A geometric description of domains whose Hardy constant is equal to~1/4

F. G. Avkhadiev

Izvestiya. Mathematics , Tome 78 (2014) no. 5, pp. 855-876.

Résumé

We give a geometric description of families of non-convex planar and spatial domains in which the following Hardy inequality holds: the Dirichlet integral of any smooth compactly supported function $f$ on the domain is greater than or equal to one quarter of the integral of $f^2(x)/\delta^2(x)$, where $\delta(x)$ is the distance from $x$ to the boundary of the domain. Our geometric description is based analytically on new one-dimensional Hardy-type inequalities with special weights and on new constants related to these inequalities and hypergeometric functions.

Détail

Keywords: Hardy inequalities, hypergeometric functions, torsional rigidity.
Mots-clés : non-convex domains

Bibliographie

[1] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952, xii+324 pp. | MR | MR | Zbl

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

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

[4] H. Brezis, M. Marcus, “Hardy's inequalities revisited”, Dedicated to E. De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:1-2 (1998), 217–237 | MR | Zbl

[5] M. Marcus, V. J. Mitzel, Y. Pinchover, “On the best constant for Hardy's inequality in $\mathbb{R}^n$”, Trans. Amer. Math. Soc., 350:8 (1998), 3237–3255 | DOI | MR | Zbl

[6] E. B. Davies, “A review of Hardy inequalities”, The Maz'ya anniversary collection (Rostock, 1998), v. 2, Oper. Theory Adv. Appl., 110, Birkhäuser, Basel, 1999, 55–67 | DOI | MR | Zbl

[7] G. Barbatis, S. Filippas, A. Tertikas, “A unified approach to improved $L^p$ Hardy inequalities with best constants”, Trans. Amer. Math. Soc., 356:6 (2004), 2169–2196 (electronic) | DOI | MR | Zbl

[8] D. H. Armitage, Ü. Kuran, “The convexity of a domain and the superharmonicity of the signed distance function”, Proc. Amer. Math. Soc., 93:4 (1985), 598–600 | DOI | MR | Zbl

[9] F. Avkhadiev, A. Laptev, “Hardy inequalities for nonconvex domains”, Around the research of Vladimir Maz'ya, v. I, Int. Math. Ser. (N. Y.), 11, Springer, New York, 2010, 1–12 | DOI | MR | Zbl

[10] F. G. Avkhadiev, “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., 21 (2006), 3–31 (electronic) | MR | Zbl

[11] F. G. Avkhadiev, “Hardy-type inequalities on planar and spatial open sets”, Proc. Steklov Inst. Math., 255:1 (2006), 2–12 | DOI | MR

[12] S. Fillipas, V. Maz'ya, A. Tertikas, “On a question of Brezis and Marcus”, Calc. Var. Partial Differential Equations, 25:4 (2006), 491–501 | DOI | MR | Zbl

[13] 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

[14] F. G. Avkhadiev, K.-J. Wirths, “Weighted Hardy inequalities with sharp constants”, Lobachevskii J. Math., 31:1 (2010), 1–7 | DOI | MR | Zbl

[15] F. G. Avkhadiev, K.-J. Wirths, “Sharp Hardy-type inequalities with Lamb's constants”, Bull. Belg. Math. Soc. Simon Stevin, 18:4 (2011), 723–736 | MR | Zbl

[16] F. G. Avkhadiev, R. G. Nasibullin, I. K. Shafigullin, “Hardy-type inequalities with power and logarithmic weights in domains of the Euclidean space”, Russian Math. (Iz. VUZ), 55:9 (2011), 76–79 | DOI | MR | Zbl

[17] F. G. Avkhadiev, K.-J. Wirths, “On the best constants for the Brezis–Marcus inequalities in balls”, J. Math. Anal. Appl., 396:2 (2012), 473–480 | DOI | MR | Zbl

[18] D. V. Prokhorov, V. D. Stepanov, “Weighted estimates for the Riemann–Liouville operators and applications”, Proc. Steklov Inst. Math., 243 (2003), 278–301 | MR | Zbl

[19] F. G. Avkhadiev, “Families of domains with best possible Hardy constant”, Russian Math. (Iz. VUZ), 57:9 (2013), 49–52 | DOI | Zbl

[20] H. Lamb, “Note on the induction of electric currents in a cylinder placed across the lines of magnetic force”, Proc. London Math. Soc., s1-15:1 (1883), 270–275 | DOI | MR

[21] J. E. Littlewood, Lectures on the theory of functions, Oxford Univ. Press, Oxford, 1944, 243 pp. | MR | Zbl

[22] G. M. Goluzin, “Nekotorye teoremy pokrytiya v teorii analiticheskikh funktsii”, Matem. sb., 22(64):3 (1948), 353–372 | MR | Zbl

[23] J. A. Hempel, “The Poincaré metric on the twice punctured plane and the theorems of Landau and Schottky”, J. London Math. Soc. (2), 20:3 (1979), 435–445 | DOI | MR | Zbl

[24] J. A. Jenkins, “On explicit bounds in Landau's theorem. II”, Canad. J. Math., 33:3 (1981), 559–562 | DOI | MR | Zbl

[25] F. G. Avkhadiev, K.-J. Wirths, Schwarz–Pick type inequalities, Front. Math., Birkhäuser Verlag, Basel, 2009, viii+156 pp. | DOI | MR | Zbl

[26] E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen, v. I, Gewöhnliche Differentialgleichungen, 9 Aufl., B. G. Teubner, Stuttgart, 1977, xxvi+668 pp. | MR | MR | Zbl | Zbl

[27] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Grundlehren Math. Wiss., 93, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1957, xiii+312 pp. | MR | MR | Zbl | Zbl

[28] G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., 27, Princeton Univ. Press, Princeton, N. J., 1951, xvi+279 pp. | MR | MR | Zbl | Zbl

[29] C. Bandle, Isoperimetric inequalities and applications, Monogr. Stud. Math., 7, Pitman (Advanced Publishing Program), Boston, Mass.–London, 1980, x+228 pp. | MR | Zbl

[30] F. G. Avkhadiev, “Solution of the generalized Saint Venant problem”, Sb. Math., 189:12 (1998), 1739–1748 | DOI | DOI | MR | Zbl

[31] M. Bañuelos, M. van den Berg, T. Carrol, “Torsional rigidity and expected lifetime of Brownian motion”, J. London Math. Soc. (2), 66:2 (2002), 499–512 | DOI | MR | Zbl