4. CHARACTERIZATIO N (1)
Q
AN D FLATNES S 5
B(r,Q)
FIGURE
1. Reifenber g fla t domain .
Note that thi s definitio n i s significant onl y fo r S small. W e will always assum e
S | an d motivat e thi s choic e later . Se t
(1.15) 0(r , Q) = in f ( -D [ B(r, Q) n dQ ; S(r , Q) n L(r , Q) ]
QeL(r,Q) [ r
and
(1.16) 0 K(r)= su p 0(r,Q) .
QednnK
We say that f 2 is vanishing Reifenberg, i f it i s 6—Reifenberg flat (S 1/8), an d
moreover, fo r ever y compac t se t K C
Mn+1
w e hav e
(1.17) limsup9
K
(r) = 0 .
In a certai n sense , suc h domain s ar e "locally flat" . T o simplif y th e notation ,
when ther e i s no ambiguit y w e will denote L(r, Q) b y LQ.
To hel p clarif y th e definition s o f Reifenber g flat an d vanishin g Reifenber g do -
mains w e present a fe w examples .
We begi n wit h th e tw o mos t basi c examples , namel y th e dis c an d th e wedge ,
that wil l emphasize th e relevanc e of the vanishing condition (1.17) . Suc h example s
serve a s the prototyp e o f the classe s o f smooth an d Lipschit z domain s respectivel y
(see Definition 3.15).
Example 1^.1. Fo r th e wedg e centered a t a point w an d wit h angl e 0 ip 7r,
we let (3 = (7T—^)/2 , see figure 2. Elementar y computations yield infL(r?lt;) 0(r , w) =
rs'mfi. Thi s show s tha t a s th e wedg e open s up , convergin g t o a line , th e domai n
becomes vanishin g Reifenberg . O n th e othe r hand , th e rati o 6(r,w)/r canno t b e
made smalle r b y decreasin g th e scal e r .
Example ^.2. I n th e cas e o f th e dis c o f radiu s R 0 w e le t w denot e a
boundary point , se e figure 3 . Choosin g L(r,w) t o b e th e tangen t w e deriv e th e
formula
6(r, w) = ma x —^ ; V 'R? + r
2
- R ,
from whic h i t immediatel y follow s tha t th e dis c i s vanishing Reifenberg .
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