A new route to finding bounds on the generalized spectrum of many physical operators

Here we obtain bounds on the generalized spectrum of that operator whose inverse, when it exists, gives Green’s function. We consider the wide range of physical problems that can be cast in a form where a constitutive equation J(x) = L(x)E(x) − h(x) with a source term h(x) holds for all x in some do...

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Veröffentlicht in:Journal of mathematical physics 2018-06, Vol.59 (6)
1. Verfasser: Milton, Graeme W.
Format: Artikel
Sprache:eng
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Zusammenfassung:Here we obtain bounds on the generalized spectrum of that operator whose inverse, when it exists, gives Green’s function. We consider the wide range of physical problems that can be cast in a form where a constitutive equation J(x) = L(x)E(x) − h(x) with a source term h(x) holds for all x in some domain Ω and relates fields E and J that satisfy appropriate differential constraints, symbolized by E∈EΩ0 and J∈J¯Ω, where EΩ0 and J¯Ω are orthogonal spaces that span the space HΩ of square-integrable fields in which h lies. Boundedness and coercivity conditions on the moduli L(x) ensure that there exists a unique E for any given h, i.e., E = GΩh, which then establishes the existence of Green’s function GΩ. We show that the coercivity condition is guaranteed to hold if weaker conditions, involving generalized quasiconvex functions, are satisfied. The advantage is that these weaker conditions are easier to verify, and for multiphase materials, they can be independent of the geometry of the phases. For L(x) depending linearly on a vector of parameters z = (z1, z2, …, zn), we obtain constraints on z that ensure that Green’s function exists and hence which provide bounds on the generalized spectrum.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.5032204