Farit Avkhadie, Ari Laptev
Hardy Inequalities for Non-Convex Domains
Preprint series: ESI preprints

MSC:

35P15 Estimation of eigenvalues, upper and lower bounds

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Abstract: We obtain a number of Hardy type inequalities
for domains involving both the distance to the boundary and
the distance to the origin. In particular, this
implies a Hardy-Sobolev inequality for the class of symmetric
functions in a ball. Moreover, we prove that if the dimension
$d\ge3$ then the Hardy inequality involving the distance to the boundary is true
with the constant $1/4$ in a large family of domains which are not convex.
At the end we give an example where we show that for any positive fixed constant
there is an ellipsoid layer, such that Hardy's inequality with the distance to
the boundary fails.