Uniform anti-maximum principles

Consider a second or higher order elliptic partial differential equation Vu = lu + l on an open bounded domain Omega of R-n with homogeneous boundary conditions Bu = 0. If there exists a simple eigenvalue for which the corres ponding eigenfunction is positive and satisfies appropriate boundary estima tes, then an anti-maximum principle holds. For positive f is an element of L-p (Omega) with p large enough there exists sigma(f) > 0 such that for lam bda is an element of (lambda(1), lambda(1) + sigma(f)) the solution is nega tive and for lambda is an element of (lambda(1) - delta(f), lambda(1)) the solution is positive. We give conditions such that this sign reversing prop erty is uniform: there is delta > 0 such that for all positive f the soluti on u is negative for l is an element of (lambda(1), lambda(1), + delta) and positive for lambda is an element of (lambda(1) - delta, lambda(1)). Two c lasses of higher order boundary value problems that satisfy these condition s will be given. (C) 2000 Academic Press.