There are a lot of regularity theorems for regularity of second order elliptic PDEs of divergence form $(a^{ij}u_i)_j=f$. Here the equality holds in the weak sense. If $a^{ij}, f$ satisfy some kind of regularity, then we can expect some regularity for $u$. Like De Giorgi's method. My question is, how bad the solutions can be? For example, if $a^{ij}, f$ are bad, can there be discontinuous solutions?
Another slightly different viewpoint is, if $a^{ij}$ are $C^1$, by regularity theory $u$ will be in $H^2$.($f$ will aotumatically be in $L^2$ to satisfy the definition of weak solution.) Under such assumption, the question will be, is there any "bad" function in $H^2$. Sobolev embedding theory gives me some guess that such function might exist, but I'm not sure.
Any such examples will be appreciated. Like discontinuous, unbounded, or not $C^{\alpha}$. Thanks for advance.