The Dirichlet problem for elliptic equation with several singular   coefficients

Tuhtasin Ergashev

arXiv:1902.04234·math.AP·February 13, 2019

The Dirichlet problem for elliptic equation with several singular coefficients

Tuhtasin Ergashev

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TL;DR

This paper derives explicit solutions for the Dirichlet problem of a multidimensional singular elliptic equation using Lauricella hypergeometric functions, advancing understanding of such equations with multiple singular coefficients.

Contribution

It provides a unique explicit solution to the Dirichlet problem for elliptic equations with multiple singular coefficients, utilizing Lauricella hypergeometric functions and their properties.

Findings

01

Explicit solutions expressed via Lauricella hypergeometric functions

02

Use of decomposition formulas and adjacent relations for solution derivation

03

Advancement in solving multidimensional singular elliptic equations

Abstract

Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the Dirichlet problem for an elliptic equation with several singular coefficients in explicit form. When finding a solution, we use decomposition formulas and some adjacent relations for the Lauricella hypergeometric function in many variables.

Equations220

u_{xx} + u_{y y} + u_{z z} + \frac{2 α}{x} u_{x} + \frac{2 β}{y} u_{y} = 0, 0 < 2 α, 2 β < 1

u_{xx} + u_{y y} + u_{z z} + \frac{2 α}{x} u_{x} + \frac{2 β}{y} u_{y} = 0, 0 < 2 α, 2 β < 1

u_{xx} + u_{y y} + u_{z z} + \frac{2 α}{x} u_{x} + \frac{2 β}{y} u_{y} + \frac{2 γ}{z} u_{z} = 0, 0 < 2 α, 2 β, 2 γ < 1

u_{xx} + u_{y y} + u_{z z} + \frac{2 α}{x} u_{x} + \frac{2 β}{y} u_{y} + \frac{2 γ}{z} u_{z} = 0, 0 < 2 α, 2 β, 2 γ < 1

H_{α_{1}, ..., α_{n}}^{m, n} (u) \equiv i = 1 \sum m u_{x_{i} x_{i}} + k = 1 \sum n \frac{2 α _{k}}{x _{k}} u_{x_{k}} = 0,

H_{α_{1}, ..., α_{n}}^{m, n} (u) \equiv i = 1 \sum m u_{x_{i} x_{i}} + k = 1 \sum n \frac{2 α _{k}}{x _{k}} u_{x_{k}} = 0,

Γ (a + m) = Γ (a) (a)_{m}; Γ (a + \frac{1}{2}) = \frac{π Γ ( 2 a )}{2 ^{2 a - 1} Γ ( a )}, Γ (\frac{1}{2}) = π .

Γ (a + m) = Γ (a) (a)_{m}; Γ (a + \frac{1}{2}) = \frac{π Γ ( 2 a )}{2 ^{2 a - 1} Γ ( a )}, Γ (\frac{1}{2}) = π .

(a)_{m + n} = (a)_{m} (a + m)_{n}

(a)_{m + n} = (a)_{m} (a + m)_{n}

F\left({a,b;c;x}\right)\equiv F{\left[{{\begin{array}[]{*{20}c}{a,b;\hfill}\\ {c;\hfill}\\ \end{array}}x}\right]}:={\sum\limits_{m=0}^{\infty}{{\frac{{\left({a}\right)_{m}\left({b}\right)_{m}}}{{\left({c}\right)_{m}m!}}}x^{m},\,\,\,{\left|{x}\right|}<1\,}}

F\left({a,b;c;x}\right)\equiv F{\left[{{\begin{array}[]{*{20}c}{a,b;\hfill}\\ {c;\hfill}\\ \end{array}}x}\right]}:={\sum\limits_{m=0}^{\infty}{{\frac{{\left({a}\right)_{m}\left({b}\right)_{m}}}{{\left({c}\right)_{m}m!}}}x^{m},\,\,\,{\left|{x}\right|}<1\,}}

F (a, b; c; 1) = \frac{Γ ( c ) Γ ( c - a - b )}{Γ ( c - a ) Γ ( c - b )} [c \neq = 0, - 1, - 2, ...; R e (c - a - b) > 0]

F (a, b; c; 1) = \frac{Γ ( c ) Γ ( c - a - b )}{Γ ( c - a ) Γ ( c - b )} [c \neq = 0, - 1, - 2, ...; R e (c - a - b) > 0]

F (a, b; c; x) = (1 - x)^{- b} F (c - a, b; c; \frac{x}{x - 1})

F (a, b; c; x) = (1 - x)^{- b} F (c - a, b; c; \frac{x}{x - 1})

F_{A}^{(n)}\left({a,b_{1},...,b_{n};c_{1},...,c_{n};x_{1},...,x_{n}}\right)\equiv F_{A}^{(n)}{\left[{{\begin{array}[]{*{20}c}{a,b_{1},...,b_{n};\hfill}\\ {c_{1},...,c_{n};\hfill}\\ \end{array}}x_{1},...,x_{n}}\right]}

F_{A}^{(n)}\left({a,b_{1},...,b_{n};c_{1},...,c_{n};x_{1},...,x_{n}}\right)\equiv F_{A}^{(n)}{\left[{{\begin{array}[]{*{20}c}{a,b_{1},...,b_{n};\hfill}\\ {c_{1},...,c_{n};\hfill}\\ \end{array}}x_{1},...,x_{n}}\right]}

= m_{1}, ... m_{n} = 0 \sum \infty \frac{( a ) _{m_{1} + ... + m_{n}} ( b _{1} ) _{m_{1}} ... ( b _{n} ) _{m_{n}}}{( c _{1} ) _{m_{1}} ... ( c _{n} ) _{m_{n}}} \frac{x _{1}^{m_{1}}}{m _{1} !} ... \frac{x _{n}^{m_{n}}}{m _{n} !} [c_{k} \neq = 0, k = \overline{1, n}; ∣ x_{1} ∣ + ... + ∣ x_{n} ∣ < 1] .

= m_{1}, ... m_{n} = 0 \sum \infty \frac{( a ) _{m_{1} + ... + m_{n}} ( b _{1} ) _{m_{1}} ... ( b _{n} ) _{m_{n}}}{( c _{1} ) _{m_{1}} ... ( c _{n} ) _{m_{n}}} \frac{x _{1}^{m_{1}}}{m _{1} !} ... \frac{x _{n}^{m_{n}}}{m _{n} !} [c_{k} \neq = 0, k = \overline{1, n}; ∣ x_{1} ∣ + ... + ∣ x_{n} ∣ < 1] .

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y) = m, n = 0 \sum \infty \frac{( a ) _{m + n} ( b _{1} ) _{m} ( b _{2} ) _{n}}{( c _{1} ) _{m} ( c _{2} ) _{n}} \frac{x ^{m}}{m !} \frac{y ^{n}}{n !} [c_{1}, c_{2} \neq = 0; ∣ x ∣ + ∣ y ∣ < 1]

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y) = m, n = 0 \sum \infty \frac{( a ) _{m + n} ( b _{1} ) _{m} ( b _{2} ) _{n}}{( c _{1} ) _{m} ( c _{2} ) _{n}} \frac{x ^{m}}{m !} \frac{y ^{n}}{n !} [c_{1}, c_{2} \neq = 0; ∣ x ∣ + ∣ y ∣ < 1]

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y) = i = 0 \sum \infty \frac{( a ) _{i} ( b _{1} ) _{i} ( b _{2} ) _{i}}{i ! ( c _{1} ) _{i} ( c _{2} ) _{i}} x^{i} y^{i} F (a + i, b_{1} + i; c_{1} + i; x) F (a + i, b_{2} + i; c_{2} + i; y) .

F_{2} (a, b_{1}, b_{2}; c_{1}, c_{2}; x, y) = i = 0 \sum \infty \frac{( a ) _{i} ( b _{1} ) _{i} ( b _{2} ) _{i}}{i ! ( c _{1} ) _{i} ( c _{2} ) _{i}} x^{i} y^{i} F (a + i, b_{1} + i; c_{1} + i; x) F (a + i, b_{2} + i; c_{2} + i; y) .

\nabla (h) = \frac{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}, Δ (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )},

\nabla (h) = \frac{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}, Δ (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + h )}{Γ ( h ) Γ ( δ _{1} + δ _{2} + h )},

\tilde{\nabla}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )},

\tilde{\nabla}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )}{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )},

\tilde{Δ}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )}{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )},

\tilde{Δ}_{z_{1}; z_{2}, ..., z_{n}} (h) = \frac{Γ ( δ _{1} + h ) Γ ( δ _{2} + ... + δ _{n} + h )}{Γ ( h ) Γ ( δ _{1} + ... + δ _{n} + h )},

F_{A}^{(n)} (a, b_{1}, ..., b_{n}; c_{1}, ..., c_{n}; z_{1}, ..., z_{n})

F_{A}^{(n)} (a, b_{1}, ..., b_{n}; c_{1}, ..., c_{n}; z_{1}, ..., z_{n})

= m_{2}, ..., m_{n} = 0 \sum \infty \frac{( a ) _{m_{2} + ... + m_{n}} ( b _{1} ) _{m_{2} + ... + m_{n}} ( b _{2} ) _{m_{2}} ... ( b _{n} ) _{m_{n}}}{m _{2} ! ... m _{n} ! ( c _{1} ) _{m_{2} + ... + m_{n}} ( c _{2} ) _{m_{2}} ... ( c _{n} ) _{m_{n}}} z_{1}^{m_{2} + ... + m_{n}} z_{2}^{m_{2}} ... z_{n}^{m_{n}}

= m_{2}, ..., m_{n} = 0 \sum \infty \frac{( a ) _{m_{2} + ... + m_{n}} ( b _{1} ) _{m_{2} + ... + m_{n}} ( b _{2} ) _{m_{2}} ... ( b _{n} ) _{m_{n}}}{m _{2} ! ... m _{n} ! ( c _{1} ) _{m_{2} + ... + m_{n}} ( c _{2} ) _{m_{2}} ... ( c _{n} ) _{m_{n}}} z_{1}^{m_{2} + ... + m_{n}} z_{2}^{m_{2}} ... z_{n}^{m_{n}}

\cdot F (a + m_{2} + ... + m_{n}, b_{1} + m_{2} + ... + m_{n}; c_{1} + m_{2} + ... + m_{n}; z_{1})

\cdot F (a + m_{2} + ... + m_{n}, b_{1} + m_{2} + ... + m_{n}; c_{1} + m_{2} + ... + m_{n}; z_{1})

\cdot F_{A}^{(n - 1)} (a + m_{2} + ... + m_{n}, b_{2} + m_{2}, ..., b_{n} + m_{n}; c_{2} + m_{2}, ..., c_{n} + m_{n}; z_{2}, ..., z_{n}), n \in N \ {1} .

\cdot F_{A}^{(n - 1)} (a + m_{2} + ... + m_{n}, b_{2} + m_{2}, ..., b_{n} + m_{n}; c_{2} + m_{2}, ..., c_{n} + m_{n}; z_{2}, ..., z_{n}), n \in N \ {1} .

F_{A}^{(n)} (a, b_{1}, b_{2}, ...., b_{n}; c_{1}, c_{2}, ...., c_{n}; z_{1}, ..., z_{n})

F_{A}^{(n)} (a, b_{1}, b_{2}, ...., b_{n}; c_{1}, c_{2}, ...., c_{n}; z_{1}, ..., z_{n})

= (2 \leq i \leq j \leq n) m_{i, j} = 0 \sum \infty \frac{( a ) _{N (n, n)}}{m _{ij} !} k = 1 \prod n [\frac{( b _{k} ) _{M (k, n)}}{( c _{k} ) _{M (k, n)}} z_{k}^{M (k, n)} F (a + N (k, n), b_{k} + M (k, n); c_{k} + M (k, n); z_{k})],

= (2 \leq i \leq j \leq n) m_{i, j} = 0 \sum \infty \frac{( a ) _{N (n, n)}}{m _{ij} !} k = 1 \prod n [\frac{( b _{k} ) _{M (k, n)}}{( c _{k} ) _{M (k, n)}} z_{k}^{M (k, n)} F (a + N (k, n), b_{k} + M (k, n); c_{k} + M (k, n); z_{k})],

M (k, n) = i = 2 \sum k m_{i, k} + i = k + 1 \sum n m_{k + 1, i}, N (k, n) = i = 2 \sum k + 1 j = i \sum n m_{i, j} .

M (k, n) = i = 2 \sum k m_{i, k} + i = k + 1 \sum n m_{k + 1, i}, N (k, n) = i = 2 \sum k + 1 j = i \sum n m_{i, j} .

k = 2 \sum n i = 2 \sum k m_{i, k} = k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i}

k = 2 \sum n i = 2 \sum k m_{i, k} = k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i}

k = 1 \sum n M (k, n) = 2 k = 2 \sum n i = 2 \sum k m_{i, k} = 2 k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i} .

k = 1 \sum n M (k, n) = 2 k = 2 \sum n i = 2 \sum k m_{i, k} = 2 k = 1 \sum n - 1 i = k + 1 \sum n m_{k + 1, i} .

R_{m}^{n +} := {(x_{1}, ..., x_{m}) : x_{1} > 0, ..., x_{n} > 0, 1 \leq n \leq m, m \geq 2} .

R_{m}^{n +} := {(x_{1}, ..., x_{m}) : x_{1} > 0, ..., x_{n} > 0, 1 \leq n \leq m, m \geq 2} .

q_{n} (x; ξ) = γ_{n} i = 1 \prod n [(x_{i} ξ_{i})^{1 - 2 α_{i}}] \cdot r^{- 2 \tilde{α}_{n}} F_{A}^{(n)} (\tilde{α}_{n}, 1 - α_{1}, ..., 1 - α_{n}; 2 - 2 α_{1}, ..., 2 - 2 α_{n}; σ),

q_{n} (x; ξ) = γ_{n} i = 1 \prod n [(x_{i} ξ_{i})^{1 - 2 α_{i}}] \cdot r^{- 2 \tilde{α}_{n}} F_{A}^{(n)} (\tilde{α}_{n}, 1 - α_{1}, ..., 1 - α_{n}; 2 - 2 α_{1}, ..., 2 - 2 α_{n}; σ),

x := (x_{1}, ..., x_{m}), ξ := (ξ_{1}, ..., ξ_{m}), σ := (σ_{1}, ..., σ_{n});

x := (x_{1}, ..., x_{m}), ξ := (ξ_{1}, ..., ξ_{m}), σ := (σ_{1}, ..., σ_{n});

0 < 2 α_{1}, ..., 2 α_{n} < 1, \tilde{α}_{n} = \frac{m - 2}{2} + n - α_{1} - ... - α_{n};

0 < 2 α_{1}, ..., 2 α_{n} < 1, \tilde{α}_{n} = \frac{m - 2}{2} + n - α_{1} - ... - α_{n};

γ_{n} = 2^{2 \tilde{α}_{n} - m} \frac{Γ ( α ~ _{n} )}{π ^{m /2}} j = 1 \prod n \frac{Γ ( 1 - α _{j} )}{Γ ( 2 - 2 α _{j} )},

γ_{n} = 2^{2 \tilde{α}_{n} - m} \frac{Γ ( α ~ _{n} )}{π ^{m /2}} j = 1 \prod n \frac{Γ ( 1 - α _{j} )}{Γ ( 2 - 2 α _{j} )},

r^{2} = i = 1 \sum m (x_{i} - ξ_{i})^{2}, r_{k}^{2} = (x_{k} + ξ_{k})^{2} + i = 1, i \neq = k \sum m (x_{i} - ξ_{i})^{2}, σ_{k} = 1 - \frac{r _{k}^{2}}{r ^{2}} .

r^{2} = i = 1 \sum m (x_{i} - ξ_{i})^{2}, r_{k}^{2} = (x_{k} + ξ_{k})^{2} + i = 1, i \neq = k \sum m (x_{i} - ξ_{i})^{2}, σ_{k} = 1 - \frac{r _{k}^{2}}{r ^{2}} .

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Full text

Ergashev T.G.

THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATION WITH SEVERAL SINGULAR COEFFICIENTS

Abstract. Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the Dirichlet problem for an elliptic equation with several singular coefficients in explicit form. When finding a solution, we use decomposition formulas and some adjacent relations for the Lauricella hypergeometric function in many variables, as well as the values of some multidimensional improper integrals.

Keywords $:$ Dirichlet problem, multidimensional elliptic equations with several singular coefficients, decomposition formulas, Lauricella hypergeometric function in many variables.

AMS Mathematics Subject Classification: 35A08

1 Introduction

It is known that the theory of boundary value problems for degenerate equations and equations with singular coefficients is one of the rapidly developing parts of the modern theory of partial differential equations, which is encountered in solving many important questions of an applied nature, for example, [3, 10]. A detailed bibliography and summary of studies of the basic boundary-value equations for degenerate equations of various types, in particular, for elliptic equations with singular coefficients, can be found in monographs [4, 13, 32, 33]. In addition, generalized axisymmetric potentials have been studied using various methods [2, 9, 11, 12, 18, 20, 22, 34]. Omitting a huge bibliography in which various local and non-local boundary-value problems for mixed-type equations containing elliptic equations with singular coefficients are studied, we note some papers which are close to the present work. In the work [14], fundamental solutions were constructed for the bi-axially symmetric Helmholtz equation, and in [28, 29, 30] the explicit solutions of the Dirichlet and Dirichlet-Neumann problems in one quarter of a circle was found.

Dirichlet and Dirichlet-Neumann problems for elliptic equation with one singular coefficient in some part of ball were investigated by Agostinelli [1] and Olevskii [26]. Recently, Nazipov published a paper devoted to the investigation of the Tricomi problem in a mixed domain consisting of hemisphere and cone [23]. Fundamental solutions for the following three-dimensional elliptic equations with two and three singular coefficients

[TABLE]

and

[TABLE]

were constructed, respectively, in [24] and [15]. For equations (1.1) and (1.2), the Dirichlet, Neumann and Holmgren problems [19, 31, 25] were solved in some parts of the ball.

In this paper, we study the Dirichlet problem for the equation

[TABLE]

where $m\geq 2,0<n\leq m;\quad\alpha_{k}$ are constants with $0<2\alpha_{k}<1.$ Hereinafter in the present work, unless there are other reservations, the natural number $k$ will vary from 1 to $n$ , inclusive.

2 Preliminaries

Below we give some formulas for Euler gamma-function, Gauss hypergeometric function, multiple Lauricella hypergeometric function (that is, Lauricella hypergeometric function in several variables), which will be used in the next sections.

It is known that the Euler gamma-function $\Gamma\left({a}\right)$ has properties [7, pp. 17-19, (2), (10), (15)]

[TABLE]

Here $\left({a}\right)_{m}$ is a Pochhammer symbol, for which an equality

[TABLE]

is true [7, p.67, (5)].

A function

[TABLE]

is known as the Gauss hypergeometric function and an equality

[TABLE]

holds [7, c.73, (14)]. Moreover, the following autotransformer formula [7, p.76, (22)]

[TABLE]

is valid.

The Lauricella hypergeometric function in $n$ variables has a form [21]

[TABLE]

For a given multiple hypergeometric function, it is useful to fund a decomposition formula which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. Burchnall and Chaundy [5, 6] systematically presented a number of expansion and decomposition formulas for some double hypergeometric functions in series of simpler hypergeometric functions. For example, the Appell function

[TABLE]

has the expansion [5]

[TABLE]

The Birchnell-Cendi method, which is limited to functions of two variables, is based on the following mutually inverse symbolic operators [5]

[TABLE]

where ${\rm\delta}_{1}=x{\frac{{\partial}}{{\partial x}}}$ ${\rm\delta}_{2}=y{\frac{{\partial}}{{\partial y}}}$ .

In order to generalize the operators $\nabla\left({h}\right)$ and $\Delta\left({h}\right)$ , defined in (2.6), A.Hasanov and H.M.Srivastava [16, 17] introduced the operators

[TABLE]

where ${\rm\delta}_{k}=z_{k}{\frac{{\partial}}{{\partial z_{k}}}}\,,$ with the help of which they managed to find decomposition formulas for a whole class of hypergeometric functions in several variables. For example, the hypergeometric Lauricelli function $F_{A}^{\left({n}\right)}$ , defined by formula (2.5) has the decomposition formula [16]

[TABLE]

However, due to the recurrence of formula (2.9), additional difficulties may arise in the applications of this expansion. Further study of the properties of operators (2.7) and (2.8) showed that formula (2.9) can be reduced to a more convenient form.

Lemma 1 [8]. The following decomposition formula holds true at $n\in{\rm N}\backslash{\left\{{1}\right\}}$

[TABLE]

where

[TABLE]

The formula (2.10) is proved by the method mathematical induction [8].

It should be noted here that the sum ${\sum\limits_{k=1}^{n}{M(k,n)}}$ has the parity property, which plays an important role in the calculation of the some values of hypergeometric functions. In fact, by virtue of equality

[TABLE]

we obtain

[TABLE]

In the present paper, ${\rm R}_{m}^{n+}$ denotes $1/2^{n}$ part of the Euclidean space ${\rm R}^{m}$ :

[TABLE]

All the fundamental solutions of equation (1.3) in the domain ${\rm R}_{m}^{n+}$ were found in [8], and we will use one of these solutions in the study of the problem:

[TABLE]

where

[TABLE]

It is easy to verify that the fundamental solution $q_{n}\left({{\rm x};{\rm\xi}}\right)$ has the property

[TABLE]

3 Formulation of the problem and the uniqueness of the solution

Let $\Omega\subset{\rm R}_{m}^{n+}$ be a finite simple-connected domain bounded by planes $x_{1}=0,...,x_{n}=0$ and by smooth $m-$ dimensional surface $S$ . The intersection of this surface with plane $x_{k}=0$ is denoted by $\chi_{k}$ . Designate as the domain $S_{k},$ a hyperplane $Ox_{1}...x_{k-1}x_{k+1}...x_{m},$ bounded by $x_{k}=0$ ( $0<x_{l}<a_{l},\quad-b_{s}<x_{s}<c_{s},\quad l=\overline{1,n},\,\,l\neq k;\,\,n<s\leq m)$ and by a curve $\chi_{k}$ . Here $a_{l},\,\,b_{s}$ and $c_{s}$ are positive constants. We introduce the notation:

[TABLE]

Dirichlet problem. To find a function $u\left({{\rm x}}\right)\in C\left({\bar{\Omega}}\right)\cap C^{2}\left({\Omega}\right)$ , satisfying equation (1.3) in $\Omega$ and conditions

[TABLE]

where $\tau_{k}\left({\tilde{x}_{k}}\right)$ and $\varphi\left({{\rm x}}\right)$ are given continuous functions fulfilling the following matching conditions:

[TABLE]

One can readily check the validity of the following relation

[TABLE]

Let $\Omega_{\varepsilon}$ be a sub-domain of $\Omega$ at a distance $\varepsilon>0$ from its boundary $\partial\Omega={\bigcup\limits_{i=1}^{n}{S_{i}}}\cup S$ and

[TABLE]

${\rm{\bf n}}$ is outer normal to $\partial\Omega$ .

Integrate both sides of above given equality on the domain $\Omega_{\varepsilon}$ and use the classical formula of Gauss-Ostrogadsky:

[TABLE]

Using the equality

[TABLE]

we obtain

[TABLE]

Applying again the formula of Gauss-Ostrogradsky to this equality and letting $\varepsilon\to 0$ , we get

[TABLE]

where

[TABLE]

To prove the uniqueness of the solution, as usual, we suppose that the problem has two $v,\quad w$ solutions. Denoting $u=v-w$ we have that satisfies homogeneous Dirichlet problem ( $\tau_{k}=0,\quad\varphi=0)$ . Further we have to prove that the homogeneous problem has only trivial solution. In this case from (3.5) one can easily get

[TABLE]

Hence, it follows that $u_{x_{1}}=...=u_{x_{m}}=0,$ which implies that $u$ is a constant function. Considering homogeneous conditions (3.1) and (3.2), we conclude that $u\left({{\rm x}}\right)\equiv 0$ in $\overline{\Omega}$ .

4 The existence of the solution

We prove the existence of the solution in a special case of the domain $\Omega$ in order to get the solution in an explicit form. Assume $R=a_{k}=b_{k}=c_{k}$ and let

[TABLE]

We find a solution of considered problem using method Green’s functions [27] . Therefore, first we give a definition of Green’s function for the formulated problem.

Definition. We call the function $G\left({{\rm x};{\rm\xi}}\right)$ as Green’s function of the Dirichlet problem, if it satisfies the following conditions:

this function is a regular solution of equation (1.3) in the domain $\Omega$ , expect at the point ${\rm\xi}$ , which is any fixed point of $\Omega$ ;

it satisfies boundary conditions

[TABLE]

it can be represented as

[TABLE]

where $q_{n}\left({{\rm x};{\rm\xi}}\right)$ is the fundamental solution found earlier (see a formula (2.13)), function

[TABLE]

is a regular solution of equation (1.3) in the domain $\Omega$ . Here

[TABLE]

Excise a small ball with its center at ${\rm\xi}$ and with radius $\rho>0$ from the domain $\Omega$ . Designate the sphere of the excised ball as $C_{\rho}$ and by $\Omega_{\rho}$ denote the remaining part of $\Omega$ .

Applying formula (3.4), we obtain

[TABLE]

where

[TABLE]

First, we consider an integral

[TABLE]

Taking (4.1) into account we rewrite it as follows

[TABLE]

Using the formula of differentiation

[TABLE]

and the following adjacent relation

[TABLE]

we calculate

[TABLE]

Below we get detailed evaluations for ${\frac{{\partial q_{n}\left({{\rm x};{\rm\xi}}\right)}}{{\partial x_{i}}}}$ , when $1\leq i\leq n$ . Indeed, using the formula of differentiation (4.3), we get

[TABLE]

Considering adjacent relation (4.4) we obtain

[TABLE]

Similarly we calculate ${\frac{{\partial q_{n}\left({{\rm x};{\rm\xi}}\right)}}{{\partial x_{i}}}}$ , when $n+1\leq i\leq m:$

[TABLE]

Taking (4.5), (4.6) and (4.7) into account we calculate

[TABLE]

Now consider the integral

[TABLE]

where

[TABLE]

We use the following generalization spherical system of coordinates:

[TABLE]

Then we have

[TABLE]

where

[TABLE]

First we evaluate $F_{A}^{\left({n}\right)}\left({\tilde{\sigma}}\right)$ . For this aim we use decomposition formula (2.10) and then auto-transformation formula (2.4):

[TABLE]

where $M(k,n)$ and $N(k,n)$ are an expressions defined in (2.11).

After the elementary evaluations we find

[TABLE]

where

[TABLE]

It is easy to see that when $\rho\to 0$ the function $\aleph$ becomes an expression that does not depend on ${\rm x}$ and ${\rm\xi}$ . Indeed, taking into account the equality (2.12), we have

[TABLE]

Applying now the summation formula (2.3) to each hypergeometric function $F\left({a,b;c;1}\right)$ in the sum (4.9), we get

[TABLE]

Taking into account the identity

[TABLE]

we obtain

[TABLE]

Using the properties (2.1) of gamma-function $\Gamma\left({z}\right)$ , property (2.2) of the Pochhammer symbol and summation formula (2.3) for hypergeometric function $F\left({a,b;c;z}\right)$ , the formula (4.10) is proved by the method mathematical induction.

Now we consider an integral

[TABLE]

with elementary transformations it is not difficult to establish that

[TABLE]

If we take into account (4.8), (4.11), (4.12) and (2.14), then we will have

[TABLE]

By similar evaluations one can get that

[TABLE]

If we consider an integral

[TABLE]

using above given algorithm for evaluations (in this case calculations will be more simple), we can prove that

[TABLE]

Now from (4.2) we can write the solution of the Dirichlet problem as follows:

[TABLE]

The particular values of Green’s function are given by

[TABLE]

where

[TABLE]

Constant $\gamma_{n}$ has the form (2.14).

Hence, the main result of the paper is formulated as the following theorem:

Theorem. If $\tau_{k}\left({\tilde{x}_{k}}\right)\in C^{2}\left({S_{k}}\right)$ and $\varphi\left({{\rm x}}\right)\in C^{2}\left({S}\right)$ are given functions fulfilling the matching conditions (3.3), then the Dirichlet problem has unique solution represented by formula (4.13).

REFERENCES

Список литературы

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Ergashev Tuhtasin Gulamjanovich

Doctoral student, Uzbekistan Academy of Sciences, V.I.Romanovskiy Institute of Mathematics,

81, Mirzo Ulugbek street, Tashkent, 100170

[email protected]

Bibliography34

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] C.Agostinelli, Integrazione dell’equazione differenziale u x x + u y y + u z z + x − 1 u x = f subscript 𝑢 𝑥 𝑥 subscript 𝑢 𝑦 𝑦 subscript 𝑢 𝑧 𝑧 superscript 𝑥 1 subscript 𝑢 𝑥 𝑓 u_{xx}+u_{yy}+u_{zz}+x^{-1}u_{x}=f e problema analogo a quello di Dirichlet per un campo emisferico, Atti della Accademia Nazionale dei Lincei, s. 6, vol. XXVI (1937) 7-8.
2[2] A.Altin, Solutions of type r m superscript 𝑟 𝑚 r^{m} for a class of singular equations, Intern. Jour. of Math. Sc., 5(3)(1982) 613-619
3[3] L.Bers, Mathematical aspects of subsonic and transonic gas dynamics, New York, London (1958).
4[4] A.V.Bitsadze, Some classes of partial differential equations, Nauka, Moscow, (1981) (In Russian).
5[5] J.L.Burchnall, T.W.Chaundy, Expansions of Appell’s double hypergeometric functions, Quart. J. Math. Oxford, 11(1940) 249-270.
6[6] J.L.Burchnall, T.W.Chaundy, Expansions of Appell’s double hypergeometric functions, II, Quart. J. Math. Oxford, 12(1941) 112-128.
7[7] A.Erdelyi, W.Magnus, F.Oberhettinger, F.G.Tricomi, Higher Transcendental Functions, Vol.I, Mc Graw-Hill Book Company, New York, Toronto and London (1953).
8[8] T.G.Ergashev, Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients, Ar Xiv.org 1805.03826(2018) 1-9.