SPECTRAL ASYMPTOTICS FOR DIRICHLET ELLIPTIC OPERATORS WITH NON-SMOOTH COEFFICIENTS

We consider a 2m-th-order elliptic operator of divergence form in a domain Q of R(n), assuming that the coefficients are Holder continuous of exponent r is an element of (0, 1]. For the self-adjoint operator associated with the Dirichlet boundary condition we improve the asymptotic formula of the spectral function e(tau(2m), x, y) for x = y to obtain the remainder estimate O(tau(n-0) + dist(x, partial derivative Omega)(-1)tau(n-1)) with any theta is an element of (0, r), using the L(p) theory of elliptic operators of divergence form. We also show that the spectral function is in C(m-1,) (1-epsilon) with respect to (x, y) for any small epsilon > 0. These results extend those for the whole space R(n) obtained by Miyazaki [19] to the case of a domain.