Asymptotics of the spectrum of degenerate-elliptic differential operators that are far from self-adjoint
Let \(\Omega\subset \mathbb{R}^n\) be a finite domain with a boundary that locally satisfies the Lipschitzian condition. Consider the bilinear form \[ {\mathcal A}[u,v]= \sum_{|\alpha|= |\beta|\in J}\int_\Omega \rho^{\theta_{|\alpha|}}(x) a_{\alpha\beta}(x) D^\beta u(x) \overline{D^\alpha v(x)} dx \] in the space \(L_2(\Omega)\), where \(D= (\partial/i\partial x_1),\dots, \partial/i\partial x_n)\), \(J\subset \{0,1,\dots, m\}\), \(m\in J\). Let \(A\) be an \(m\)-sectorial operator that is associated with the bilinear form \({\mathcal A}[u,v]\). In this work, the spectral asymptotics of the operator \(A\) is calculated in the case when the condition \[ a_{\alpha\beta}(x)\equiv \overline {a_{\beta\alpha}(x)}, \qquad x\in\partial\Omega, \qquad |\alpha|= |\beta|=r, \] can be satisfied even for \(r=m\).
scientific article; zbMATH DE number 1099399