Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Helmholtz Operator

The goal of this paper is to develop a unified approach to radiation conditions for the entire class of null-solutions of the Helmholtz operator which are Clifford algebra-valued. The latter is an algebraic context which permits the simultaneous consideration of scalarvalued and vector-valued functi...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2018-06, Vol.231 (3), p.367-472
Main Authors: Marmolejo-Olea, E., Mitrea, I., Mitrea, D., Mitrea, M.
Format: Article
Language:English
Subjects:
Algebra
CLIFFORD ALGEBRA
DIRAC OPERATORS
HELMHOLTZ THEOREM
INTEGRALS
Mathematical analysis
MATHEMATICAL METHODS AND COMPUTING
MATHEMATICAL SOLUTIONS
Mathematics
Mathematics and Statistics
Operators (mathematics)
Radiation (Physics)
SCALARS
VECTORS
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Summary:The goal of this paper is to develop a unified approach to radiation conditions for the entire class of null-solutions of the Helmholtz operator which are Clifford algebra-valued. The latter is an algebraic context which permits the simultaneous consideration of scalarvalued and vector-valued functions, as well as differential forms of any mixed degree. In such a setting, we provide a multitude of novel radiation conditions which naturally contain the classical Sommerfeld and Silver–Müller radiation conditions in the case of null-solutions for the scalar Helmholtz operator and the Maxwell system respectively, and which also encompass as a particular case the radiation condition introduced by McIntosh and Mitrea for perturbed Dirac operators.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-018-3826-9