# 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.