Geodesic
Lower bound for the Hardy constant for an arbitrary domain in $\mathbb{R}^n$
Ufa mathematical journal, Tome 9 (2017) no. 2, pp. 102-108 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper we consider the conjecture by E.B. Davies on an uniform lower bound for the Hardy constant. We provide the known counterexamples rebutting this conjecture for the dimension $4$ and higher. In the work we obtain non-zero lower bounds for the Hardy constants. These estimates are order sharp as $n\to+\infty$, where $n$ is the space dimension. Moreover, these estimates are independent of the properties of the considered domains and are true for all domains not coinciding with the entire space. In the proof of the main theorem we reduce the multidimensional case to the one-dimensional case by choosing special classes of functions. As a result, the considered inequalities are reduced to the well-known Poincaré inequality.
Keywords: Hardy constant, lower bounds, Hardy inequalities, variational inequalities. 
@article{UFA_2017_9_2_a7,
     author = {I. K. Shafigullin},
     title = {Lower bound for the {Hardy} constant for an arbitrary domain in $\mathbb{R}^n$},
     journal = {Ufa mathematical journal},
     pages = {102--108},
     year = {2017},
     volume = {9},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_2_a7/}
}
TY  - JOUR
AU  - I. K. Shafigullin
TI  - Lower bound for the Hardy constant for an arbitrary domain in $\mathbb{R}^n$
JO  - Ufa mathematical journal
PY  - 2017
SP  - 102
EP  - 108
VL  - 9
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2017_9_2_a7/
LA  - en
ID  - UFA_2017_9_2_a7
ER  - 
%0 Journal Article
%A I. K. Shafigullin
%T Lower bound for the Hardy constant for an arbitrary domain in $\mathbb{R}^n$
%J Ufa mathematical journal
%D 2017
%P 102-108
%V 9
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2017_9_2_a7/
%G en
%F UFA_2017_9_2_a7
I. K. Shafigullin. Lower bound for the Hardy constant for an arbitrary domain in $\mathbb{R}^n$. Ufa mathematical journal, Tome 9 (2017) no. 2, pp. 102-108. http://geodesic.mathdoc.fr/item/UFA_2017_9_2_a7/

[1] V.G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin–New York, 1985 | MR | Zbl

[2] G. Talenti, “Osservazione sopra una classe di disuguaglianze”, Rend. Semin. mat. efis. Milano, 39 (1969), 171–185 | DOI | MR | Zbl

[3] G. Tomaselli, “A class of inequalities”, Boll. Unione mat. ital. Ser. 4, 1969, no. 6, 622–631 | MR | Zbl

[4] A. Kufner, L.-E. Persson, Weighted inequalities of Hardy type, World Scientific Pub Co Inc., 2003, 376 | MR

[5] Dubinskii Yu.A., “Ob odnom neravenstve tipa Khardi i ego prilozheniyakh”, Teoriya funktsii i differentsialnye uravneniya, Sbornik statei. K 105-letiyu so dnya rozhdeniya akademika Sergeya Mikhailovicha Nikolskogo, Tr MIAN, 269, MAIK, M., 2010, 112–132 | Zbl

[6] Prokhorov D.V., Stepanov V.D., “O vesovykh neravenstvakh Khardi v smeshannykh normakh”, Teoriya funktsii i uravneniya matematicheskoi fiziki, Sbornik statei. K 90-letiyu so dnya rozhdeniya chlena-korrespondenta RAN Lva Dmitrievicha Kudryavtseva, Tr. MIAN, 283, MAIK, M., 2013, 155–170 | Zbl

[7] F.G. Avkhadiev, K.-J. Wirths, “Unified Poincaré and Hardy inequalities with sharp constants for convex domains”, Z. Angew. Math. Mech., 87:8–9 (2007), 632–642 | DOI | MR | Zbl

[8] M. Marcus, V.J. Mizel, Y. Pinchover, “On the best constants for Hardy's inequality in $\mathbb{R}^n$”, Trans. Amer. Math. Soc., 350 (1998), 3237–3250 | DOI | MR

[9] E.B. Davies, “The Hardy constant”, Quart. J. Math. Oxford (2), 46:4 (1995), 417–431 | DOI | MR | Zbl

[10] T. Matskewich, P.E. Sobolevskii, “The best possible constant in a generalized Hardy's inequality for convex domains in $\mathbb{R} ^n$”, Nonlinear Anal., 28:9 (1997), 1601–1610 | DOI | MR | Zbl

[11] H. Brezis, M. Marcus, “Hardy's inequality revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:1–2 (1997), 217–237 | MR | Zbl

[12] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, “A geometrical version of Hardy's inequality”, J. Funct. Anal., 189:2 (2002), 539–548 | DOI | MR | Zbl

[13] V. Opic, A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Math., 219, 1990 | MR | Zbl

[14] Avkhadiev F.G., Shafigullin I.K., “Tochnye otsenki konstant Khardi dlya oblastei so spetsialnymi granichnymi svoistvami”, Izvestiya vuzov. Matem., 2014, no. 2, 69–73

[15] Avkhadiev F.G., Shafigullin I.K., “Otsenki konstant Khardi pri trubchatom rasshirenii mnozhestv i v oblastyakh s konechnymi granichnymi momentami”, Matematicheskie trudy, 16:2 (2013), 3–12

[16] Avkhadiev F.G., Nasibullin R.G., Shafigullin I.K., “Neravenstva tipa Khardi so stepennymi i logarifmicheskimi vesami v oblastyakh evklidova prostranstva”, Izvestiya vuzov. Matem., 2011, no. 9, 90–94