Asymptotic distribution of eigenvalues of nonsymmetric elliptic operators (Q1114865)

The author considers an integro-differential sesquilinear form B of \(V\times V\) of order m \((2m>n)\) with bounded coefficients; namely: \[ B[u,\nu]=\int_{\Omega}\sum_{| \alpha | | \beta | \leq m}a_{\alpha \beta}(x)D^{\alpha}u \overline{D^{\beta}\nu} dx, \] where: \(x=(x_ 1,...,x_ n)\in {\mathbb{R}}^ n\); \(\alpha =(\alpha_ 1,...,\alpha_ n)\) is a multi-index of nonnegative integers; \(D^{\alpha}=D_ 1^{\alpha_ 1},...,D_ n^{\alpha_ n}\) with \(D_ k=-i\partial /\partial x_ k\); \(\Omega \in {\mathbb{R}}^ n\) (n\(\geq 2)\) is a bounded domain in possessing the restricted cone property and V is a closed subspace of \(H_ m(\Omega)\) containing \(H^ 0_ m(\Omega)\), and defines an operator A associated with the form B as follows: An element u of V belongs to D(A) and \(Au=f\in L_ 2(\Omega)\) if \(B[u,\nu]=(f,\nu)_{L_ 2}\) is valid for any \(\nu\in V\). Then he establishes an asymptotic formula (with optimal remainder estimate) for N(t) which holds as \(t\to \infty\) and gives the number of eigenvalues of A whose real parts are smaller or equal to \(t>0\) with repetition according to the multiplicities. His method is based upon the resolvent kernel estimates and Tauberian theorems. The importance of the result is that B has non-symmetric top terms.