Multivariable Derivatives/Elliptic PDE

[Axiomer]

From http://en.wikipedia.org/wiki/Elliptic_operator:
"A nonlinear operator $$L(u) = F(x, u, (\partial^\alpha u)_{|\alpha| \le 2k})$$ is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator."

I'm a bit confused by what we mean by taylor expansion w.r.t. u and derivatives... So say we have an (made up) example where $u:\mathbb{R}^3→\mathbb{R}$, and we have a second order operator given by $L(u) = F(x, y, z, u, (\partial^\alpha u)_{|\alpha| \le 2}) = z^2uΔu$, what are the derivatives and taylor expansion of interest?

Are there simpler ways of classifying quasilinear/non-linear PDEs as being elliptic?

 

[gtoguy97]

The Taylor expansion of interest is the expansion of the function F(x, y, z, u, (\partial^\alpha u)_{|\alpha| \le 2}) around a given point. This expansion is a sum of terms involving powers of u and its derivatives up to second order. The coefficients of these terms must be such that the resulting operator is an elliptic operator.For example, if we consider the given function F, its Taylor expansion can be written as F(x, y, z, u, (\partial^\alpha u)_{|\alpha| \le 2}) = c_0 + c_1u + c_2(\partial^2u) + c_3u^2 + c_4(\partial^2u)^2 + ...where the coefficients c_i must be chosen so that the resulting operator is elliptic.Another way of classifying quasilinear/non-linear PDEs as elliptic is to check whether the equation satisfies certain criteria for ellipticity. These criteria include the Gâteaux conditions, the Legendre conditions, and the Hörmander conditions.