CHAPTER 2
Preliminaries o n Capacitie s
Let Lipl
oc(Rn)
b e th e se t o f al l comple x value d locall y Lipschit z function s
in R
n
(wit h exponen t 1). Th e fundamenta l solutio n $ fo r th e Laplac e equatio n
Af = 0 in R
n
i s define d b y $(x) = i JtT-2 ? n 3 , wher e a
n
0 is a constant .
$
W
=
- ^
l o
g R for n = 2 .
We now introduce the class of admissible functions fo r the definition o f Lipschitz
harmonic capacity . Fo r a compact se t E i n R
n,
se t
L(E, 1) := { / Lipl
c(Rn)
supp(A/) c E, ||V/||o o 1, V/(oo) = 0},
where supp(A/ ) i s the suppor t o f the distributio n A/ .
We shal l conside r function s modul o constant s i n L(E, 1), meanin g tha t w e
shall writ e / = g fo r / , g G L(E, 1) i f / g i s constant . Not e tha t eac h functio n
/ G L(JE7, 1) is harmonic i n R
n
\ E an d / = $ * A/ - f constant .
The Lipschit z harmoni c capacit y o f the se t E i s defined b y
7 ( £ ) : = s u p { | A / , l | : / L ( i 5 , l ) } ,
where (a s usual) (5 , /?) means th e actio n o f the distributio n wit h compac t suppor t
on a smooth tes t function .
Letting a(n 1) be the volum e o f the uni t bal l in R n _ 1 , w e define th e ( n 1)-
dimensional Hausdorf f measur e fo r a subset E i n R n b y
oo
Hn-l{E) := liminf{V a(n-
1 ) T ? - 1
: E C
1
5(a;
i
,r
i
), n 5} .
i=l
Then th e restrictio n o f 7Y n_1 t o sufficientl y regula r hypersurface s give s the surfac e
measure.
A se t E i n R n i s called removabl e fo r Lipschit z harmoni c function s i f fo r eac h
domain D i n MJ 1 ever y locall y Lipschit z functio n / : D C whic h i s harmonic i n
D \ E i s harmonic i n D. I t i s proved i n [35 ] tha t
A set E i s removable fo r Lipschit z harmoni c function s i f and onl y i f y(E) = 0 .
A set E i n M71 is called removable for subharmonic Lipschitz harmoni c function s
if fo r eac h domai n D i n R n ever y locall y Lipschit z functio n / : D C whic h i s
harmonic i n D \ E an d subharmoni c i n D i s harmonic i n D.
Such set s als o ca n b e characterize d usin g a natura l capacity . Le t u s conside r
7 +
( £ ) : = sup{|(A/, 1)| :/ L(E, 1), A/ =
M
e M +(E)} ,
where M+(E) i s the famil y o f positive Bore l measure s o n E. Similarl y t o [35] , one
can se e tha t
A set E i s removable fo r subharmoni c Lipschit z harmoni c function s
if and onl y i f 7+(£) = 0 .
7
http://dx.doi.org/10.1090/cbms/100/02
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