Estimate solution of Poisson equation on unit ball Consider the following boundary value problem where $U=\{x \in \mathbb{R}^3 \mid |x|<1\}$ and $g$ is some nice bounded function,
$$\Delta u = g ~~~ \text{on}~U\\
u=0 ~~~\text{on} ~\partial U.$$
Assume that $x_0 \in U$ with $|x_0|=r$ for some $0<r<1$ such that $x_0$ lies (very) close to the boundary of $U$.
Is it possible to find an upper bound for $|u(x_0)|$?
Any idea is much appreciated!
 A: Let $w(x)=a(|x|^2-1)$, where $a$ is a constant whose value will be adjusted shortly. By construction, we have
$$
w|_{\partial U} = 0.
$$
On the other hand, we compute
$$
\Delta w = 6a,
$$
and so by choosing
$$
a=\frac16\inf_Ug,
$$
we ensure
$$
\Delta u\geq\Delta w\qquad\textrm{in}\,\, U.
$$
Thus the comparison principle gives
$$
u\leq w\qquad\textrm{in}\,\, U,
$$
that is,
$$
u(x)\leq\frac{|x|^2-1}6\inf_Ug.
$$
A: Yes you can.
In fact, there is a theorem :

Theorem (a priori bound) : Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$ be a solution of the Dirichlet problem $Lu=f$ in $\Omega$, $u=g$ on $\partial\Omega$, with $\Omega$ bounded, and $Lu=-\sum_{i,j=1}^na_{ij}(x)\partial_{x_i,x_j}u(x)+\sum_{i=1}^nb_i(x)\partial_{x_i}u(x)+c(x)u(x)$ is uniformly elliptic with $c\geq 0$. Then $$||u||_{C^0(\overline{\Omega})}\leq||g||_{C^0(\partial\Omega)}+C\sup_{x\in\Omega}\frac{|f(x)|}{\lambda(x)}$$ with $C=e^{2(B+1)}-1$, and $B:=\sup_\Omega \frac{|b(x)|}{\lambda(x)}$.

Where uniform ellipticity of the operator is defined as :

Definition : $L$ is said to be elliptic in $\Omega$ if there exists positive functions $\lambda,\Lambda :\Omega\rightarrow \mathbb{R}_+$ such that $$0<\lambda(x)|\xi|^2\leq \xi^TA(x)\xi=\sum_{i,j}a_{i,j}(x)\xi_i\xi_j\leq\Lambda(x)|\xi^2|,$$ for all $x\in\Omega$, $\xi=(\xi_1,\cdots,\xi_n)\in\mathbb{R}^n$. $L$ is uniformly elliptic if there exists $\lambda_0,M\in\mathbb{R}$ such that $$\lambda(x)\geq \lambda_0 >0\text{ and }\frac{\Lambda(x)}{\lambda(x)}\leq M\leq \infty,$$ for all $x\in\Omega$. 

So In this case, (by the other theorem we know the solution exists) $g=0$, $\Omega=S^3$, and $L=\Delta$, which is uniformly elliptic with $\lambda(x)=1$. Therefore, we have a priori bound : $$||u||_{C^0(\overline{\Omega})}\leq (e^2-1)||g||_{C^0(\Omega)}.$$
But this bound covers all $x\in\Omega$, so almost surely not a optimal bound for $x_0$. Similar to @timur did, you can have a super solution that is explicit, and try to get a better bound.
