Elliptic Equations, Principal Eigenvalue and Dependence on the Domain

We consider a general second order uniformly elliptic diierential operator L and also the set of all open sets (not neccessarily smooth) in the unit ball of I R n. We deene a metric d in this set (up to an equivalence relation) that makes the space ((= ; d) a complete metric space. We show that the principal eigenvalue and eigenfunction of L are continuous with the metric d. Similar results are obtained for the solutions of the equation Lv = f. 1 Introduction. The main motivation of this paper comes from the work of Beresticki, Nirenberg and Varadhan in 6]. In this paper, for a general bounded domain (not neccessarily smooth) the relation between the sign of the principal eigenvalue and a reened version of the maximum principle is obtained. Also, the existence of a principal eigenfunction is established along with a general existence theorem of bounded solutions of an elliptic equation in a general bounded domain. To be more especiic let us denote by L an elliptic operator of the form L = a ij @ ij + b i @ i + c (1.1) where there exist positive numbers c 0 and C 0 with c 0 jj 2 a ij i j C 0 jj 2 (1.2) Also,