Consider $\Omega\subset\mathbb{R}^n$, open bounded,
$$
Lu=f\text{ in }\Omega,\quad u=0\text{ on }{\partial\Omega},
$$
with $Lu=a^{ij}D_{ij}u+b^{i}(x)D_iu+c(x)u=f(x)$, $a^{ij}=a^{ji}$, $L$: strictly elliptic.
Q: How smooth can $u$ get if $f$, $a$, $b$, $c$ is just continuous?
Boundary can be smooth.
I am asking this because: I was studying Gilbarg–Trudinger (and Evans), and thought if there is $C$ equivalent of Sobolev estimates, and bumped into these questions, answers of which says we cannot get $u\in C^2$ if $f$ is just $C$.
Counterexample for the solvability of $-\Delta u = f$ for $f\in C^{0}$
Then I found Theorem 11.1.2 (a) in Partial Differential Equations, 2nd by Jost, GTM 214, which says pretty much
$\Omega\subset{\mathbb{R}^n}$: open, bounded.
$\Omega_0\subset\subset\Omega$. Let $u$ be a weak solution of $\Delta u=f$ in $\Omega$.If $f\in C(\Omega)$, then $u\in C^{1,\alpha}(\Omega)$ ($0<\alpha<1$) and
$$
\|u\|_{C^{1,\alpha}}(\Omega_0)\le c(\|f\|_{C(\Omega)}+\|u\|_{L^2(\Omega)})
$$
Then Jost proceed to the discussion on variable coefficients, where $f\in C^{\alpha}$ ($0<\alpha<1$) is assumed. Can we have a similar result for non-Poisson case as well? Not only interior but also global estimate?
Best Answer
It's possible to construct an $f$ that is continuous such that that the equation $-\Delta u = f$ does not admit a $C^2$ solution. Rather than write out the proof, let me give you the reference you asked for above. Take a look at Exercise 4.9 in Elliptic Partial Differential Equations by Gilbarg and Trudinger.