FREDHOLM'S alternative for the DIRICHLET problem generalized according to WIENER of the reduced wave equation
✍ Scribed by Günter Albinus
- Publisher
- John Wiley and Sons
- Year
- 1977
- Tongue
- English
- Weight
- 168 KB
- Volume
- 79
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
FREDHOLM'S alternative for the DIRICHLET problem generalized according to WIENER of the reduced wave equation By GUNTER ALBINUS in Berlin (Eingegangen am 15.8.1975) 0. Introduction. Let QcR" be a bounded region. The DIRICHLET problem ( A , 9, f ) for the LAPLACE operator in the region 9 with boundary values f is uniquely solvable only in a generalized sense. N. WIENER has constructed a bounded harmonic function Hf in 9 in such way that for certain points x on the boundary aQ of Q the condition lim Hf (2) = f ( 2 ) 2 -Z is fulfilled. The set K = { z ~a 9 : limHf(x)=f(z) for uZZ fcC(%Q)} 2-2 is called the set of regular points of 9. Let y be a real number and let f be a continuous function on aQ. We denote by ( A + y , 9, f ) K the generalized DIRICHLET problem according to WIENER. It consists in determinating a bounded classical solution u of the equation AU + y~ = 0 in 9, which fulfils the boundary condition lim u ( z ) = f (2) X -z in each regular point of 9. Let us denote by @ the fundamental solution Ic, 1x12-n n > 2 , of the negative LAPLACE operator and by G the GREEN function G ( z , Y ) : = @ ( x -Y ) -H [@(.-Y) I 891 (x)=: @ ( x -~) -H ( x , y )