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!
Best Answer
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, $$ u(x)\leq\frac{|x|^2-1}6\inf_Ug. $$