1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What makes you think that the operator is not elliptic? $\endgroup$ – mkl314 May 13 '14 at 21:02
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.