PDE simplification Is there a simplification for the following expression in terms of a positive symmetric definite matrix $A(x)$ and the Laplacian $\Delta u$ of a function $u \in \mathrm{R}^{n}\times (0,\infty)$:
$u_{t} = \sum_{i,j=1}^{n} a_{ij}(x) \frac{\partial^{2}u}{\partial x_{i} \partial x_j}$
where $a_{ij}(x)$ is the i,jth element of $A(x)$?  
I know that for:
$u_{t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_{i}}\left(a_{ij}(x) \frac{\partial u}{\partial x_j}\right) = \text{div}(A(x)\nabla u)$.  I'm looking for something similar.  
 A: This is the only thing I can think of(I'm using summation over repeated indices):
$$
a_{ij} \partial_i \partial_ju = \partial_i(a_{ij}\partial_ju) - (\partial_ia_{ij})(\partial_j u)
$$
This can be rewritten as
$$
\text{Tr}(A\nabla(\nabla u))=a_{ij} \partial_i \partial_ju =\partial_i(a_{ij}\partial_ju) - (\partial_ia_{ij})(\partial_j u) = \text{div}(A\nabla u) - \text{div}(A)\cdot\nabla u
$$
Here is what divergence of tensor is. Basically $\text{div}(T)_j = \partial_i T_{ij}$

Edit: I think the term $\text{div}(A\nabla u)$ might be interpreted as Laplace-Beltrami operator. So you are basically solving Laplace equation on some Riemann surface plus you have some first order terms in $\text{div}(A)\cdot\nabla u$. But I don't have real knowledge about this. So someone better extend this idea.
A: If the coefficients are smooth, you can switch between divergence and non-divergence forms at will. Indeed,
$$
 \sum_{i,j=1}^{n} a_{i,j}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}  =
\sum_{i=1}^n\frac{\partial  }{\partial x_{i}} \left(\sum_{ j=1}^{n} a_{i,j}(x)\frac{\partial u}{ \partial x_{j}} \right)
- \sum_{i,j=1}^{n}\frac{\partial a_{i}}{\partial x_i}\frac{\partial u}{\partial x_{j}}
$$
and the first-order term can be combined with what you already have in the PDE.
But you should also consider whether you really want to use the energy method. Generally, maximum principle works better for non-divergence type equations, while the energy method works better for the divergence-type equations.
