Singular perturbation approach to Legendre type operators

Let Q be a bounded domain in ℝN with compact smooth boundary (N ∈ N). Then this paper is concerned with the nonnegative selfadjointness in L2(Q) of the maximal realization T2 of N-dimensional second-order differential operators in divergence form with diffusion coefficients vanishing on the boundary r - dQ. The operators may be called Legendre type operators over Q. The key to the proof is a singular perturbation argument developed in [9]. In particular, the resolvent of T2 is given as the uniform limit of (ξ + n-1( - Δ) + T2)-1 as n ^ro, for every ξ > 0, where -A is the Neumann-Laplacian inL2(Q). It should be noted that if N - 1 then (ξ + n-1( -A) + Tp)-1 converges strongly to (ξ + Tp)-1 in LP(I), where Tp is the one-dimensional analog constructed by Campiti, Metafune and Pallara [2],.