ON COURANTS NODAL DOMAIN THEOREM

We study the validity of Courant's nodal domain theorem for eigenfunct ions of selfadjoint second order elliptic operators with low regularit y assumptions on the coefficients. We prove that in two dimensions Cou rant's theorem holds also when the coefficients are just bounded and m easurable. In the higher dimensional case, we prove a weakened version of Courant's theorem when the coefficients in the principal part are Holder continuous.