1208.2342 (B. Devyver et al.)

Optimal Hardy-type inequality for second-order elliptic operator and
applications    [PDF]

B. Devyver, M. Fraas, Y. Pinchover

For a general subcritical second-order elliptic operator P in a domain Omega in R^n (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense. The operator P - lambda W is subcritical in Omega for all lambda < 1, null-critical in Omega for lambda = 1, and supercritical near any neighborhood of infinity in Omega for any lambda > 1. Moreover, in the symmetric case, if W>0, then the spectrum and the essential spectrum of W^{-1}P are equal to [1, infinity). Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu=0, the existence of which depends on the subcriticality of P in Omega.

View original: http://arxiv.org/abs/1208.2342