For a function $f(x,t)$ I'm aware of a fairly strong result about the PDE:

$\sigma^2 f_{xx} + f_t = 0,$

$f(x,0) = h(x)$,

which guarantees that any local maxima of $f$ or any of its derivatives in $x$ can only exist along the boundary $t = 0$. I'm wondering if a similar result holds for a more complicated version of the PDE:

$\sigma^2(g_{xx},g_x,g,x,t) g_{xx} + g_t = 0,$

$g(x,0) = h(x)$.

I've been able to show that any local maxima of $g$ and $g_x$ exist only along the boundary $t = 0$ but I am having trouble with $g_{xx}$.

I'm wondering whether or not the maximum principle can be extended this far, and if so what conditions I would need on the function $\sigma$ to make it work.

  • $\begingroup$ Have you derived the differential equations satisfied by $u_x$, $u_{xx}$ etc. and isolated the general form, i.e. a form which is so general that all the others are but special cases, yet so specific that you can still derive a maximum principle for this general form? $\endgroup$ – Carl Christian Feb 16 '16 at 17:19
  • $\begingroup$ Not sure if I follow your comment completely, but that sounds like the approach I've tried using. The problem is that differentiation of the PDE introduces a bunch of new terms resulting from the partial derivatives of $\sigma$. With $g_x$ the cost is minimal but with, say, $u = g_{xx}$ the PDE includes a factor with the second derivative of $\sigma^2$ in $x$ multiplying $u$ itself. $\endgroup$ – jjfunk Feb 16 '16 at 17:27
  • $\begingroup$ I can not promise you that it will work, but you need ignore the fact that the expressions are nasty and bundle as many terms together as you can in the effort to isolate the general form. A small example: $u_t = a(u)u_{xx}$. Then $(u_x)_t = a(u) (u_x)_{xx} + (a'(u)u_x) (u_x)_x$. Defining $v = u_x$ we have $v_t = a(x,t) v_{xx} + b(x,t) v_x$. For the purpose of estimating $v$, the complex expression for $b$ is irrelevant, as it dies at all stationary points. $\endgroup$ – Carl Christian Feb 16 '16 at 17:37
  • $\begingroup$ Agreed, but my understanding is that successive derivatives in $x$ do not respect this new form. For instance if we take your example one step further so that $w = v_x$, then $w_t = a(x,t) w_{xx} + b(x,t) w_x + c(x,t) w$. The inclusion of $c(x,t)w$ has been the dead end for me so far, not to mention any further derivatives which would no longer be PDEs. $\endgroup$ – jjfunk Feb 16 '16 at 17:48
  • $\begingroup$ I am exhausted, so I will just comment briefly. The creation term $cw$ gives exponential growth unless is is always negative in which case it does not matter. You can find maximum principles for the last form you named in Fritz John's book "Partial differential equations". When you differentiate once more, then you get the term $c_x w$ but you can treat that as an inhomogenous term with respect to $w_x$! And then apply your previous maximum principle for $w$ to estimate $w_x$. Admittedly, it get nasty :) $\endgroup$ – Carl Christian Feb 16 '16 at 18:05

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