**Idea**

Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) continuous. If you have better regularity away from one isolated point, say you are $C^1$ on the puncutered ball, can the solution still fail to be differentiable at that isolated point?

The situations I am most familiar with tend to be ones in which one can go from boundedness and continuity to smoothness via Schauder theory and bootstrapping. Here, we already have continuity: But can differentiability fail at a single point? TIt seems out of reach of all the theorems I've seen, which makes me suspect it is false, but I cannot be sure without a counterexample. Does anyone have any ideas?

**Details**

In the specific situation I am interested in, I know a bit more. I am considering the following:

$u \in W^{1,\infty}(B_1(0)) \cap C^{1,1}_{loc}(B_1(0)\setminus ${0}$)$ satisfies weakly the equation

$D_i(A_{ij}(x)D_ju) = D_ig^i$

in $B_1(0)$, where $A_{ij},g^i \in L^{\infty}(B_1(0))\cap W^{1,2}_{loc}(B_1(0)\setminus ${0}$)$.

**Questions**

1) Must $u$ in fact be a $C^1$ solution on $B_1(0)$?

2) What about just being differentiable at 0?

3) How about even just $u \in W^{2,p}(B_1(0))$ for some $p > 1$?