Optimal Hardy inequalities for general elliptic operators with improvements
Craig Cowan  Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2, Canada (email) Abstract: We establish Hardy inequalities of the form $ \int_\Omega  \nabla u_A^2 dx \ge \frac{1}{4} \int_\Omega \frac{ \nabla E_A^2}{E^2}u^2dx, \qquad u \in H_0^1(\Omega) \qquad\qquad (1)$ where $ E$ is a positive function defined in $ \Omega$, div$(A \nabla E)$ is a nonnegative nonzero finite measure in $ \Omega$ which we denote by $ \mu$ and where $ A(x)$ is a $ n \times n$ symmetric, uniformly positive definite matrix defined in $ \Omega$ with $  \xi _A^2:= A(x) \xi \cdot \xi$ for $ \xi \in \mathbb{R}^n$. We show that (1) is optimal if $ E=0$ on $ \partial \Omega$ or $ E=\infty$ on the support of $ \mu$ and is not attained in either case. When $ E=0$ on $\partial \Omega$ we show $ \int_\Omega  \nabla u_A^2dx \ge \frac{1}{4} \int_\Omega \frac{ \nabla E_A^2}{E^2}u^2dx + \frac{1}{2} \int_\Omega \frac{u^2}{E} d \mu, \qquad u \in H_0^1(\Omega)\qquad (2) $
is optimal and not attained.
Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1) and (2) are established for $p$≠ 2 which, in the case that $ \mu$ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar (see [1]).
$\int_\Omega  \nabla u_A^2dx \ge \frac{1}{4} \int_\Omega \frac{ \nabla E_A^2}{E^2} u^2dx + \int_\Omega V(x) u^2dx, \qquad u \in H_0^1(\Omega).\qquad (3)$
Necessary and sufficient conditions on $V$ are obtained (in terms of the solvability of a linear pde)
for (3) to hold. Analogous results involving improvements are obtained for the weighted versions.
Keywords: Hardy inequalities, general elliptic operators.
Received: December 2008; Revised: July 2009; Published: October 2009. 
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