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

Abstract

Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a constitutive equation ${\bf J}({\bf x})={\bf L}({\bf x}){\bf E}({\bf x})-{\bf h}({\bf x})$ with a source term ${\bf h}({\bf x})$ holds for all ${\bf x}$ in some domain $\Omega$, and relates fields ${\bf E}$ and ${\bf J}$ that satisfy appropriate differential constraints, symbolized by ${\bf E}\in\cal{E}_\Omega$ and ${\bf J}\in\cal{J}_\Omega$ where $\cal{E}_\Omega$ and $\cal{J}_\Omega$ are orthogonal spaces that span the space $\cal{H}_\Omega$ of square-integrable fields in which ${\bf h}$ lies. Boundedness and coercivity conditions on the moduli ${\bf L}({\bf x})$ ensure there exists a unique ${\bf E}$ for any given ${\bf h}$, i.e., ${\bf E}={\bf G}_\Omega{\bf h}$ which then establishes the existence of the Green's function ${\bf G}_\Omega$. 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 ${\bf L}({\bf x} )$ depending linearly on a vector of parameters ${\bf z}=(z_1, z_2,\ldots, z_n)$, we obtain constraints on ${\bf z}$ that ensure the Green's function exists, and hence which provide bounds on the spectrum.