Anisotropic Diffusion

Many of the results derived so far can be generalized to the case where the conduction coefficient is not constant, but depends on spatial coordinates: . This spatial variation leads to the so-called anisotropic diffusion equation:

Note that this reduces to (2.1) if c is constant. Two important points to note is that does not depend on time, and that we require that c is strictly positive: c>0.

Proof

The proof closely parallels that of Theorem 6, and basically requires some more careful arithmetic.

First, let . Then . On the other hand, the divergence theorem implies where is the outward normal from the boundary of the region . Hence:

We apply the divergence theorem again to :

 

1. Let . Then:

   

   In particular, (strict), unless everywhere, which in turn requires u to be constant.

2. Suppose u attains its maximum value at some point in the interior of for some time , say at . Now, it may be that over some time interval . Then there is a neighborhood of , say , , and some time interval , in which both and . But then the diffusion equation implies in this neighborhood. In particular, for some interval . Let be the earliest time for which this holds.
3. If at all points on , then (2.22) implies . Let . Then: