Let Ω be an open set in Rn such that
Ω ≠ Rn.
For 1 ≤ p < ∞, 1 < s <
∞ and δ = dist(x, ∂Ω) we estimate the Hardy
constant
cp(s, Ω)= sup {||f/
δs/p||Lp(Ω):
f ∈ C0∞(Ω),
||(∇ f)/δs/p - 1||Lp(Ω)=1}
and some related quantities.
For open sets Ω ⊂ R2 we prove the following bilateral estimates
min {2, p} M0(Ω) ≤ cp (2, Ω)
≤ 2 p
(π M0 (Ω) + a0)2, a0=4.38,
where M0(Ω) is the geometrical parameter defined as the
maximum modulus of ring domains in Ω with center on
∂Ω. Since the condition M0 (Ω) < ∞
means the uniformly perfectness of ∂Ω, these
estimates give a direct proof of the following Ancona-Pommerenke
theorem:
c2(2, ∂Ω) is finite if and only if the boundary set
∂Ω is uniformly perfect.
Moreover, we obtain the following direct extension of the
one-dimensional
Hardy inequality to the case n ≥ 2: if
s > n,
then
for arbitrary open sets Ω ⊂ Rn (Ω
≠ Rn)
and any p ∈ [1, ∞) the sharp
inequality
cp (s, Ω) ≤ p/(s-n)
is valid. This gives a solution of a known problem due to
J.L.Lewis and A.Wannebo.
Estimates of constants in certain other Hardy and Rellich type inequalities
are also considered. In particular, we obtain an improved version
of a Hardy type inequality by H.Brezis and M.Marcus
for convex domains and give its generalizations.
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