Boundary Value Problems for the Three-Dimensional Helmholtz Equation in the Unbounded Octant, Square and Half Space

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Introduction

It’s known, that the Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis [15].

The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieu’s differential equation.

Two- and more-dimensional Helmholtz equations

$\sum _{j=1}^n\frac{{\partial }^{2}u}{\partial x_j^2}+\mu u=0$

and their related boundary-value problems have been investigated in a large number of papers [1–3, 12–14].

On the other hand, the equation has important applications. In 1952 Kapilevich [18] has solved Dirichlet and Neumann problems for multidimensional Helmholtz equation with singular coefficient

$\sum _{j=1}^n\frac{{\partial }^{2}u}{\partial x_j^2}+\frac{2\alpha }{x_1}\frac{\partial u}{\partial x_1}+\mu u\text{\hspace{0.05em}=0, \hspace{0.05em}}0<2\alpha <1$ (1)

in the half-space. In 1978 Marichev [19] has investigated two-dimensional Helmholtz equation with two singular coefficients

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{2\alpha }{x}\frac{\partial u}{\partial x}+\frac{2\beta }{y}\frac{\partial u}{\partial y}+\mu u=0,\text{\hspace{0.05em} 0<\hspace{0.05em}2}\alpha \mathrm{,}2\beta <1$

There are many works [6–9, 11] devoted to the Helmholtz equation (1). For instance, in the work [10] the Dirichlet problem fot the singular Helmholtz equation (1) for $\mu =-{\lambda }^2$ is solved explicitly.

Generally speaking, our further goal is to pose and investigate boundary value problems for Helmholtz equation with three singular coefficients

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}+\frac{2\alpha }{x}\frac{\partial u}{\partial x}+\frac{2\beta }{y}\frac{\partial u}{\partial y}+\frac{2\gamma }{y}\frac{\partial u}{\partial z}+\mu u=0, 0\le 2\alpha \mathrm{,}2\beta \mathrm{,2}\gamma <1$ (2)

 in some infinite domains.

For beginning, in the present paper, we study the Dirichlet, Neumann and Dirichlet-Neumann boundary value problems for equation (2) at $\alpha =\beta =\gamma =0$ and $\mu =-{\lambda }^2$ in the unbounded domains – in an octant, square of the space and half-space.

The Dirichlet problem $D_3^3$ for the Helmholtz equation in the first octant

Let us consider the following Helmholtz equation

$u_{xx}+u_{yy}+u_{zz}-{\lambda }^{2}u\text{\hspace{0.05em}=0\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}}$ (3)

in the infinite domain ${\Omega }_3\equiv \left\{\left(x\mathrm{,}y\mathrm{,}z\right):x>0,y>0,z>0\right\}$

The Dirichlet problem $D_3^3$. Find a regular solution $u\left(x\mathrm{,}y\mathrm{,}z\right)$ to the Helmholtz equation (3) in the class of functions $C\left(\overline{{\Omega }_3}\right)\cup C^2\left({\Omega }_3\right)$, satisfying the conditions

$u\left(x\mathrm{,}y,0\right)={\tau }_1\left(x\mathrm{,}y\right), 0\le x\mathrm{,}y<\infty \mathrm{,}$ (4)

$u\left(x\mathrm{,0,}z\right)={\tau }_2\left(x\mathrm{,}z\right),0\le x\mathrm{,}z<\infty \mathrm{,}$ (5)

$u\left(0,y\mathrm{,}z\right)={\tau }_3\left(y\mathrm{,}z\right),0\le y\mathrm{,}z<\infty \mathrm{,}$ (6)

$\underset{R\to \infty }{\lim }u\left(x\mathrm{,}y\mathrm{,}z\right)=0, R=\sqrt{x^2+y^2+z^2}\mathrm{,}$ (7)

作者简介

Zafarjon Arzikulov

编辑信件的主要联系方式.
Email: zafarbekarzikulov1984@gmail.com
ORCID iD: 0009-0004-2965-4566

Doktorant, Fac. of Phys. & Department of Higher Mathematics

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