maximum principle – The first eigenvalue of Dirichlet is the largest of the eigenvalues ​​corresponding to the sub-resolutions.

Let $ Omega $ to be an open domain, delimited in $ R ^ n $. Let $ Lu = sum (a_ {ij} (x) u_ {x_i}) _ {x_j} + c (x) u $ to be the operator where $ a_ {ij} = a_ {ji} $ are in $ C ^ 1 $ and $ c (x) $ is continuous and $ L $ is uniformly elliptical operator. $ Omega_0 $ to be an open set bounded with $ C ^ 2 $ limit, whose closure is compact in $ Omega $. CA watch $$ lambda ( omega_0) = sup { mu in mathbb {R}: text {there is a positive positive} phi in C ^ 2 ( Omega_0) st (L + mu ) phi le0 } $$ Right here $ lambda ( omega_0) $ is the first eigenvalue of Dirichlet of $ -L $.

I could prove it $ lambda ( omega_0) $ is less than the supremum but could not get the opposite inequality. Any help is welcome.