Hardy and Rellich inequalities on the complement of convex sets
| Autor: | Derek W. Robinson |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Mathematics::Functional Analysis Inequality General Mathematics media_common.quotation_subject 010102 general mathematics Regular polygon Mathematics::Analysis of PDEs Boundary (topology) 01 natural sciences 31C25 47D07 39B62 010101 applied mathematics Range (mathematics) Mathematics - Analysis of PDEs Hausdorff dimension FOS: Mathematics 0101 mathematics Mathematics Complement (set theory) media_common Analysis of PDEs (math.AP) |
| ISSN: | 1446-7887 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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