Communications on Pure and Applied Analysis (CPAA)

Optimal Hardy inequalities for general elliptic operators with improvements

Pages: 109 - 140, Volume 9, Issue 1, January 2010      doi:10.3934/cpaa.2010.9.109

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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]).
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.

Keywords:  Hardy inequalities, general elliptic operators.
Mathematics Subject Classification:  Primary: 35Jxx; Secondary: 46E35.

Received: December 2008;      Revised: July 2009;      Published: October 2009.