Rellich inequalities in bounded domains
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Mathematische Annalen
Rellich inequalities in bounded domains G. Metafune1
· L. Negro1 · M. Sobajima2 · C. Spina1
Received: 9 April 2019 / Revised: 2 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We find necessary and sufficient conditions for the validity of weighted Rellich inequalities in L p , 1 ≤ p ≤ ∞, for functions in bounded domains vanishing at the boundary. General operators like L = + c |x|x 2 · ∇ − |x|b 2 are considered. Critical cases and remainder terms are also investigated. Mathematics Subject Classification 26D10 · 35PXX · 47F05
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic results and methods . . . . . . . . . . . . . . . . . . . . 3 The operator A = |x|2 + cx · ∇ . . . . . . . . . . . . . . . . p 3.1 The Laplace-Beltrami operator 0 on L J (S N −1 ) . . . . . p N −1 dr ) . . . . . . . . . . . . . 3.2 The operator on L (I , r p p 3.3 The operator A = |x|2 + cx · ∇ on L J (R N ) and L J (B) 2 p 3.4 The operator A = |x| + cx · ∇ on L () . . . . . . . . 4 Rellich inequalities in R N and in B . . . . . . . . . . . . . . 5 Rellich inequalities in general domains . . . . . . . . . . . . . 5.1 The operator L in L 2 (, dμ) . . . . . . . . . . . . . . . .
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Communicated by Y. Giga.
B
G. Metafune [email protected] L. Negro [email protected] M. Sobajima [email protected] C. Spina [email protected]
1
Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100 Lecce, Italy
2
Department of Mathematics, Tokyo University of Science, Tokyo, Japan
123
G. Metafune et al. 5.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Rellich inequalities in exterior domains . . . . . . . . . . 6 Critical cases in L p (R N ) . . . . . . . . . . . . . . . . . . . 7 Best constants and remainder terms . . . . . . . . . . . . . . 8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Approximation on Sobolev spaces on domains . . . . . . 8.2 Some results on spectral theory . . . . . . . . . . . . . 8.3 Spectrum of a second order ordinary differential operator References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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