On Some Quasimetrics and Their Applications
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Research Article On Some Quasimetrics and Their Applications 2 ˆ Imed Bachar1 and Habib Maagli 1
Mathematics Department, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 2 D´epartement de Math´ematiques, Facult´e des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia Correspondence should be addressed to Imed Bachar, [email protected] Received 27 September 2009; Accepted 8 December 2009 Recommended by Shusen Ding We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain in Rn . Copyright q 2009 I. Bachar and H. Mˆaagli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Let D be a bounded smooth domain in Rn , n ≥ 1, or D Rn : {x , xn ∈ Rn : xn > 0} the half space. We denote by GD the Green’s function of the operator u → −Δα u with Dirichlet or Navier boundary conditions, where α is a positive integer or 0 < α < 1. The following inequality called 3G inequality has been proved by several authors see 1, 2 for α 1 or 3, 4 for α ≥ 1 or 5 for 0 < α < 1. There exists a constant C > 0 such that for each x, y, z ∈ D, GD x, zGD z, y λz λz GD x, z GD y, z , ≤C λx GD x, y λ y
1.1
where x → λx is a positive function which depends on the Euclidean distance between x and ∂D and the exponent α. More precisely, to prove this inequality, the authors showed that the function x, y → ρx, y λxλy/GD x, y is a quasi-metric on D see Definition 2.1. We emphasis that the generalized 3G inequality is crucial for various applications see e.g., 1, Theorem 1.2, 6, Lemma 7.1. It is also very interesting tools for analysts working on pde’s. In 7, Theorem 5.1, the authors used the standard 3G inequality see 7, Proposition 4.1 to prove that on the unit ball B in Rn , the inverse of polyharmonic operators that are
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Journal of Inequalities and Applications
perturbed by small lower order terms is positivity preserving. They also obtained similar results for systems of these operators. On the other hand, local maximum principles for solutions of higher-order differential inequalities in arbitrary bounded domains are obtained in 8, Theorem 2 by using estimates on Green’s functions. Recently, a refined version of the standard 3G inequality for polyharmonic operator is obtained in 3, Theorem 2.8 and 4, Theorem 2.9. This allowed the authors to introduce and study an interesting functional Kato class, which permits them to investigate the existence of positive solutions for some polyharmonic nonlinear problems. In the present manuscript we aim at giving a generalization of these known 3G inequalities by proving a rich class of quasimetrics see Theorem 2.8 which in particular includes the one ρx, y λxλy/GD
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