Maximum principle, quantitative version

Question

Is it true that

$$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?

Answer 1

Let $M=\|\Delta f\|_{L^\infty(\Omega)}$. Consider $f^{\pm}(x):=f\pm \frac{M}{2}|x-x_0|^2$ for some $x_0\in \Omega$. Then $f^{\pm}$ is sub/super harmonic, and by the maximum principle

$$ \sup_{x\in\Omega} f\leq \sup_{x\in\Omega} f^{+}\leq \sup_{x\in\partial\Omega} f^+\leq \|f\|_{L^\infty(\partial\Omega)}+\frac{M}{2}\text{diam}(\Omega)^2 $$

and

$$ \inf_{x\in\Omega}f\geq\inf_{x\in\Omega} f^{-}\geq \inf_{x\in\partial\Omega} f^-\geq -\|f\|_{L^\infty(\partial\Omega)}-\frac{M}{2}\text{diam}(\Omega)^2 $$

thus

$$ \|f\|_{L^\infty(\Omega)}\leq \|f\|_{L^\infty(\partial\Omega)}+C\|\Delta f\|_{L^\infty(\Omega)}. $$