Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$: $$ \begin{cases} -\Delta u &= 0, \quad \text {in} \quad B(r), \\ \ \ \ \ \ \, u&= g, \quad \text {in}\quad \partial B(r). \end{cases} $$ Suppose $g$ can be extended to a function $\overline g$ defined in $B((1+\sigma)r)$ such that $\overline g \in C^{1,\alpha}(B((1+\sigma)r))$ for some $\alpha, \sigma >0$ and $g=\overline g$ on $\partial B(r)$.

Now my question is that is it possible to extend $u$ to a function $\overline u$ defined in some ball $B((1+\delta)r)=:B_\delta$, for $\delta >0$, such that $\overline u$ is a subsolution in $B_\delta$. Moreover, if possible, I would also like to have $|\overline u_{x_i}|$ to be a subsolution in $B_\delta$, for every $i=1, \dots, n$.