Extending a harmonic function in a ball to subharmonic in a larger ball 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 $i=1, \dots, n$.
 A: One can build a distributional subsolution which is a Lipschitz extension of $u$ by making an extension with a positive jump in radial derivative across the boundary. Say $u$ is harmonic on $B_1$ and we want to subharmonically extend to $B_2$.
Let $v$ be a harmonic function on $B_2-B_1$ with boundary data $g$ on $\partial B_1$ and, say, constant on $\partial B_2$.
Since the boundary data are $C^{1,\alpha}$ we have that $|\nabla u|,|\nabla v| < C$ globally. Let
$$w(x) = \begin{cases}
          u(x), \quad x \in B_1 \\
          v(x) + C(|x|^2-1), x \in B_2 - B_1
         \end{cases}$$
Then $w$ is a Lipschitz extension of $v$ with a positive jump in the radial derivative across $\partial B_1$. Let $w_r^+$ denote the radial derivative from outside and $w_r^-$ denote the radial derivative from inside.
We can now show that $w$ is distributionally subharmonic. Since it is harmonic in $B_1$ and subharmonic outside $B_1$, we just need to check on $\partial B_1$.
Take any ball $B_{\eta}$ centered on $\partial B_1$ and let $\nu$ be the outer normal to $\partial B_{\eta}$. We compute (using that $w$ is harmonic in $B_1$)
$$\int_{\partial B_{\eta}} w_{\nu} = \int_{\partial B_{\eta} \cap B_1^c} w_{\nu} - \int_{B_{\eta} \cap \partial B_1} w_r^-.$$
Since 
$$0 < \int_{B_{\eta} \cap B_1^c} \Delta w = \int_{\partial B_{\eta} \cap B_1^c} w_{\nu} - \int_{ B_{\eta} \cap \partial B_1} w_r^+,$$
and $w_r^- \leq w_r^+$ we conclude that
$$\int_{\partial B_{\eta}} w_{\nu} \geq 0$$
showing the desired property. 
