Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Abstract

We consider linear elliptic equations 1u + q(x)u = u + f in bounded Lipschitz domainsD R N with mixed boundary conditions@u/@n =(x)u +g on@D. The main feature of this boundary value problem is the appearance of both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient (x) . We study positivity principles and anti-maximum principles. One of our main results states that if is somewhere negative, q 0 and R D q(x)dx > 0 then there exist two eigenvalues 1, 1 such the positivity principle holds for 2 ( 1, 1) and the anti-maximum principle holds if 2 ( 1, 1 + ) or 2 ( 1 , 1). A similar, but more complicated result holds if q 0. This is due to the fact that 0 = 0 becomes an eigenvalue in this case and that 1() as a function of connects to 1() when the mean value of crosses the value 0 = | D|/|@D|. In dimension N = 1 we determine the optimal -interval such that the anti-maximum principles holds uniformly for all right-hand sides f,g 0. Finally, we apply our result to the problem 1u +q(x)u = u +f inD,@u/@n =u +g on@D with constant coefficients , 2 R.