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: tex2html_wrap_inline1579 . This spatial variation leads to the so-called anisotropic diffusion equation:

displaymath1581

Note that this reduces to (2.1) if c is constant. Two important points to note is that tex2html_wrap_inline1585 does not depend on time, and that we require that c is strictly positive: c>0.

theorem531

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

 

 
 
 

First, let tex2html_wrap_inline1601 . Then tex2html_wrap_inline1603 . On the other hand, the divergence theorem implies tex2html_wrap_inline1515 where tex2html_wrap_inline1517 is the outward normal from the boundary tex2html_wrap_inline1417 of the region tex2html_wrap_inline1207 . Hence:

displaymath1613

We apply the divergence theorem again to tex2html_wrap_inline1615 :

equation553

  1. Let tex2html_wrap_inline1527 . Then:

    2.  

     
     
     

    displaymath1619

    In particular, tex2html_wrap_inline1531 (strict), unless tex2html_wrap_inline1533 everywhere, which in turn requires u to be constant.

  2.  Suppose u attains its maximum value at some point in the interior of tex2html_wrap_inline1207 for some time tex2html_wrap_inline1541 , say at tex2html_wrap_inline1351 . Now, it may be that tex2html_wrap_inline1545 over some time interval tex2html_wrap_inline1547 . Then there is a neighborhood of tex2html_wrap_inline1351 , say tex2html_wrap_inline1551 , tex2html_wrap_inline1553 , and some time interval tex2html_wrap_inline1555 , in which both tex2html_wrap_inline1557 and tex2html_wrap_inline1559 . But then the diffusion equation implies tex2html_wrap_inline1561 in this neighborhood. In particular, tex2html_wrap_inline1545 for some interval tex2html_wrap_inline1547 . Let tex2html_wrap_inline1567 be the earliest time for which this holds.
  3.  If tex2html_wrap_inline1455 at all points on tex2html_wrap_inline1417 , then (2.22) implies tex2html_wrap_inline1663 . Let tex2html_wrap_inline1575 . Then:

    4.  

     
     
     

    displaymath1667