Rellich type inequalities in domains of the Euclidean space

被引：7

作者：

Avkhadiev F.G. ^{[1
]}

机构：

[1] Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan

来源：

Russian Mathematics | 2016年 / 60卷 / 1期

基金：

俄罗斯基础研究基金会;

关键词：

distance function; non-convex domain; Rellich type inequality; uniformly perfect set;

D O I：

10.3103/S1066369X16010060

中图分类号：

学科分类号：

摘要：

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres. © 2016, Allerton Press, Inc.

引用

页码：60 / 63

页数：3

共 19 条

[1]

Rellich F., Perturbation Theory of Eigenvalue Problems, (1969)

[2]

Davies E.B., Hinz A.M., Explicit Constants for Rellich Inequalities in Lp(Ω), Math. Z., 227, 3, pp. 511-523, (1998)

[3]

Owen M.P., The Hardy–Rellich Inequality for Polyharmonic Operators, Proc. Royal Soc. Edinburgh, A, 129, 9, pp. 825-835, (1999)

[4]

Barbatis M.G., Improved Rellich Inequalities for the Polyharmonic Operator, Indiana UniversityMath. J., 55, 4, pp. 1401-1422, (2006)

[5]

Barbatis M.G., Tertikas A., On a Class of Rellich Inequalities, J. Comp. Appl. Math., 194, 1, pp. 156-172, (2006)

[6]

Adimurthi M., Santra S., Optimal Hardy–Rellich Inequalities, Maximum Principle and Related Eigenvalue Problem, J. Func. Anal., 240, 1, pp. 36-83, (2006)

[7]

Evans W.D., Lewis R.T., Hardy and Rellich Inequalities with Remainders, J. Math. Inequal., 1, 4, pp. 473-490, (2007)

[8]

Avkhadiev F.G., Hardy Type Inequalities in Higher Dimensions with Explicit Estimate of Constants, Lobachevskii J. Math., 21, pp. 3-31, (2006)

[9]

Avkhadiev F.G., Hardy-Type Inequalities in Planar and Spacial Open Sets, Proc. Steklov Inst. Math., 255, 1, pp. 2-12, (2006)

[10]

Avkhadiev F.G., Wirths K.-J., Unified Poincaréand Hardy Inequalities with sharp constants for convex domains, Z. Angew. Math. Mech, 87, 8-9, pp. 632-642, (2007)

← 1 2 →