Skip to main content
×
×
Home

HARDY AND RELLICH INEQUALITIES ON THE COMPLEMENT OF CONVEX SETS

  • DEREK W. ROBINSON (a1)
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.

Copyright
References
Hide All
[Avk15a] Avkhadiev, F. G., ‘Sharp constants in Hardy type inequalities’, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (2015), 6165.
[Avk15b] Avkhadiev, F. G., ‘Hardy type L p -inequalities in r-close-to-convex domains’, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (2015), 8488.
[BEL15] Balinsky, A. A., Evans, W. D. and Lewis, R. L., The Analysis and Geometry of Hardy’s Inequality, Universitext (Springer, New York, 2015).
[BFT04] Barbatis, G., Filippas, S. and Tertikas, A., ‘A unified approach to improved L p Hardy inequalities’, Trans. Amer. Math. Soc. 356 (2004), 21692196.
[Dav95] Davies, E. B., ‘The Hardy constant’, Q. J. Math. 46 (1995), 417431.
[DH98] Davies, E. B. and Hinz, A. M., ‘Explicit constants for Rellich inequalities in L p (𝛺)’, Math. Z. 227 (1998), 511523.
[EG92] Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1992).
[Hör94] Hörmander, L., Notions of Convexity, Progress in Mathematics, 127 (Birkhäuser, Boston–Basel–Berlin, 1994).
[LR16] Lehrbäck, J. and Robinson, D. W., ‘Uniqueness of diffusion on domains with rough boundaries’, Nonlinear Anal. Theory Methods Appl. 131 (2016), 6080.
[LV16] Lehrbäck, J. and Vähäkangas, A. V., ‘In between the inequalities of Sobolev and Hardy’, J. Funct. Anal. 271 (2016), 330364.
[MSS15] Metafune, G., Sobajima, M. and Spina, C., ‘Weighted Calderón–Zygmund and Rellich inequalities in L p ’, Math. Ann. 361 (2015), 313366.
[Rob18] Robinson, D. W., ‘Hardy inequalities, Rellich inequalities and local Dirichlet forms’, J. Evol. Equ. 18 (2018), 15211541.
[RS10] Robinson, D. W. and Sikora, A., ‘Degenerate elliptic operators in one dimension’, J. Evol. Equ. 10 (2010), 731759.
[Sim11] Simon, B., Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics, 187 (Cambridge University Press, Cambridge, 2011).
[SSW03] Secchi, S., Smets, D. and Willem, M., ‘Remarks on a Hardy–Sobolev inequality’, C. R. Acad. Sci. Paris Sér. I 336 (2003), 811815.
[War14] Ward, A. D., ‘On essential self-adjointness, confining potentials and the -Hardy inequality’, PhD Thesis, Massey University, Albany, New Zealand, 2014,http://hdl.handle.net/10179/5941.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed