Elliptic Regularity Theorem

I want to collect some results on elliptic regularity. The problem I consider is \begin{align} Lu&=f,&in \quad U,\\ u&=g,&on \quad \partial U.\tag{1} \end{align} where $Lu:=a_{ij}(x)u_{x_ix_j}+b_i(x)u_{x_i}+c(x)u.$ is a strictly elliptic operator.

I have known that the $C^{2,\alpha}$-regularity from Gilbarg&Trudinger's book and the $H^2$-regularity from Evans'book. Now I wonder that can the $C^2$-regularity is also available？Namely，can we take $\alpha=0$ in the $C^{2,\alpha}$-regularity. More precisely，I want to make clear that is the following theorem valid？

THEOREM ($C^2$-elliptic regularity ) Let $U$ is $C^2$ bounded domain, $g\in C^2(\bar U)$,$u\in C(\bar U)\cap C^2(U)$ is a classical solution of the Dirichlet problem $(1)$, where $a_{ij},b_i,c,f\in C(\bar U)$. Then $u\in C^2(\bar U)$.

In addition, I also wonder the solvability of $(1)$ in function space $C^2(\bar U)$.Namely,is the following existence theorem valid？

THEOREM ($C^2$-existence) Let $U$ is $C^2$ bounded domain, $g\in C^2(\bar U)$, $c\leq 0$,$a_{ij},b_i,c,f\in C(\bar U)$. Then the Dirichlet problem $(1)$ has a unique solution $u\in C^2(\bar U)$.

Any answer or reference is appreciated! :)

Solution 1:

"$C^2$-existence theorem" is false, even if $L$ is the Laplacian. This is in Gilbarg & Trudinger, problem 4.9. Recently discussed here, where a reference to an older thread on this topic is found.

"$C^2$-elliptic regularity" is also false. For example, the harmonic extension of a $C^2$-smooth function on the boundary of the unit disk $\mathbb D$ is not necessarily in $C^2(\overline{\mathbb D})$. This is discussed in Chapter II of Harmonic Measure by Garnett and Marshall, although for 1st derivatives instead of 2nd. Below I take their example as a starting point.

Consider a conformal map $f:\mathbb D\to \Omega$ where $\Omega=\{x+iy:0<x<\frac{1}{1+|y|}\}$. Clearly, $f$ is unbounded. On the other hand, $\operatorname{Re} f$ has a continuous extension to $\overline{\mathbb D}$ because it has a finite limit even at the points which are mapped by $f$ to infinity.

Write $f(z)=\sum_{n=0}^\infty c_n z^n$ and define $F(z)=\sum_{n=1}^\infty c_n n^{-2}z^n$. I claim that $\operatorname{Re} F$ is the counterexample. Indeed:

• $\operatorname{Re} F$ is $C^2$ smooth on the boundary, because writing $z=e^{it}$ and differentiating in $t$ twice, we get $-\operatorname{Re} f+\operatorname{Re} f(0)$. (To be more rigorous, we can integrate the latter twice to get the former.)
• If all second-order partials of $\operatorname{Re} F$ were bounded in $\mathbb D$, then $F''$ would be bounded. But this is impossible because $z(zF')'=f(z)-f(0)$, which is unbounded.