7.2. COMPARABLE DIRICHLET INTEGRALS 897.2. Comparable Dirichlet integralsWe observed in Theorem 5.2.3 that if D C R^2 is a simply connected domain,
then each function u in Li(D) can be extended to a function v in Li(R^2 ) if and
only if D is a quasidisk. We consider here a similar problem, namely to determine
the Jordan domains D c R^2 for which functions harmonic in D and D with equal
boundary values on aD have comparable Dirichlet integrals.
If Dis a disk or half-plane and if u and v are harmonic in D and D, respectively,
with continuous and equal boundary values, then
(7 .2.1) j [grad u[^2 dm = j [grad v[^2 dm.
D D
The converse is also true. If D is a Jordan domain and if (7.2.1) holds for
each pair of functions u and v harmonic in D and D with continuous and equal
boundary values, then D is a disk or half-plane (Hag [77]).
The following example shows what we should expect if D is a quasidisk.
EXAMPLE 7.2.2. If D is a sector of angle a, then there exist functions u and
v harmonic in D and D*, respectively, with continuous and equal boundary values
such that
(7.2.3) j^2
2 7f - a + sin a j 2
[grad v(z)[ dm =. [grad u(z)[ dm.
D* a - sma D
For a proof of this suppose that D = S (a) and let
1-z
f(z) =
1
+ z, u(z) = Re(f(z))for z E R^2 \ { -1} and let
( )
1 + z
g z ---- 1-z' v( z ) = Re(g(z))for z E R^2 \ {l}. Then u is harmonic in D, vis harmonic in D*, and u(z ) = v(z)
for z E 3 D \ { oo}.
Next f(D) = f(S(a)) is a domain that contains the interval I= (-1, 1) and
is bounded by two circular arcs which meet I at angles ±a/2 at its endpoints. An
elementary calculation shows that
(7.2.4) m(f(D)) = a - sin a.
sin^2 ( a/2)
Similarly g(D*) = f (S(2 7f - a)) is bounded by circular arcs which meet I at angles
±(7r - a/2) at its endpoints and
( 7 · 2 · 5 ) m(g(D*)) = 27f - a+ sin a
sin^2 ( a/2)
as above. Then
l [grad u(z)[^2 dm = l [f'(z)[^2 dm = m(f(D)),
j [grad v(z)[
(^2) dm = j [g'(z) [ (^2) dm = m(g(D))
D D*
and we obtain (7.2.3) from (7.2.4) and (7 .2.5).