On discrete boundary value problems with nonlocal conditions in a quarter-plane

Let $\mathbb{Z}^{2}$ be an integer lattice in a plane. Let $K=\{x\in\mathbb{R}^{2}:x=(x_{1},x_{2}),x_{1}>0,x_{2}>0\}$ be a quadrant, $K_{d}=h\mathbb{Z}^{2}\cap K,h>0$. We consider functions of discrete variable $u_{d}(\tilde{x}),\tilde{x}=(\tilde{x}_{1},\tilde{x}_{2})\in h\mathbb{Z}^{2}$.

Let us denote $\mathbb{T}^{2}=[-\pi,\pi]^{2},\hbar=h^{-1}$. We consider functions defined in $\hbar\mathbb{T}^{2}$ as periodic functions defined in $\mathbb{R}^{2}$ with basic square of periods $\hbar\mathbb{T}^{2}$.

One can define the discrete Fourier transform for the function $u_{d}$

if the latter series converges, and the function $\tilde{u}_{d}(\xi)$ is a periodic function in $\mathbb{R}^{2}$ with basic square of periods $\hbar\mathbb{T}^{2}$. Such discrete Fourier transform preserves all properties of integral Fourier transform, and the inverse discrete Fourier transform looks as follows

The discrete Fourier transform gives one-to-one correspondence between spaces $L_{2}(h\mathbb{Z}^{2})$ and $L_{2}(\hbar\mathbb{T}^{2})$ with norms

We need more general discrete functional spaces and we introduce such spaces using divided differences [13].

The divided differences of first order look as follows

and their discrete Fourier transforms are given by formulas

The divided difference of second order is a divided difference of first order from divided difference of first order

with the Fourier transform

Discrete analogue of the Laplacian is the following

so that its Fourier transform is

We use such discrete objects for constructing discrete Sobolev–Slobodetskii spaces to study wide class of discrete equations.

First, we introduce discrete analogue of the Schwartz space $S(h\mathbb{Z}^{2})$ as a set of discrete functions with finite semi-norms

for arbitrary $l\in\mathbb{N},{\bf k}=(k_{1},k_{2}),k_{r}\in\mathbb{N},r=1,2$,

Definition 1. A discrete distribution is called a linear continuous functional defined on the space $S(h\mathbb{Z}^{2})$.

A set of such distributions will be denoted by $S^{\prime}(h\mathbb{Z}^{2})$, and a value of the discrete distribution $f_{d}$ on the test discrete function $u_{d}\in S(h\mathbb{Z}^{2})$ will be denoted by $(f_{d},u_{d})$.

One can introduce a concept of a support for a discrete distribution. Namely, a support of the discrete function $u_{d}\in S(h\mathbb{Z}^{2})$ is a subset of the set $h\mathbb{Z}^{2}$ such that $u_{d}(\tilde{x})\neq 0$ for all points $\tilde{x}$ from this subset. For an arbitrary set $M\subset\mathbb{R}^{2}$ we denote $M_{d}=M\cap h\mathbb{Z}^{2}$, and then one says that $f_{d}=0$ in the discrete domain $M_{d}$ if $(f_{d},u_{d})=0,\forall u_{d}\in S(M_{d}),$ where $S(M_{d})\subset S(h\mathbb{Z}^{2})$ consists of discrete functions with supports in $M_{d}$. If $\widetilde{M}_{d}$ is a union of such $M_{d}$ where $f_{d}=0$ then support of the discrete distribution $f_{d}$ is the set $h\mathbb{Z}^{2}\setminus\widetilde{M}_{d}$.

Similarly [26] we can define standard operations in the space $S^{\prime}(h\mathbb{Z}^{2})$, but differentiation will be changed by divided difference of first order. These operations are described in [21] in details, a convergence is meant as a weak convergence in the space $S^{\prime}(h\mathbb{Z}^{2})$.

Example 1. If the function $f_{d}(\tilde{x})$ is locally summable then it generates the discrete distribution

But there are different possibilities, for example, analogue of the Dirac mass-function

which can not be represented by the formula (1).

Let $\zeta^{2}=h^{-2}((e^{-ih\cdot\xi_{1}}-1)^{2}+(e^{-ih\cdot\xi_{2}}-1)^{2})$. We introduce the following definition.

Definition 2. The space $H^{s}(h\mathbb{Z}^{2})$ consists of discrete distributions and it is a closure of the space $S(h\mathbb{Z}^{2})$ with respect to the norm

Let us remind that a lot of properties of such discrete spaces were studied in [7], Varying the parameter $h$ in (2) we obtain different norms which are equivalent to the $L_{2}$-norm. But constants in this equivalence depend on $h$. In our constructions all constants do not depend on $h$.

Definition 3. T͡he space $H^{s}(K_{d})$ consists of discrete distributions from $H^{s}(h\mathbb{Z}^{2})$ such that their supports belong to the set $\overline{K_{d}}$. A norm in the space $H^{s}(K_{d})$ is induced by the norm of the space $H^{s}(h\mathbb{Z}^{2})$. The space $H^{s}_{0}(K_{d})$ consists of discrete distributions $f_{d}\in S^{\prime}(h\mathbb{R}^{2})$ with supports inside of $K_{d}$, and these discrete distributions must admit a continuation into the space $H^{s}(h\mathbb{Z}^{2})$. A norm in the space $H^{s}_{0}(K_{d})$ is given by the formula

where infimum is taken for all continuations $\ell$.

The Fourier image of the space $H^{s}(K_{d})$ will be denoted by $\widetilde{H}^{s}(K_{d})$.

Let $\widetilde{A}_{d}(\xi)$ be a measurable periodic function in $\mathbb{R}^{2}$ with basic square of periods $\hbar\mathbb{T}^{2}$. Such functions we call symbols.

Definition 4. A digital pseudo-differential operator $A_{d}$ with the symbol $A_{d}(\xi)$ in the discrete quadrant $K_{d}$ is called an operator of the following type

We say that the operator $A_{d}$ is elliptic one if

A more general digital pseudo-differential operator with the symbol $\widetilde{A}_{d}(\tilde{x},\xi)$ depending on a spatial variable $\tilde{x}$

can be defined in the same way, but here we consider only operators of type (3).

We consider symbols satisfying the condition

with constants $c_{1},c_{2}$ non-depending on $h$. The number $\alpha\in\mathbb{R}$ is called an order of digital pseudo-differential operator $A_{d}$.

The following simple result can be proved easily.

Lemma 1. A digital pseudo-differential operator $A_{d}$ with the symbol $\widetilde{A}_{d}(\xi)$ is a linear bounded operator $H^{s}(h\mathbb{Z}^{2})\to H^{s-\alpha}(h\mathbb{Z}^{2})$ with a norm non-depending on $h$.

We study a solvability of the discrete equation

in the space $H^{s}(K_{d})$ assuming that $v_{d}\in H^{s-\alpha}_{0}(K_{d})$.

We will use certain special domain in two-dimensional complex space $\mathbb{C}^{2}.$ A domain of the type ${\mathcal{T}}_{h}(K)=\hbar\mathbb{T}^{2}+iK$ is called a tube domain over the quadrant $K$, and we will consider analytical functions $f(x+i\tau)$ in the domain ${\mathcal{T}}_{h}(K)=\hbar\mathbb{T}^{2}+iK$.

Let us introduce the periodic Bochner kernel similar [26]

and corresponding integral operator

Lemma 2. For the quadrant $K$ the operator $B_{h}$ has the following form

and $B_{h}$ is a linear bounded operator $H^{s}(\hbar\mathbb{T}^{2})\rightarrow H^{s}(\hbar\mathbb{T}^{2})$ for $|s|<1/2$. Moreover, the operator $B_{h}$ is a projector $\widetilde{H}^{s}(h\mathbb{Z}^{2})\rightarrow\widetilde{H}^{s}(K_{d})$.

Proof. Corresponding calculations for one-dimensional discrete cone were done in [20]. We use these evaluations adapting to our two-dimensional case. Since

then multiplying two factors and applying the Fourier property on correspondence between a product and convolution we obtain the assertion.

Boundedness of the one-dimensional operator with the kernel $h\cot\frac{hz}{2}$for $|s|<1/2$ was proved in [7], Theorem 6; two-dimensional case can be considered by the same method. $\blacksquare$

Remark 1. The operator $B_{h}$ is so called periodic bi-singular operator. Using classical results for Cauchy type integral [8, 10] one can evaluate the boundary value, but it is not important this time. Since these formulas are very huge we can do some simplifications without lost of generality. For example, we can consider the space $S_{1}(h\mathbb{Z}^{2})\subset S_{1}(h\mathbb{Z}^{2})$ with zeroes in coordinate axes and consider the space $H^{s}(h\mathbb{Z}^{2})$ as closure of the set $S_{1}(h\mathbb{Z}^{2})$ assuming that all functions of discrete variable vanish on coordinate axes. For this case the first three summands in $B_{h}$ will be zero.

Lemma 3. If $|s|<1/2$ then the space $\widetilde{H}^{s}(h\mathbb{Z}^{2})$ is uniquely represented as the direct sum

Proof. It is simple consequence of Lemma 2. Indeed, the unique representation of the function $\tilde{f}\in\widetilde{H}(h\mathbb{Z}^{2})$ is the following

A uniqueness of the such representation is possible only for $|s|<1/2$. $\blacksquare$

To describe a solvability picture for the discrete equation (5) we need some additional elements of multidimensional complex analysis. We give it in the next section.

This section is devoted to most simple case when a solution of the equation (5) exists and it is unique.

Theorem 1. Let $|\ae-s|<1/2$. Then the equation (5) has a unique solution for arbitrary right hand side $v_{d}\in H^{s-\alpha}_{0}(K_{d})$, and it is given by the formula

where $\ell v_{d}$ is an arbitrary continuation of $v_{d}$ into $H^{s-\alpha}(h\mathbb{Z}^{2})$.

Proof Let $\ell v_{d}$ be an arbitrary continuation of $v_{d}\in H^{s-\alpha}_{0}(K_{d})$ into $H^{s-\alpha}(h\mathbb{Z}^{2})$. Let us introduce the function

so that $w(\tilde{x})=0$ for $\tilde{x}\notin K_{d}$.

Now we write the equation (5) in the form

and after applying the discrete Fourier transform and periodic wave factorization we obtain

We have the following inclusions according to Lemma 1 and Lemma 2

and then according to Lemma 3 the right hand side of the equality (6) is uniquely represented by the sum

where

.

Further, we rewrite the equality (6)

and using the uniqueness of the representation as the direct sum $\widetilde{H}^{s-\ae}(K_{d})\oplus\widetilde{H}^{s-\ae}(h\mathbb{Z}^{2}\setminus K_{d})$ we conclude that both left hand side and right hand side should be zero. Thus,

and Theorem 1 is proved. $\blacksquare$

This section uses some results from [21] concerning a form of a discrete distribution supported at the origin.

Theorem 2. Let $\ae-s=n+\delta,n\in\mathbb{N},|\delta|<1/2$. Then a general solution of the equation (5) has the following form

where $Q_{n}(\xi)$ is an arbitrary polynomial of order $n$ of variables $\zeta_{k}=\hbar(e^{-ih\xi_{k}}-1),k=1,2,$ satisfying the condition (4) for $\alpha=n$, $\tilde{c}_{k}(\xi_{1}),\tilde{d}_{k}(\xi_{2}),k=0,1,\cdots,n-1,$ – are arbitrary functions from $H^{s_{k}}(h\mathbb{T}),s_{k}=s-\ae+k-1/2$.

The a priori estimate

holds, where $[\cdot]_{s_{k}}$ denotes a norm in $H^{s_{k}}(h\mathbb{T})$, and $const$ does not depend on $h$.

Proof. We start from the equality (6). Let $Q_{n}(\xi)$ be an arbitrary polynomial of order $n$ of variables $\zeta_{k}=\hbar(e^{-ih\xi_{k}}-1),k=1,2,$ satisfying the condition (4) for $\alpha=n$. We multiply the equality (6) by $Q^{-1}_{n}(\xi)$

We have in view of Lemma 1

and since $s-\ae+n=-\delta$ then according to Lemma 3 we write the unique decomposition

where

Taking into account this fact we rewrite the equality (7) in the form

and further,

Since $F^{+}_{d}(\xi)\in\widetilde{H}^{s-\ae+n}(K_{d}),F^{-}_{d}(\xi)\in\widetilde{H}^{s-\ae+n}(h\mathbb{Z}^{2}\setminus K_{d})$ then according to Lemma 1 we conclude $Q_{n}(\xi)F^{+}_{d}(\xi)\in\widetilde{H}^{s-\ae}(K_{d}),Q_{n}(\xi)F^{-}_{d}(\xi)\in\widetilde{H}^{s-\ae}(h\mathbb{Z}^{2}\setminus K_{d})$. Applying the inverse discrete Fourier transform we obtain an equality for two discrete distributions. The left hand side vanishes at least under one condition $\tilde{x}_{1}<0$ or $\tilde{x}_{2}<0$, and the right hand side vanishes under the condition .$\tilde{x}_{1}>0,\tilde{x}_{2}>0$, Thus, it should be a discrete distribution supported on sides of the discrete quadrant $\{(\tilde{x}_{1},\tilde{x}_{2})\in h\mathbb{Z}^{2}:\{\tilde{x}_{1}>0,\tilde{x}_{2}=0\}\cup\{\tilde{x}_{1}=0,\tilde{x}_{2}>0\}\}$. Using corresponding result from [21] we obtain the following form for this discrete distribution

where all summands should be elements of the space $H^{s-\ae}(h\mathbb{Z}^{2})$.

The left question is how much summands we need in the right-hand side. Counting principle is a very simple because every summand should belong to the space $\widetilde{H}^{s}(\hbar\mathbb{T}^{2})$.

Let us consider the summand $c_{k}(\xi_{1})\zeta_{2}^{k}$. Taking into account that order of $A^{-1}_{d,+}(\xi)$ is $-\ae$ we need to verify the finiteness of the $H^{s-\ae}$-norm for $c_{k}(\xi_{1})\zeta_{2}^{k}$. We have