Maximum principle for heat eq. with boundary conditions on derivatives

The Maximum principle for parabolic eq. is based on the fact that the boundary conditions are given on u. How can this Maximum principle be used, when having boundary conditions including derivatives.

i.e.: the heat equation , x in [0,1], t>0. and boundary conditions given: u(0,t)=0; du/dx (1,t) = 0; u(x,0)=f(x);

flag	asked Jun 29 2011 at 10:27 Shira 11●1

Used what for? Perhaps the question should be is where an analogue to maximum principle fot this problem? – Andrew Jun 29 2011 at 11:26

The maximum principle is still the same. It just might be harder to apply if one does not know the boundary values exactly... – Yakov Shlapentokh-Rothman Jun 29 2011 at 13:54

In the particular case where you prescribe Neumann type conditions, you can also appeal to the parabolic Hopf lemma. – Willie Wong Jun 29 2011 at 14:40

The problem with applying Hopf lemma, is that in the current case the boundary conditions are given in a corner (i.e. at x=1),while the Hopf lemma requires a circle tangent to the boundary. Is there an analogical Lemma in this case? A different direction that I am considering, is to define w=du/dx. w satisfies the heat eq. with boundary conditions given on w (and not its derivatives). and I will be able to apply the maximum principle. Is this direction is a valid one? (my concern is that we are given u(0,t) = 0. and I can not take x derivative from this) – Shira Jul 4 2011 at 9:42