For a smooth, bounded domain $\mathcal{O} \subset \mathbb{R}^n$, consider $u$ the solution to $$ \begin{cases} \Delta u &= f \qquad \text{ on } \mathcal{O}\\ u &= 0 \qquad \text{ on } \partial \mathcal{O}. \end{cases} $$ We assume $f$ is such that a solution $u$ exists, but we do not assume anything on its sign (e.g. being strictly positive/negative). Can we bound the supremum norm of $u$ by the $L^2$ norm of its Laplacian, i.e. does there exist a constant $C \geq 0$, depending on the domain $\mathcal{O}$, such that $$ \|u\|_{\infty} \leq C \|f\|_{L^2(\mathcal{O})} $$ for all $f \in L^2(\mathcal{O})$ for which the PDE has a solution?
In one dimension ($n = 1$) this is trivial, but I don't know if this is the case for higher-dimensional domains.
This is a classical result due to Stampacchia. It is Theorem B.2 in
Kinderlehrer; Stampacchia: An introduction to variational inequalities and their applications. Pure and Applied Mathematics, 88. Academic Press, Inc.
There $f\in L^p(\mathcal O)$ with $p>n$ was required. One can improve the proof easily to allow for $f\in L^p(\mathcal O)$ with $p>n/2$, see Thm 4.5 in
Tröltzsch: Optimal control of partial differential equations. Theory, methods and applications. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010.
So the answer to your question is: The result is true for $n\le3$.
The method of proof is very elementary, no assumptions on smoothness of the domain or coefficients are necessary. It can be applied for monotone elliptic pdes and variational inequalities as well.