Solvability of a nonlocal boundary value problem. (English. Russian original) Zbl 1016.35021
Differ. Equ. 37, No. 11, 1643-1646 (2001); translation from Differ. Uravn. 37, No. 11, 1565-1567 (2001).
From the text: The present paper continues the series [the author, Differ. Uravn. 35, No. 8, 1094-1100 (1999; Zbl 0973.35085); M. E. Lerner and O. A. Repin, ibid. 35, No. 8, 1087-1093 (1999; Zbl 0976.35028); ibid. 37, No. 11, 1562-1564 (2001; Zbl 1016.35016)]. For the degenerate elliptic equation
\[
y^m u_{xx}+ u_{yy}- b^2 y^m u= 0,\qquad 0< x< 1,\quad y> 0,\quad b>0,
\]
with the nonlocal condition
\[
u(0,y)= u(1,y),\qquad u_x(0,y)= 0,\quad y> 0,
\]
and the condition
\[
u(x,0)= f(x),\quad f\in C^{2+z}[0,1],\quad f(0)= f(1),\quad f'(0)= 0,
\]
it is shown that there exists a unique solution
\[
u\in C^0([0,1]\times [0,\infty])\cap C^2((0,1)\times (0,\infty)
\]
tending to zero at infinity. Moreover, a series representation of this solution is given as follows:
\[
u(x,y)= u_0(y)+ \sum^\infty_{n=1} u_n(y)\cos 2\pi nx+ \sum^\infty_{n=1} v_n(y)x\sin 2\pi nx,
\]
where for example
\[
v_n(y)= {8\over\Gamma(1/(2q))} \Biggl({\sqrt{(2\pi u)^2+ b^2}\over 2q}\Biggr)^{1/(2q)} \int^1_0 f(t)\sin 2\pi nt dt\sqrt{y} K_{1/(2q)}\Biggl({\sqrt{2\pi n)^2+ b^2}y^q\over q}\Biggr),
\]
here \(K_{1/(2q)}\) is a so-called Macdonald function.
MSC:
35J25 | Boundary value problems for second-order elliptic equations |
35J70 | Degenerate elliptic equations |
35C10 | Series solutions to PDEs |
35M10 | PDEs of mixed type |