Maximum principle type problem

Question

Suppose $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ is sufficiently smooth and is such that $(1/2) u_{xx} + u_{xy} + 2u_{yy} = 0$ in a ball $B$ centered at the origin. Must $u$ attain a maximum inside $B$? This seems like a maximum principle type problem, but one can't apply directly the elliptic maximum principle since the differential operator is not elliptic.

Answer 1

You can apply the standard maximum principle (see, cf. here), if you write your PDE in the form $$ \sum_{i=1}^{2} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} = 0, $$ where $$ a_{11} = \frac{1}{2}, \quad a_{12} = \frac{1}{2}, \\ a_{21} = \frac{1}{2}, \quad a_{22} = 2. $$ You can apply it, since the matrix $\{a_{ij}\}$ is symmetric and positive definite. You can check it by Sylvester's criterion, or something similar.