On Certain Elliptic Problems in Sectorial Domains

Abstract

A certain conjugation problem for an elliptic pseudo-differential equation in a plane sector is studied in Sobolev–Slobodetskii spaces. Using wave factorization for an elliptic symbol with concrete index we consider Dirichlet and Neumann conditions on sector sides. It permits to reduce the considered boundary value problem to a system of one-dimensional linear integral equations. For a special case it is possible further to reduce the mentioned system to a system of linear algebraic equations with respect to $8$ unknown functions.

Appendices

Appendix

PROPERTIES OF THE MELLIN TRANSFORM

For convenience of a reader we will give here certain facts on the Mellin transform and will show how it can be applied to special integral equations. The Mellin transform is defined by formula

$$\hat{f}(s)=\int\limits_{0}^{\infty}f(x)x^{s-1}dx,\quad s=\sigma+i\tau,$$

at least for functions $f(x)\in C_{0}^{\infty}(\mathbb{R}_{+}).$ The integral converges for all complex $s$ and it is an entire analytic function. If we change variable $x=e^{t}$, then the Mellin transform passes into the Fourier transform of function $f(e^{t})$:

$$\hat{f}(s)=\int\limits_{-\infty}^{\infty}e^{(\sigma+i\tau)}f(e^{t})dt,\quad s=\sigma+i\tau.$$

Thus, all properties of the Mellin transform can be obtained from corresponding properties of the Fourier transform. Particularly, the inversion formula of the Mellin transform for $f(x)\in C_{0}^{\infty}(\mathbb{R})$ has the following form

$$f(x)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\hat{f}(s)t^{-s}d\tau,\quad s=\sigma+i\tau.$$

Parceval equality for Mellin transform

$$\int\limits_{0}^{+\infty}t^{2\sigma-1}|f(t)|^{2}dt=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}\left|\hat{f}(s)\right|^{2}d\tau,\quad s=\sigma+i\tau,$$

particularly, for $\sigma=1/2$ we have

$$\int\limits_{0}^{+\infty}|f(t)|^{2}dt=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}\left|\hat{f}(s)\right|^{2}d\tau,\quad s=1/2+i\tau,$$

or, in other words,

$$\int\limits_{0}^{+\infty}|f(t)|^{2}dt=\frac{1}{2\pi i}\int\limits_{1/2-i\infty}^{1/2+i\infty}\left|\hat{f}(s)\right|^{2}ds,$$

meaning the right integral as

$$\lim\limits_{y\to\infty}\int\limits_{1/2-iy}^{1/2+iy}\left|\hat{f}(s)\right|^{2}ds.$$

If we have the integral

$$\int\limits_{0}^{+\infty}K(t_{1},t_{2})u(t_{1})dt_{2},$$

in which the kernel $K(t_{1},t_{2})$ is a homogeneous function of order $-1$, then after applying the Mellin transform we obtain the following expression

$$\int\limits_{0}^{+\infty}t_{1}^{\lambda-1}\left(\int\limits_{0}^{+\infty}K(t_{1},t_{2})u(t_{1})dt_{2}\right)dt_{1}.$$

The change of variable in the inner integral $t_{1}=xt_{2}$ leads to the following integral

$$\int\limits_{0}^{+\infty}t_{2}^{\lambda-1}x^{\lambda-1}\left(\int\limits_{0}^{+\infty}t_{2}K(xt_{2},t_{2})u(t_{2})dt_{1}\right)dx,$$

and after rearrangements of integrals we obtain the following product

$$\int\limits_{0}^{+\infty}t_{2}^{\lambda-1}u(t_{2})dt_{2}\int\limits_{0}^{+\infty}x^{\lambda-1}K(x,1)dx=\hat{u}(\lambda)\hat{K}(\lambda),$$

where $\hat{u}$ denotes the Mellin transform of $u$.