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Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying $$ 0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2 $$ where $\Delta$ denotes the Laplace operator. I would like to find a constant $C>0$ such that for every $z \in \mathbb R^2$ one has $$ \max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)} $$ where $B(z)$ denotes the ball of radius one around $z$. The point is here that I want to make $C$ independent of the point $z$.

Can I find such a constant? Thank you in advance!

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  • $\begingroup$ why do you write $e^{f(w)}\le C e^{f(z)}$ instead of $f(w)\le f(z)+C_1$, where $C_1=\log C$? $\endgroup$
    – Fedor Petrov
    Commented Dec 16, 2023 at 11:04

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This is not possible in general: take $f(z)=\|z\|^2$, then $f(x+1/2,0)$ is much greater than $f(x,0)$ for large $x$.

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