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

1. 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?
2. Similarly, if for a $C^2$ domain $\Omega$ we have $$ A f=0\;\text{ in } \Omega, $$ 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?

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

1. 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?
2. 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?

I'm learning elliptic PDE and a natural question came to me. For a constant linear differential operator and $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$

$$A=\sum a_\alpha\partial^\alpha$$

where $a_\alpha\in\mathbb{R}$ are constants with respect to $\alpha$ and $\alpha=(\alpha_1,\cdots,\alpha_n)\in\mathbb{Z}^n$ such that $\partial^\alpha$ is defined to be $\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}$. My questions is, if $f\in C^{\infty}(B_1)$ satisfies

$$A f=0\text{ in } B_1,$$

when can we have $f$ is analytic in $B_{1/2}$? I know if $A$ is $\Delta$ then we have the result. Furthermore for $A=\Delta+1$ we still have the result. The regularity can even apply to distribution solutions. Do we have the same results for other operators? When can we raise the regularity of a distribution to an analytic function?

Similarly, if for a $C^2$ domain $\Omega$ we have

$$A f=0\text{ in } \Omega,$$

when can we have $f$ is $C^2$ up to $\partial\Omega$? I know if $A$ is $\Delta$ then we have the result. Furthermore for $A=\Delta+1$ we still have the result. The regularity can even apply to distribution solutions. Do we have the same results for other operators? When can we raise the regularity of a distribution to a $C^2$ function up to boundary?

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

1. 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?
2. Similarly, if for a $C^2$ domain $\Omega$ we have $$ A f=0\;\text{ in } \Omega, $$ 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?