Best Constants in the Hardy-Rellich Inequalities and Related Improvements

By A. Tertikas, N. Zographopoulos. 2006

151 citations recorded by the source corpus.

Abstract We consider Hardy–Rellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. Rellich–Sobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and higher order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer expected to consist of radial functions. Finally, the successively use of the Rellich inequalities lead to various new higher order Rellich inequalities.

Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/c84cb658ab16ce2913c7254398d5a1170f2a8dc2

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In Applied Mathematics

1. M. Grossi
2. I. Kombe
3. M. Ozaydin
4. D. Ganguly
5. William Beckner
6. Adimurthi
7. Elvise Berchio
8. G. Grillo

In Mathematical Physics

1. M. Grossi
2. I. Kombe
3. M. Ozaydin
4. D. Ganguly
5. William Beckner
6. Adimurthi
7. Elvise Berchio
8. G. Grillo

In Mechanics of Materials

1. I. Kombe
2. D. Ganguly
3. William Beckner
4. M. Ozaydin
5. Adimurthi
6. Elvise Berchio
7. G. Grillo

Citations in the corpus

• Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space (2015, methodology): “without any claim for completeness [6, 12, 16, 22, 23, 33, 39].”
• Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem (2006, background): “Furthermore, Tertikas and Zographopolous [17] have improved this by obtaining an optimal inequality, ∫”
• Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds (2011, background): “MR0240668 (39:2014) [21] E. Stein, “Harmonic Analysis, Real-Variable Methods, Orthgonality, and Oscillatory Inte- grals”, Princeton University Press, Princeton, NJ. MR1232192 (95c:42002) [22] A. Tertikas and N. Zographopoulos, Best constants in the Hardy-Rellich Inequalities and Related Improvements”
• Weighted inequalities and Stein-Weiss potentials (2006, background): “in the family of gradient estimates, Pitt’s inequality has been characterized as a HardyRellich inequality in some parts of the recent literature with alternative proofs and extensions (see [1], [7], [14], and [17]). Some natural generalizations of Pitt’s inequality can now be viewed in the context ”