show/hide this revision's text 2 added 456 characters in body

I have

Idea

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

The precise 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 is 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 $p > 1$?)

This seems just out of reach of regularity theory/removable singularity theorems that I've seen so far and I'm starting to doubt that it is true, but it would be nice to be sure.1$?
show/hide this revision's text 1

Divergence form Elliptic PDE Removable Singularity/Regularity Question

I have a weak, Lipschitz solution to a divergence form elliptic PDE with bounded coefficients in the unit ball. If I know that the solution is in fact C^1 on the punctured unit ball, can it still fail to be differentiable at the origin?

The precise situation I am considering is 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:

Must $u$ in fact be a $C^1$ solution on $B_1(0)$? What about just being differentiable at 0? (How about even just $u \in W^{2,p}(B_1(0))$ for some $p > 1$?)

This seems just out of reach of regularity theory/removable singularity theorems that I've seen so far and I'm starting to doubt that it is true, but it would be nice to be sure.