CHAPTE R 1
Motivation an d statemen t o f th e mai n result s
This chapte r ca n b e rea d separatel y fro m th e res t o f th e book ; i t i s intende d
to giv e a quic k overvie w o f th e result s presente d i n th e nex t chapters , an d t o plac e
them withi n th e curren t stat e o f knowledg e i n potentia l theor y an d boundar y valu e
problems.
As motivatio n fo r th e th e topic s t o b e discusse d i n thes e notes , le t u s conside r
the solutio n t o th e classica l Dirichlet Problem fo r a connected , ope n se t ft C M n,
i.e. th e unique , smoot h functio n u G C°°(£2), satisfyin g
j Au = 0 i n ft
( ' j \u\
dn
= fe C
b
(dft).
Here Cb(dft) denote s th e spac e o f continuous , bounde d function s define d o n
dft. Th e boundar y dft i s calle d regular i f i n additio n u G Cb(ft) an d achieve s
the boundar y dat a continuously . Th e maximu m principl e yields , vi a th e Ries z
representation theorem , th e representatio n formul a
(1.2) u(x*)= / f{Q)du x*(Q), fo r ever y x * G fi,
J an
where th e famil y o f probability , positiv e measure s {dw x*} i s th e harmonic measure
corresponding t o th e Dirichle t proble m i n ft. Fo r th e precis e definitio n o f {duu x*}
see [42 , Definitio n 1.2.6]. Whe n ther e i s n o ris k o f ambiguity , w e fix x * G ft an d
denote duj duox*. Roughl y speaking , th e smoothnes s o f the domai n determine s th e
smoothness o f solution s t o th e Dirichle t problem . I f th e domai n ft i s "sufficientl y
regular" (se e fo r instanc e Theore m 1.14 , Theore m 6.11, [16] o r [18]) the n harmoni c
measure an d surfac e measur e da ar e mutuall y absolutel y continuous . I n thi s cas e
we denot e th e Poisson kernel fo r th e domai n ft b y
k{''X*) = ^a~'
For instance , i f ft i s a smoot h domai n the n
c)C
(1-3) du x*{Q) = ^{Q,x*)do-{Q),
dnQ
where G denote s th e Gree n functio n fo r ft, UQ is th e oute r norma l a t Q G dft an d
da i s th e surfac e measur e o n dft. Fo r mor e detail s se e e.g . [30 , Sec . 2.4] . Le t ft b e
l
http://dx.doi.org/10.1090/ulect/035/01
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