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On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces
Author(s) -
Colton David,
Kress Rainer
Publication year - 2001
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.277
Subject(s) - mathematics , sobolev space , curl (programming language) , helmholtz equation , mathematical proof , bounded function , mathematical analysis , jump , norm (philosophy) , pure mathematics , boundary value problem , geometry , quantum mechanics , physics , computer science , law , political science , programming language
Let D ⊂ℝ 3 be a bounded domain with connected boundary δD of class C 2 . It is shown that Herglotz wave functions are dense in the space of solutions to the Helmholtz equation with respect to the norm in H 1 ( D ) and that the electric fields of electromagnetic Herglotz pairs are dense in the space of solutions to curl curl E = k 2 E with respect to the norm in H curl ( D ). Two proofs are given in each case, one based on the denseness of the traces of Herglotz wave functions on δD and the other on variational methods. Copyright © 2001 John Wiley & Sons, Ltd.