$\displaystyle \min_{B} u \geq -\|\Delta u\|_{L^{2}}$

Question

Could someone help me with a reference to get the following estimate? I am looking in standard textbooks but without any success.

Let $B=\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\}$ be the unit ball in $\mathbb{R}^{2}$. Let $u\in C^{\infty}(\overline{B})$ be a subharmonic function in a neighbourhood of $B$ such that $u=0$ on the boundary of $B$. Then one has

$$\displaystyle \min_{B} u \geq - C_{0}\|\Delta u\|_{L^{2}(B)}.$$Here $C_{0}$ is some positive absolute constant (for example $C_{0}=10$).

Thanks so much.

Answer 1

This follows from the Sobolev regularity result for the Laplace equation,

$$ \|u\|_{H^2}\leq C\|\Delta\|_{L^2}, $$ and the Sobolev Morrey embedding theorem

$$ \|u\|_{L^\infty}\leq C\|u\|_{H^2} $$

(wich can even be strengthened to $\|u\|_{C^{0,1}}\leq C\|u\|_{H^2}$ in two dimensions). In particular, you don't need the subharmonicity assumption.

As for references, you'll find both results in any introductory PDE book, such as Evans' Partial Differential Equations.