On the spectral asymptotic of non-selfadjoint elliptic systems of differential operators on manifolds with finite dimension (Q1409812)

The authors consider a bounded domain \(\Omega_0\subset\mathbb{R}^n\), a closed manifold \(S< \Omega_0\) with non-empty boundary and with co-dimension \(\mu\in \{1,2,\dots, n-1\}\). They investigate spectral properties and find an asymptotic formula for distribution of the eigenvalues of the non-selfadjoint elliptic differential operator \[ (Au)(x)= -\sum^n_{i,j=1} \Biggl({\partial\over\partial x_j}\Biggl(\rho^{2\alpha}(x) a_{ij}(x) q(x){\partial u\over\partial x_i}(x)\Biggr)\Biggr), \] defined on \(L^2(\Omega)^\ell\), associated with the bilinear form \[ {\mathcal A}[u,v]= \int_\Omega \Biggl\langle \rho^\alpha(x) a_{ij}(x) q(x){\partial u\over\partial x_i}(x),\,\rho^\alpha(x){\partial v\over\partial x_j}(x)\Biggr\rangle_{C^\ell}\,dx, \] where \(0\leq\alpha< \mu/n\), \(\rho(x)= \text{dist}\{x, S\}\), \(q(x)\in C^2(\overline\Omega, \text{End\,}C^\ell)\) (\(q(x)\) is a matrix function defined on \(\overline\Omega\)).