Extend a continuous function on $\partial\Omega$ to inside with higher regularity?

Question

Let $\Omega$ be a $C^3$ (or, $C^k$) bounded domain. Say a continuous function $\phi$ is defined on the boundary, i.e., $\phi\in C^0(\partial\Omega)$.

My question is

i) can we find an extension $\widetilde{\phi}$ such that $\widetilde{\phi}\in C^3(\Omega)$ and $\widetilde{\phi}\big|_{\partial\Omega}=\phi$? I'm not requiring the higher regularity $C^3$ even upto the boundary, because it's gonna be too much.

ii) can we ask the same for $C^2$ extensions? for $C^1$ extensions?

iii) any related results?

Thanks!

Answer 1

Just take any differential operator for which you establish the regularity of the solution of the boundary value problem... Determine $u$ such that \begin{cases} {\mathcal L} u = 0, &\Omega \\ u=\phi, & \partial \Omega\end{cases} and take $u$ as the extension of $\phi$. Example: the Laplace operator ${\mathcal L}:= \Delta$.