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

TL;DR

This paper investigates discrete boundary value problems with nonlocal conditions in a quarter-plane, establishing solvability criteria and comparing discrete solutions to their continuous counterparts using Sobolev--Slobodetskii spaces.

Contribution

It introduces a discrete analogue of pseudo-differential equations in a quarter-plane and develops solvability conditions based on wave factorization, bridging discrete and continuous solutions.

Findings

Derived solvability conditions for discrete boundary value problems

Compared discrete solutions with continuous solutions in Sobolev--Slobodetskii spaces

Established a framework for analyzing nonlocal boundary conditions in discrete settings

Abstract

We consider discrete analogue of model pseudo-differential equations in discrete plane sector using discrete variant of Sobolev--Slobodetskii spaces. Starting from the concept of wave factorization for elliptic periodic symbol we describe solvability conditions for the equations and corresponding discrete boundary value problems. We give also a comparison between discrete and continuous solutions in appropriate discrete normed spaces.

Equations

$$(F_d{u}_d)(\xi )\equiv {\tilde{u}}_d(\xi )=\sum _{\tilde{x}\in h{\mathbb{Z}}^2}{e}^{-i\tilde{x}\cdot \xi }{u}_d(\tilde{x})h^2,\text{leavevmode}\text{leavevmode}\text{leavevmode}\xi \in \hslash {\mathbb{T}}^2,$$

$$(F_d^{-1}{\tilde{u}}_d)(\tilde{x})=\frac{1}{(2\pi {)}^2}\underset{\hslash {\mathbb{T}}^2}{\int }{e}^{i\tilde{x}\cdot \xi }{\tilde{u}}_d(\xi )d\xi ,\text{leavevmode}\text{leavevmode}\text{leavevmode}\tilde{x}\in h{\mathbb{Z}}^2.$$

$$||u_d|{|}_2={\left(\sum _{\tilde{x}\in h{\mathbb{Z}}^2}|u_d(\tilde{x}){|}^2{h}^2\right)}^{1/2},||{\tilde{u}}_d|{|}_2={\left(\underset{\xi \in \hslash {\mathbb{T}}^2}{\int }|{\tilde{u}}_d(\xi ){|}^{2}d\xi \right)}^{1/2}.$$

$$({\Delta }_1^{(1)}{u}_d)(\tilde{x})=h^{-1}(u_d({\tilde{x}}_1+h,{\tilde{x}}_2)-u_d({\tilde{x}}_1,{\tilde{x}}_2)),$$

$$({\Delta }_2^{(1)}{u}_d)(\tilde{x})=h^{-1}(u_d({\tilde{x}}_1,{\tilde{x}}_2+h)-u_d({\tilde{x}}_1,{\tilde{x}}_2)),$$

$$\widetilde{({\Delta }_k^{(1)}{u}_d)}(\xi )=h^{-1}(e^{-ih\cdot {\xi }_k}-1){\tilde{u}}_d(\xi ),k=1,2.$$

$$({\Delta }_1^{(2)}{u}_d)(\tilde{x})=h^{-2}(u_d({\tilde{x}}_1+2h,{\tilde{x}}_2)-2u_d({\tilde{x}}_1+h,{\tilde{x}}_2)+u_d({\tilde{x}}_1,{\tilde{x}}_2)),$$

$$({\Delta }_2^{(2)}{u}_d)(\tilde{x})=h^{-2}(u_d({\tilde{x}}_1,{\tilde{x}}_2+2h)-2u_d({\tilde{x}}_1,{\tilde{x}}_2+h)+u_d({\tilde{x}}_1,{\tilde{x}}_2)),$$

$$\widetilde{({\Delta }_k^{(2)}{u}_d)}(\xi )=h^{-2}(e^{-ih\cdot {\xi }_k}-1{)}^2{\tilde{u}}_d(\xi ),k=1,2.$$

$$({\Delta }_d{u}_d)(\tilde{x})=({\Delta }_1^{(2)}{u}_d)(\tilde{x})+({\Delta }_2^{(2)}{u}_d)(\tilde{x}),$$

$$\widetilde{({\Delta }_d{u}_d)}(\xi )=h^{-2}((e^{-ih\cdot {\xi }_1}-1{)}^2+(e^{-ih\cdot {\xi }_2}-1{)}^2){\tilde{u}}_d(\xi ).$$

$$|u_d|=\underset{\tilde{x}\in h{\mathbb{Z}}^2}{\sup }(1+|\tilde{x}|{)}^l|{\Delta }^{(\mathbf{k})}{u}_d(\tilde{x})|$$

$${\Delta }^{(\mathbf{k})}{u}_d(\tilde{x})={\Delta }_1^{k_1}{\Delta }_2^{k_2}{u}_d(\tilde{x}).$$

$$(f_d,u_d)=\sum _{\tilde{x}\in h{\mathbb{Z}}^2}{f}_d(\tilde{x})u_d(\tilde{x})h^2,\text{leavevmode}\text{leavevmode}\text{leavevmode}\forall u_d\in S(h{\mathbb{Z}}^2).$$

$$({\delta }_d,u_d)=u_d(0),$$

$$||u_d|{|}_s={\left(\underset{\hslash {\mathbb{T}}^2}{\int }(1+|{\zeta }^2|{)}^s|{\tilde{u}}_d(\xi ){|}^{2}d\xi \right)}^{1/2}.$$

$$||f_d|{|}_s^{+}=\inf ||\ell f_d|{|}_s,$$

$$(A_d{u}_d)(\tilde{x})=\sum _{\tilde{y}\in h{\mathbb{Z}}^2}{h}^2\underset{\hslash {\mathbb{T}}^2}{\int }{\tilde{A}}_d(\xi )e^{i(\tilde{x}-\tilde{y})\cdot \xi }{\tilde{u}}_d(\xi )d\xi ,\text{leavevmode}\text{leavevmode}\text{leavevmode}\tilde{x}\in K_d,$$

$$ess\text{leavevmode}\underset{\xi \in \hslash {\mathbb{T}}^2}{\inf }|A_d(\xi )|>0.$$

$$(A_d{u}_d)(\tilde{x})=\sum _{\tilde{y}\in h{\mathbb{Z}}^2}{h}^2\underset{\hslash {\mathbb{T}}^2}{\int }{A}_d(\tilde{x},\xi )e^{i(\tilde{x}-\tilde{y})\cdot \xi }{\tilde{u}}_d(\xi )d\xi ,\text{leavevmode}\text{leavevmode}\text{leavevmode}\tilde{x}\in K_d,$$

$$c_1(1+|{\zeta }^2|{)}^{\alpha /2}\le |A_d(\xi )|\le c_2(1+|{\zeta }^2|{)}^{\alpha /2}$$

$$(A_d{u}_d)(\tilde{x})=v_d(\tilde{x}),\text{leavevmode}\text{leavevmode}\text{leavevmode}\tilde{x}\in K_d,$$

$$B_h(z)=\sum _{\tilde{x}\in K_d}{e}^{i\tilde{x}\cdot (\xi +i\tau )}{h}^2,\text{leavevmode}\text{leavevmode}\text{leavevmode}\xi \in \hslash {\mathbb{T}}^2,\text{leavevmode}\text{leavevmode}\text{leavevmode}\tau \in K,$$

$$(B_h{\tilde{u}}_d)(\xi )=\underset{\tau \to 0,\tau \in K}{\lim }\frac{1}{4{\pi }^2}\underset{\hslash {\mathbb{T}}^2}{\int }{B}_h(\xi +i\tau -\eta ){\tilde{u}}_d(\eta )d\eta .$$

$$(B_h{\tilde{u}}_d)(\xi )=\frac{h^2}{8{\pi }^2}\underset{{\mathbb{T}}^2}{\int }{\tilde{u}}_d(\eta )d\eta +\underset{\tau \to 0+}{\lim }\frac{ih}{8{\pi }^2}\underset{{\mathbb{T}}^2}{\int }\cot \frac{h({\xi }_1-{\eta }_1+i{\tau }_1)}{2}{\tilde{u}}_d(\eta )d\eta +$$

$$+\underset{\tau \to 0+}{\lim }\frac{ih}{8{\pi }^2}\underset{{\mathbb{T}}^2}{\int }\cot \frac{h({\xi }_2-{\eta }_2+i{\tau }_2)}{2}{\tilde{u}}_d(\eta )d\eta -$$

$$-\underset{\tau \to 0+}{\lim }\frac{h^2}{8{\pi }^2}\underset{{\mathbb{T}}^2}{\int }\cot \frac{h({\xi }_1-{\eta }_1+i{\tau }_1)}{2}\cot \frac{h({\xi }_2-{\eta }_2+i{\tau }_2)}{2}{\tilde{u}}_d(\eta )d\eta ,$$

$$\sum _{{\tilde{x}}_k\in h{\mathbb{Z}}_{+}}{e}^{-i{\tilde{x}}_k{z}_k}h=\frac{h}{2}-\frac{ih}{2}\cot \frac{h{z}_k}{2},\text{leavevmode}\text{leavevmode}\text{leavevmode}{z}_k={\xi }_k+i{\tau }_k,\text{leavevmode}\text{leavevmode}\text{leavevmode}k=1,2.$$

$${\tilde{H}}^s(h{\mathbb{Z}}^2)={\tilde{H}}^s(K_d)\oplus {\tilde{H}}^s(h{\mathbb{Z}}^2\setminus K_d)$$

$$\tilde{f}=B_h\tilde{f}+(I-B_h)\tilde{f}.$$

Full text

1. Introduction

A theory of pseudo-differential operators and equations [9, 16, 17, 6] not so long history than other mathematical subjects of analysis. Nevertheless, these operators and related boundary value problems widely arise in a lot of applied problems in physics and technique (see, for example [14] and references therein). Discrete aspects of the theory are reflected in mathematical papers more weak [11, 2] although these studies are closely related to the theory of Fourier series [5]. In our opinion the discrete theory is very important since it permits to use computer calculations to solve concrete applied problems.

We interested in studying discrete pseudo-differential equations and their solvability in appropriate discrete functional spaces. There are certain approaches to studying discrete boundary value problems for partial differential equations including finite difference method [13, 12]. But these approaches are not applicable to studying discrete boundary value problems for elliptic pseudo-differential equations. According to this statement the first author with colleagues has started to develop discrete theory for elliptic pseudo-differential equations [21, 15]. This is main motivation, and we have started from certain canonical domains. First considerations were related to discrete $m$-dimensional space and half-space, and here we consider discrete quadrant.

We consider a special type of boundary conditions, namely integral conditions on a boundary. These conditions are nonlocal, and it seems, such conditions are artificial. But there are a lot of applied problems for partial differential equations with such boundary conditions [1, 4, 3], therefore it is natural way. Moreover, these conditions appear in a natural way to determine arbitrary functions in a general solution of an elliptic pseudo-differential equation.

2. Cones, periodic symbols, digital operators and equations

2.1. Discrete spaces and transforms

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 $u_{d}$

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

and their discrete Fourier transforms are given by formulas

[TABLE]

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

[TABLE]

[TABLE]

with the Fourier transform

[TABLE]

Discrete analogue of the Laplacian is the following

[TABLE]

so that its Fourier transform is

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

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

2.2. Symbols, operators and projectors

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

We consider symbols satisfying the condition

[TABLE]

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

[TABLE]

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]

[TABLE]

and corresponding integral operator

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

*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

[TABLE]

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

[TABLE]

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

[TABLE]

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.

3. Periodic wave factorization

This concept is a periodic analogue of the wave factorization [18]. Some first preliminary considerations and results were described in [25, 24, 23, 22].

Definition 5. A periodic wave factorization for the elliptic symbol $A_{d}(\xi)\in E_{\alpha}$ is called its representation in the form

[TABLE]

where the factors $A_{d,\neq}(\xi),A_{d,=}(\xi)$ admit analytical continuation into tube domains ${\mathcal{T}}_{h}(K),{\mathcal{T}}_{h}(-K)$ respectively with estimates

[TABLE]

[TABLE]

and constants $c_{1},c^{\prime}_{1},c_{2},c^{\prime}_{2}$ non-depending on $h$, where

[TABLE]

[TABLE]

*The number $\ae\in{\mathbb{R}}$ is called an index of periodic wave factorization. *

Unfortunately, we have no an algorithm to construct the factorization. But the are certain examples of periodic symbols which admit such factorization. We give one of them.

If $f$ is an arbitrary function of a discrete variable, $f\in S(h\mathbb{Z}^{2})$, $supp\leavevmode\nobreak\ f\subset K_{d}\cup(-K_{d})$ then we have

[TABLE]

where $\chi_{\pm}$ are indicators of $\pm K_{d}$. Applying the discrete Fourier transform we obtain the representation $\tilde{f}=\tilde{f}_{+}+\tilde{f}_{-}$, and $\tilde{f}_{\pm}$ admit an analytical continuation into ${\mathcal{T}}_{h}(\pm K)$ according to Lemma 2. Thus, we can write $\exp\tilde{f}=\exp\tilde{f}_{+}\cdot\exp\tilde{f}_{-}$, therefore we obtain periodic wave factorization with index zero for the function $\exp\tilde{f}$.

Everywhere below we assume existence of such periodic wave factorization for the symbol $A_{d}(\xi)$ with index $\ae$.

3.1. A unique solvability

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

[TABLE]

*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

[TABLE]

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

Now we write the equation (5) in the form

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

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

[TABLE]

where

[TABLE]

.

Further, we rewrite the equality (6)

[TABLE]

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,

[TABLE]

and Theorem 1 is proved. $\blacksquare$

4. Discrete boundary value problem

In this section we consider more interesting case when the equation (5) has a lot of solutions.

4.1. Form of a discrete solution

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

[TABLE]

[TABLE]

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

[TABLE]

*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 $Q^{-1}_{n}(\xi)$

[TABLE]

We have in view of Lemma 1

[TABLE]

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

[TABLE]

where

[TABLE]

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

[TABLE]

and further,

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

and the constants $a_{1},a_{2}$ do not depend on $h$.

The last summand should be $(n-1)$th because for $n$th summand we obtain a positive growth: for $k=n$ we have $s_{n}=s-\ae-n+1/2=-n-\delta+n+1/2=-\delta+1/2>0$.

A priori estimates can be obtained in the same way described in [21]. $\blacksquare$

4.2. The Dirichlet discrete boundary condition

We consider here first simple case with discrete Dirichlet boundary conditions. We suppose in this section that $\ae-s=1+\delta,|\delta|<1/2,v_{d}\equiv 0.$ It follows from Theorem 2 that we have the following general solution of the equation (5)

[TABLE]

where $c_{0},d_{0}\in H^{s-\ae-1/2}(\hbar\mathbb{Z})$ are arbitrary functions. To determine uniquely these functions we add the discrete Dirichlet conditions on angle sides

[TABLE]

Thus, we have the discrete Dirichlet problem (5),(9).

First, we apply the discrete Fourier transform to discrete conditions (9) and obtain the following form

[TABLE]

Substituting (10) into (8) we obtain the following relations

[TABLE]

[TABLE]

Let us denote

[TABLE]

and suppose that $\tilde{a}_{0}(\xi_{2}),\tilde{b}_{0}(\xi_{1})\neq 0,\forall\xi_{1}\neq 0,\xi_{2}\neq 0$.

Therefore, we have the following system of two linear integral equations with respect to two unknown functions $\tilde{c}_{0}(\xi_{1}),\tilde{d}_{0}(\xi_{2})$

[TABLE]

where we have used the following notations

[TABLE]

[TABLE]

Unique solvability conditions for the system (11) will be equivalent to unique solvability for the discrete Dirichlet problem (5),(9).

Thus, we obtain the following result.

Proposition 1. *Let $f_{d},g_{d}\in H^{s-1/2}(\mathbb{R}_{+}),s>1/2,v_{d}\equiv 0$. Then the discrete Dirichlet problem (5),(9) is reduced to the equivalent system of linear integral equations (11). *

4.3. Non-local discrete boundary condition

We consider here the $\ae-s=1+\delta,|\delta|<1/2$ for the equation (5) with different boundary conditions, namely

[TABLE]

These additional conditions will help us to determine uniquely the unknown functions $c_{0},d_{0}$ in the solution (8).

Indeed, using the discrete Fourier transform we rewrite the conditions (12) as follows

[TABLE]

Now we substitute the formulas (13) into (8). The first two equality are

[TABLE]

[TABLE]

It implies the following relations according to the third condition

[TABLE]

Then we have at least formally

[TABLE]

It is left to formulate and to prove exactly the obtained result.

Theorem 3. Let $f_{d},g_{d}\in H^{s+1/2}(h\mathbb{Z}),v_{d}\equiv 0$. Then the discrete problem (5),(12) has unique solution which is given by the formula (14).

The a priori estimate

[TABLE]

*holds with a const non-depending on $h$ *

Proof. We need to prove the a priori estimate only. Let us consider the first summand

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The second summand has the same estimate. $\blacksquare$

5. A comparison between discrete and continuous solutions

The continuous analogue of the discrete boundary value problem is the following [19].

Let $A$ be a pseudo-differential operator with the symbol $A(\xi),\xi=(\xi_{1},\xi_{2})$ satisfying the condition

[TABLE]

and admitting the wave factorization with respect to the quadrant $K$ with index $\ae$.

We consider the equation

[TABLE]

with the following additional conditions

[TABLE]

A solution of the problem (15),(16) is sought in the space $H^{s}(K)$ [18] and boundary functions are taken from the space $H^{s+1/2}(\mathbb{R}_{+})$. Such problem was considered in [19] and it has the solution

[TABLE]

under condition that the symbol $A(\xi)$ admits the wave factorization with respect to the quadrant $K$

[TABLE]

with index $\ae$ such that $\ae-s=1+\delta,|\delta|<1/2$.

To construct a discrete boundary value problem which is good approximation to (15),(16) we need to choose $A_{d}(\xi)$ and $f_{d},g_{d}$ in a special way. First, we introduce the operator $l_{h}$ which acts as follows. For a function $u$ defined in $\mathbb{R}$ we take its Fourier transform $\tilde{f}$ then we take its restriction on $\hbar T$ and periodically extend it to $\mathbb{R}$. Finally, we take its inverse discrete Fourier transform and obtain the function of discrete variable $(l_{h}u)(\tilde{x}),\tilde{x}\in h\mathbb{R}$. Thus, we put

[TABLE]

Second, the symbol of digital operator $A_{d}$ we construct in the same way. If we have the wave factorization for the symbol $A(\xi)$ then we take restrictions of factors on $\hbar\mathbb{T}^{2}$ and the periodic symbol $A_{d}(\xi)$ is a product of these restrictions. For such $f_{d},g_{d}$ and the symbol $A_{d}(\xi)$ we obtain the following result.

Theorem 4. Let $f,g\in S(\mathbb{R}),\ae>1$. Then we have the following estimate for solutions $u$ and$u_{d}$ of the continuous problem (15),(16) and the discrete one (5),(12)

[TABLE]

where the const $C(f,g)$ depends on functions $f,g$, $\beta>0$ can be an arbitrary number.

Proof. We need to compare two functions (14) and (17), more exactly their inverse discrete Fourier transform and inverse Fourier transform at points $\tilde{x}\in K_{d}$. We have

[TABLE]

[TABLE]

since according to our choice for $A_{d},f_{d},g_{d}$ the functions $\tilde{u}_{d}$ and $\tilde{u}$ coincide in points $\xi\in\hbar\mathbb{T}^{2}$.

We will estimate one summand.

[TABLE]

[TABLE]

since $\tilde{g}\in S(\mathbb{R})$. It implies the required estimate. $\blacksquare$

Conclusion

In this paper we have considered two-dimensional cone only, but the authors continue to work in multidimensional situations and we hope to obtain results similar to a discrete half-space [21, 15].

As first practical applications the authors plan to study discrete variant of a quarter-plane problem in diffraction theory [14] and elasticity theory [18]. We hope it will useful application of the developed theory.