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HARDY AND RELLICH INEQUALITIES ON THE COMPLEMENT OF CONVEX SETS

Part of: Miscellaneous topics - Partial differential equations Partial differential equations General theory in ordinary differential equations

Published online by Cambridge University Press:  21 December 2018

DEREK W. ROBINSON
DEREK W. ROBINSON*
Affiliation:
Mathematical Sciences Institute (CMA), Australian National University, Canberra, ACT 0200, Australia email [email protected]
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Abstract

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_{p}(\unicode[STIX]{x1D6FA})$, where $\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$ with $K$ a closed convex subset of $\mathbf{R}^{d}$. Let $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ denote the boundary of $\unicode[STIX]{x1D6FA}$ and $d_{\unicode[STIX]{x1D6E4}}$ the Euclidean distance to $\unicode[STIX]{x1D6E4}$. We consider weighting functions $c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$ with $c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$ and $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$. Then the Hardy inequalities take the form

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D711}|^{p}\geq b_{p}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-p}|\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$
and the Rellich inequalities are given by
$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|H\unicode[STIX]{x1D711}|^{p}\geq d_{p}\int _{\unicode[STIX]{x1D6FA}}|c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-2}\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$
 with $H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$. The constants $b_{p},d_{p}$ depend on the weighting parameters $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.

Keywords

Hardy inequalityRellich inequality

MSC classification

Primary: 39B62: Functional inequalities, including subadditivity, convexity, etc.
Secondary: 34A40: Differential inequalities 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35R45: Partial differential inequalities
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 1 , February 2020 , pp. 98 - 119
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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