I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$
$$A=\sum a_\alpha\partial^\alpha$$
where
- $\alpha=(\alpha_1,\cdots,\alpha_n)\in\mathbb{Z}^n$ is a multiindex such that $\partial^\alpha$ is defined to be $\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}$
- $a_\alpha\in\mathbb{R}$ are constants with depending only on the multiindex $\alpha$.
My questions
- if $f\in C^{\infty}(B_1)$ satisfies
$$
A f=0\;\text{ in } B_1,
$$
when $f$ is analytic in $B_{1/2}$?
I know that if $A$ is $\Delta$ then the result is true, as it happens also if $A=\Delta+1$. The regularity can even apply to distribution solutions. Is the same result true also for other operators? When can we say that a distributional solution is indeed an analytic function? - Similarly, if for a $C^2$ domain $\Omega$ we have $$ A f=0\;\text{ in } \Omega, $$ $$ f=g\;\text{ on } \partial\Omega, $$ where $g$ is a $C^2$ function. When can we have the $f$ is $C^2$ up to the boundary $\partial\Omega$? Again I know that if $A$ is $\Delta$ then the result is true, as it happens also if $A=\Delta+1$, and again the regularity result holds also for distribution solutions. Is the same result true also for other operators? When can we say that a distributional solution is a $C^2$ function up to the boundary?
