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

Question

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 $i=1, \dots, n$.

Answer 1

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.