Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I am trying to find references to the following kinds of estimates: $$ \Vert Lu(t_0,.)\Vert_{L^\infty (B_{1/2})} \leq C\Vert u\Vert_{L^2(B_1)}$$

To me, they look somewhat like the Moser estimates, but not quite. I would be really grateful for a reference. Thanks!

Edit: Coefficients of $L$ are smooth, and $L$ is 2nd order.

  • 1
    $\begingroup$ What are the assumptions of the (regularity of) the coefficients of $L$? $\endgroup$
    – Fan Zheng
    Apr 16 '15 at 2:21
  • $\begingroup$ @FanZheng They are smooth. $\endgroup$
    – student
    Apr 16 '15 at 13:22
  • $\begingroup$ Why can't you do Moser iteration applied to the PDE's satisfied by $u$, $(\partial_t, \nabla_x)u$, and $\partial_t^2 + L)u$? $\endgroup$
    – Deane Yang
    Apr 16 '15 at 18:50
  • 1
    $\begingroup$ @DeaneYang I don't know this theory that well. Could you please mention a reference where I can read up what you say? $\endgroup$
    – student
    Apr 16 '15 at 18:54
  • $\begingroup$ I suggest googling "Moser iteration PDE". One of the hits is a book by Jurgen Jost on PDE's. That seems like a reasonable place to look. $\endgroup$
    – Deane Yang
    Apr 16 '15 at 21:09

Integrating by parts, using a cutoff function if necessary, we have

$$ \int_{B(r)} |\nabla u|^2\ll \int_{B(r)} a_{ij}\partial_iu\partial_ju=\int_{B(R)} uLu+O(u|\nabla u|)\ll \int_{B(R)} |u\nabla u|. $$

Using Cauchy Schwarz, you can get

$$ \int_{B(r)} |\nabla u|^2 \ll \int_{B(R)} |u|^2. $$

(If $u$ does not have $W^{1,2}$ regularity to start with, you can approximate the left hand side by difference quotients.)

Now do the same for $\nabla u$, $\nabla^2 u$, etc. You will get

$$ \|u\|_{W^{k,2}(B(1/2))}\ll \|u\|_{L^2(B(1))} $$

for each $k$. Choose a large enough $k$ so that you have

$$ \|u\|_{W^{2,\infty}} \ll \|u\|_{W^{k,2}} $$

and you're done.

NOTE: The following solution that I originally posted is wrong.

This is an expansion of Yang's suggestion. Moser iteration (combined with some Maximum-principle style arguments, see Chapter 8 of Gilbarg-Trudinger) would give you $C^\alpha$ bound for $u$. Then you fix the coefficients locally and rewrite the PDE as

$$\partial_tu+\Delta u=(\Delta-L)u$$

and treat the right hand side as an inhomogeneous term (which has been shown to lie in $C^\alpha_{loc}$). Then Schauder theory for the Poisson equation (Chapter 4 of Gilbarg-Trudinger) would give you local $C^{2,\alpha}$ bound for $u$, which is more than what you have asked.

  • $\begingroup$ Why is right side in $C^\alpha$? $\endgroup$
    – Deane Yang
    Apr 19 '15 at 20:13
  • $\begingroup$ @DeaneYang Right, what I wrote was nonsense. $\endgroup$
    – Fan Zheng
    Apr 20 '15 at 0:44
  • $\begingroup$ Nice and simple. The more involved Moser iteration or Schauder is needed only if you need weaker regularity assumptions on the coefficients. For example, for a nonlinear PDE. $\endgroup$
    – Deane Yang
    Apr 20 '15 at 12:08

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