The minimal surface equation is $$ \sum D_i \left(\frac{D_i u }{\sqrt{1+ |Du|^2}}\right)=0 $$ How to show it is a non-uniform elliptic equation ?
What I try: $$ D_i \left(\frac{D_i u }{\sqrt{1+ |Du|^2}}\right) = \frac{D_{ii}u}{\sqrt{1+ |Du|^2}} -\frac{D_iu \sum D_juD_{ij}u}{2(1+ |Du|^2)^{3/2}} $$ so the coefficient of $D_{ii}u$ is $$ \frac{1}{\sqrt{1+ |Du|^2}} -\frac{(D_iu)^2 }{2(1+ |Du|^2)^{3/2}} $$ the coefficient of $D_{ij}u$ is $$ \frac{D_iu D_ju}{2(1+ |Du|^2)^{3/2}} $$ then, how to get the minimum eigenvalue and maximal eigenvalue of the coefficient matrix ? In my read book, it says the ratio of minimum eigenvalue and maximal eigenvalue is $1+ |Du|^2$. But I can't calculate it.