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 $i=1, \dots, n$.



Extending a harmonic function in a ball to subharmonic in a larger ball
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. 


