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).

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