Diffusion Equation Maximum Principle

Question

Does the general diffusion equation satisfy the maximum principle? I.e., is its maximum value obtained on the boundary of the region?

Answer 1

Without being too rigorous here - the general principle is that if it attains a maximum away from the boundary, then at that point the first derivative is zero, and the second derivative matrix is negative definite. So I think if $D_{ij}(\phi)$ is positive definite, then I think you will be OK. This is a pseudo proof, of course, because the second derivative matrix might be negative semi-definite. But the rigorous proof is a mild modification of this idea. (And you will need appropriate regularity of $D_{ij}(\phi)$ and stuff like that.)