ESTIMATES THE UPPER BOUNDS OF DIRICHLET EIGENVALUES FOR FRACTIONAL LAPLACIAN

Let Omega subset of R-n (n = 2) be a bounded domain with continuous boundary partial derivative Omega. In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to Omega with 0 < s < 1. Denoting by lambda(k) the k(th) Dirichlet eigenvalue of (-Delta)(s)vertical bar Omega, we establish the explicit upper bounds of the ratio lambda(k+1)/lambda(1), which have polynomially growth in k with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function R-sigma(z) = Sigma(k)(z -lambda(k))(sigma) + with sigma >= 1 and the trace of the Dirichlet heat kernel of fractional Laplacian.