be satisfied. Then the series

$$\sum_{n=1}^{\infty}\sum _{s=0}^{k-1}\lambda_{n}^{k-2-s}\bigl\vert f_{n}^{(s)}(a)\bigr\vert ,\quad a=0,q, $$

is convergent, and the series

$$\sum_{n=1}^{\infty}\sum _{s=0}^{k-1}\lambda _{n}^{k-2-s}\bigl\vert f_{n}^{(s)}(y)-f_{n}^{(0)}e^{-\lambda_{n}y} \bigr\vert $$

and

$$\sum_{n=1}^{\infty}\sum _{s=0}^{k-1}\lambda _{n}^{k-2-s}\bigl\vert f_{n}^{(s)}(y)-f_{n}^{(s)}(q)e^{-\lambda_{n}(q-y)} \bigr\vert $$

are absolutely and uniformly convergent.

Proof of this lemma is analogous to that of Lemma 3.7. Using the results of the presented lemmas, we get the following theorem.

Theorem 3.1

Let the following conditions be satisfied:

$$\begin{aligned} (\mathrm{i})&\quad \varphi(x)\in W_{2}^{3}(0,p),\qquad \varphi(0)=\varphi(p)=0,\qquad \varphi ''(0)= \varphi''(p)=0; \\ (\mathrm{ii})&\quad \psi\in W_{2}^{3}(0,p), \qquad \psi(0)= \psi(p)=0, \qquad \psi''(0)=\psi''(p)=0; \\ (\mathrm{iii})&\quad \frac{\partial^{k-1}f(x,y)}{\partial x^{i}\, \partial y^{j}}\in C(\overline{\Omega}), \quad i+j=k-1, i,j=0,1,\ldots,k-1, \\ &\quad\frac{\partial^{k} f(x,y)}{\partial y^{k}}\in L_{2}(\Omega),\qquad \frac {\partial^{2l}f}{\partial x^{2l}}(0,y)= \frac{\partial^{2l}f}{\partial x^{2l}}(p,y)=0, \end{aligned}$$

where

$$l=0,1,\ldots, \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{k-2}{2}& \textit{if } k \textit{ is even}, \\ \frac{k-3}{2} &\textit{if } k \textit{ is odd}. \end{array}\displaystyle \right . $$

Then the series (3.8)-(3.11) absolutely and uniformly converge in Ω̅, and solution (3.8) satisfies equation (1.1) in Ω and conditions (1.2)-(1.5), where \(u(x,y)\in C^{2}(\overline {\Omega})\), \(\frac{\partial^{k}u}{\partial y^{k}}\in C(\overline{\Omega})\).

Proof

Adding (3.9) and (3.10), we find that solution (3.8) satisfies equation (1.1) in Ω. From the properties of the functions \(X_{n}(x)\) it follows that solution (3.8) satisfies conditions (1.2) and (1.3). The absolute and uniform convergence of the series (3.8) in Ω̅ follows from Lemma 3.3 with \(m=k \) and from Lemmas 3.4 and 3.5. Therefore, \(u(x,y)\in C(\overline{\Omega })\). The absolute and uniform convergence of the series (3.9) and (3.10) in Ω̅ follows from Lemmas 3.1-3.5. Hence, we have \(\frac{\partial^{2}u}{\partial x^{2}}\in C(\overline{\Omega})\), \(\frac{\partial^{2}u}{\partial y^{2}}\in C(\overline{\Omega})\). The absolute and uniform convergence of the series (3.11) in Ω̅ follows from the Lemmas 3.2, 3.5, 3.7, and 3.8. Therefore, \(\frac{\partial^{k}u}{\partial y^{k}}\in C(\overline{\Omega})\). Consequently, \(u(x,y)\in C^{2}(\overline{\Omega})\), \(\frac{\partial ^{k}u}{\partial y^{k}}\in C(\overline{\Omega})\). Taking the limit in (3.11) as \(y \rightarrow0 \) and \(y \rightarrow q\), we conclude that solution (3.8) satisfies conditions (1.4) and (1.5).

This proves the theorem. □