Tag Archives: caccioppoli ineq

Holder continuity of weak solution and inverse holder inequality

\textbf{Thm:}  Any weak solution of

\displaystyle (a^{ij}\partial_iu)_j=0

 is automatically holder continuous.

This is a corollary of the celebrated thm of De Giorgi, John Nash and J, Moser. It is obtained from the Harnack inequality of weak solution. Actually in dimension 2, this theorem is easily known by mathematicians before the technique of Moser iteration. In the following, we will give a short expository of the proof

From the Poincare inequality

\displaystyle |u-u_B|\leq C\int_{B_r(x_0)}\frac{|\nabla u(y)|}{|x-y|^{n-1}}dy=I_{1}(\nabla u)

||I_1(\nabla u)||_2\leq C||\nabla u||_p , where \displaystyle \frac{1}{p}-\frac{1}{n}=\frac{1}{2}, p>1 when n=2, p=\frac{2n}{n+2}<2

So this means that \displaystyle \left(\frac{1}{|B_r|}\int_{B_r}|u-u_B|^2dx\right)^{1/2}\leq Cr\left(\frac{1}{|B_r|}\int_{B_r}|\nabla f|^p\right)^{1/p}

Combining with Caccippoli’s inequality, we can get

\displaystyle \left(\frac{1}{|B_\frac{r}{2}|}\int_{B_{\frac{r}{2}(x_0)}}|\nabla u|^2dx\right)^{1/2}\leq C\left(\frac{1}{|B_r|} \int_{B_r(x_0)}|\nabla u|^pdx\right)^{1/p}

with p<2. This is the reverse holder inequality

So \displaystyle |\nabla u|^p\in RH_{\frac{2}{p}}, \frac{2}{p}>1. Thus by Gehring, w\in RH_{\frac{2}{p}+\epsilon}. That is \nabla u\in L^{2+\epsilon}(B^r)

The by the Morrey’s embedding thm, in dimension 2, \displaystyle u\in C^{0,1-\frac{2}{2+\epsilon}}(B_r).

Caccioppoli inequality

A Caccioppoli inequality is of the form

\displaystyle r^p\int_{B_{\frac{r}{2}}(x_0)}|\nabla u|^pdx\leq C\int_{B_r(x_0)}\bigg|u(x)-(u)_{B_{r}(x_0)}\bigg|^pdx\quad\quad (1)

where C is a constant independent of r and u, 1<p<\infty, and \displaystyle (u)_{B_{r}(x_0)}=\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}udx.

Caccioppoli inequality can be considered as the reverse Poincare inequality

\displaystyle \int_{B_r(x_0)}\bigg|u(x)-(u)_{B_{r}(x_0)}\bigg|^pdx\leq Cr^p\int_{B_{r}(x_0)}|\nabla u|^pdx

holds for u\in W^{1,p}(B_r(x_0)).

Of course (1) does not hold for any u\in W^{1,p}(B_r(x_0)), but if u satisfies some second order elliptic equation or certain variational problems, u can satisfies (1).

For example, if u is the weak solution of the divergence form

\displaystyle \text{div}(A\nabla u)=0 or (a^{ij}\partial_ju)_i=0 in B_r(x_0)

here a^{ij} are just bounded measurable and strictly elliptic. \nu I\leq A\leq \nu^{-1}I.

Choose a cut off function \eta\in C^\infty_c(B_r(x_0)) and \eta=1 in B_{\frac{r}{2}}(x_0) and \displaystyle |\nabla\eta|\leq \frac{4}{r}. Let v=\eta^2(u-\overline{u})\in H^1_0(B_r(x_0)), here we use \overline{u}=(u)_{B_{r}(x_0)} for abbreviation. Then

\displaystyle \int_{B_r(x_0)}a^{ij}\partial_ju\partial_iv=0

\displaystyle \int_{B_r(x_0)} a^{ij}\partial_ju\eta^2\partial_iu=-\int_{B_r(x_0)} 2a^{ij}\partial_ju\eta\nabla\eta(u-\overline{u})

\displaystyle \nu\int_{B_r(x_0)}\eta^2|\nabla u|^2dx\leq C\int_{B_r(x_0)} |\partial_ju\eta\nabla\eta(u-\overline{u})|

\displaystyle \int_{B_r(x_0)}\eta^2|\nabla u|^2dx\leq C(\nu)\int_{B_r(x_0)}|\nabla\eta(u-\overline{u})|^2dx\leq \frac{C}{r^2}\int_{B_r(x_0)} |u-\overline{u}|^2dx

Noticing that \eta=1 on B_{\frac{r}{2}(x_0)}, we can get (1)

\textbf{Remark:} If u satisfies (a^{ij}\partial_ju)_i+b^i\partial_iu=0, we can also get the same conclusion.

MA6000A: Theory of Partial Differential Equations. Roger Moser