January  2010, 9(1): 109-140. doi: 10.3934/cpaa.2010.9.109

Optimal Hardy inequalities for general elliptic operators with improvements


Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2, Canada

Received  December 2008 Revised  July 2009 Published  October 2009

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]).
We examine improved versions of the above inequalities of the form

$\int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2} u^2dx + \int_\Omega V(x) u^2dx, \qquad u \in H_0^1(\Omega).\qquad (3)$

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.
In addition we obtain various results concerning the above inequalities valid for functions $ u$ which are nonzero on the boundary of $ \Omega$. We also examine the nonquadradic case ,ie. $p$ ≠2 of the above inequalities.

Citation: Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure & Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109

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