This question was sparked from the fact that, using the mean value property, you can bound the value of $u(1/2)$ given that $u$ is a positive harmonic function on $D(0, 1)$ and $u(0)=1$: \begin{align*} u(0) &= \frac{1}{\pi} \int_{D(0,1)}u \\ &\leq \frac{1}{\pi}\int_{D(1/2, 1/2)}u \\ &=4u(1/2) \end{align*} you can also get an upperbound by considering $1/u$. More generally, you get $$(1-r)^2 \leq u(r)\leq \frac{1}{(1-r)^2}$$ Now, $\operatorname{Re}\left(\frac{1}{(1-z)^2}\right)$ achieves this maximal bound but it is not positive on $D(0, 1)$. I suppose what I'm looking for is a positive harmonic function such that $$\int_{[0, 1-\epsilon)} u$$ is maximized for all $\epsilon>0$. Does such a harmonic function exist? And more generally, given a piecewise differentiable curve $\gamma$ in a plane region $\Omega$ does there exist a positive harmonic function on $\Omega$ such that $u(\gamma(0))=1$ and $$\int_{\gamma_{[0, 1-\epsilon)}} u$$ is maximized for all $\epsilon>0$?
- 2$\begingroup$ Positive harmonic functions in a region with value 1 at a fixed point form a compact family (this is a consequence of Hanrack's theorem). Therefore a function maximizing the integral exists. $\endgroup$Alexandre Eremenko– Alexandre Eremenko2025-01-19 14:36:20 +00:00Commented Jan 19 at 14:36
1 Answer
A positive harmonic function on a disc can be represented* by an integral of a Poisson kernel: $$ u(z)=\frac{1}{2\pi}\int_{\partial\mathbb{D}}\Re\left[\frac{\zeta+z}{\zeta-z}\right]d\mu(\zeta), $$ where $\mu$ is a positive measure. The condition $u(0)=1$ ensures that the total mass of $\mu$ is one. For a given $z$, the kernel $\Re\left[\frac{\zeta+z}{\zeta-z}\right]$ is maximized for $\zeta =z/|z|$, hence the whole integral is maximized when $\mu$ is a delta-mass at $z/|z|$. Since the maximizer is the same for each of the point $z\in[0,1-\epsilon),$ it also maximizes $\int_{[0,1-\epsilon)}u$.
(*) This can be shown by considering weak limits of the restrictions of $u$ to circles of radii tending to one, or circumvented by applying the argument above to these smaller circles and then passing to the limit.
