Hardy–Rellich integral inequalities in domains satisfying the exterior sphere condition

Avkhadiev, F.

doi:10.1090/spmj/1536

Hardy–Rellich integral inequalities in domains satisfying the exterior sphere condition
HTML articles powered by AMS MathViewer

by F. G. Avkhadiev
Translated by: E. Peller

St. Petersburg Math. J. 30 (2019), 161-179

DOI: https://doi.org/10.1090/spmj/1536

Published electronically: February 14, 2019

PDF | Request permission

Abstract:

Analogs of Hardy–Rellich inequalities are studied for compactly supported functions on the domains of Euclidean space in the case when weight functions are powers of the distance between a point and the boundary of the domain and the domains satisfy the exterior sphere condition. Explicit estimates of constants in this inequalities are obtained in terms of the dimension, the weight function, and two geometric characteristics: the radius in the exterior sphere condition and the inner radius of the domain.

References

Alexander A. Balinsky, W. Desmond Evans, and Roger T. Lewis, The analysis and geometry of Hardy’s inequality, Universitext, Springer, Cham, 2015. MR 3408787, DOI 10.1007/978-3-319-22870-9
Tanya Matskewich and Pavel E. Sobolevskii, The best possible constant in generalized Hardy’s inequality for convex domain in $\textbf {R}^n$, Nonlinear Anal. 28 (1997), no. 9, 1601–1610. MR 1431208, DOI 10.1016/S0362-546X(96)00004-1
Moshe Marcus, Victor J. Mizel, and Yehuda Pinchover, On the best constant for Hardy’s inequality in $\mathbf R^n$, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3237–3255. MR 1458330, DOI 10.1090/S0002-9947-98-02122-9
E. B. Davies, A review of Hardy inequalities, The Maz′ya anniversary collection, Vol. 2 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 55–67. MR 1747888
G. Barbatis, S. Filippas, and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2169–2196. MR 2048514, DOI 10.1090/S0002-9947-03-03389-0
F. G. Avkhadiev, Hardy type inequalities in higher dimensions with explicit estimate of constants, Lobachevskii J. Math. 21 (2006), 3–31. MR 2220697
F. G. Avkhadiev, Hardy-type inequalities in two- and three-dimensional open sets, Tr. Mat. Inst. Steklova 255 (2006), no. Funkts. Prostran., Teor. Priblizh., Nelineĭn. Anal., 8–18 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(255) (2006), 2–12. MR 2301606, DOI 10.1134/s008154380604002x
F. G. Avkhadiev and I. K. Shafigullin, Sharp estimates of Hardy constants for domains with special boundary properties, Russian Math. (Iz. VUZ) 58 (2014), no. 2, 58–61. Translation of Izv. Vyssh. Uchebn. Zaved. Mat. 2014, no. 2, 69–73. MR 3254462, DOI 10.3103/S1066369X14020091
Farit G. Avkhadiev and Karl-Joachim Wirths, Unified Poincaré and Hardy inequalities with sharp constants for convex domains, ZAMM Z. Angew. Math. Mech. 87 (2007), no. 8-9, 632–642. MR 2354734, DOI 10.1002/zamm.200710342
F. G. Avkhadiev and K.-J. Wirths, Sharp Hardy-type inequalities with Lamb’s constants, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 4, 723–736. MR 2907615
F. G. Avkhadiev and K.-J. Wirths, On the best constants for the Brezis-Marcus inequalities in balls, J. Math. Anal. Appl. 396 (2012), no. 2, 473–480. MR 2961240, DOI 10.1016/j.jmaa.2012.06.008
Farit Avkhadiev and Ari Laptev, Hardy inequalities for nonconvex domains, Around the research of Vladimir Maz’ya. I, Int. Math. Ser. (N. Y.), vol. 11, Springer, New York, 2010, pp. 1–12. MR 2723812, DOI 10.1007/978-1-4419-1341-8_{1}
F. G. Avkhadiev, Hardy inequalities in non-convex domains of the Euclidean space, J. Anal. 23 (2015), 1–20. MR 3491643
F. G. Avkhadiev, A geometric description of domains whose Hardy constant is equal to 1/4, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014), no. 5, 3–26 (Russian, with Russian summary); English transl., Izv. Math. 78 (2014), no. 5, 855–876. MR 3308642, DOI 10.1070/im2014v078n05abeh002710
Mark P. Owen, The Hardy-Rellich inequality for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 825–839. MR 1718522, DOI 10.1017/S0308210500013160
F. G. Avkhadiev, Hardy-Rellich inequalities in domains of the Euclidean space, J. Math. Anal. Appl. 442 (2016), no. 2, 469–484. MR 3504010, DOI 10.1016/j.jmaa.2016.05.004
F. G. Avkhadiev, Integral inequalities in hyperbolic-type domains and their application, Mat. Sb. 206 (2015), no. 12, 3–28 (Russian, with Russian summary); English transl., Sb. Math. 206 (2015), no. 11-12, 1657–1681. MR 3438572, DOI 10.4213/sm8477
F. G. Avkhadiev and R. G. Nasibullin, A Hardy type inequality in arbitrary domains with finite inner radius, Sibirsk. Mat. Zh. 55 (2014), no. 2, 239–250 (Russian, with Russian summary); English transl., Sib. Math. J. 55 (2014), no. 2, 191–200. MR 3237329, DOI 10.1134/s0037446614020013
F. G. Avkhadiev and I. K. Shafigullin, Estimates for Hardy constants in a tubular extension of sets and in domains with finite boundary moments, Mat. Tr. 16 (2013), no. 2, 3–12 (Russian, with Russian summary); English transl., Siberian Adv. Math. 24 (2014), no. 3, 153–158. MR 3156692, DOI 10.3103/s1055134414030018
O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoĭ fiziki, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579