Some problems associated with sum and integral inequalities

0044 English OPEN
Thomas, James Christian;
  • Subject: QA

In 2 , the following extension of the higher order Rellich inequality / AV(x) 2>7(n,a,i) / /(x) 2- (1) JRn lxl JRn lxl was proven by W. Allegretto for all / G C£ (Rn {0}). The constant 7 is calculated explicitly by the author for all n > 2, a > 0 and j 6 N, giving... View more
  • References (12)
    12 references, page 1 of 2

    1 Introdu ction 1 1.1 The Rellich inequality ................................................................... 2 1.2 Magnetic P o ten tials.......................................................................... 6 1.2.1 The Aharonov-Bohm effect................................................. 6 1.2.2 Repairing the Hardy inequality in two dimensions . . . 7 1.2.3 Repairing the Rellich in e q u a lity ........................................ 8 1.3 CLR type b o u n d s ............................................................................. 10 1.4 N o ta tio n .............................................................................................. 13

    2 T he higher order R ellich inequality 15 2.1 The Rellich inequality and Allegretto'sc o n s ta n t....................... 16 2.2 The higher order Rellich in e q u a lity .................................................21 2.3 Restriction of the class of fu n c tio n s .................................................24

    3 A R ellich ty p einequality w ith m agnetic p o ten tia ls 35 3.1 The magnetic L a p la c ia n ................................................................ 37 3.2 Proof of Theorem 3 . 2 .......................................................................... 46 3.3 A Rellich type inequality with a magneticp o te n tia l.....................49 3.4 A higher order Rellich type inequality......................................... 57

    4 C ounting Eigenvalues 59 4.1 An upper bound for I H I / 1 |Loc(r + ) ................................ 60 4.2 Forms and O p e ra to rs ...................................................................... 68 4.3 A CLR type inequality for + K+ - V in L2(R8) .................... 78

    [1] Y. Aharonov and D. Bohm. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. (2), 115:485-491, 1959.

    [2] W. Allegretto. Nonoscillation theory of elliptic equations of order 2n. Pacific J. Math., 64(1): 1-16, 1976.

    [15] W. D. Evans and R. T. Lewis. On the Rellich inequality with magnetic potentials. Math. Z ., 251(2):267-284, 2005.

    [16] T. Kato. Perturbation theory fo r linear operators. Classics in M athematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.

    [17] A. Kufner, L. M aligranda, and L.-E. Persson. The prehistory of the Hardy inequality. Amer. Math. Monthly, 113(8) :715-732, 2006.

    [23] M. Reed and B. Simon. Methods of m odem mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich P ub ­ lishers], New York, 1978.

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