Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions

Cranston, Fabes and Zhao ([26], [5]) established the uniform bound sup x; y 2 x 6= y R G1;n (x; z)G1;n (z; y) dz G1;n (x; y) M < 1; (1) where G1;n (x; y) is the Green function for the Laplacian - with Dirichlet boundary conditions on a Lipschitz domain - Rn with n 3 (see [27] for n = 2). This estimate was used in [23] and [18] to obtain positivity, uniformly with respect to f 0, for noncooperative elliptic systems as