Estimate of minimum of the Poisson integrals corresponding to a convergent Hausdorff sequence of smooth bounded domains from below

Question

Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be a sequence of smooth bounded domains in $\mathbb{C}^{n}$ such that $\Omega_{j}$ converges to a smooth bounded domain $\Omega$ in the sense that the defining function of $\Omega_{j}$, say $\rho_{j}$ converges to the defining function of $\Omega~~$,say $\rho$ uniformly on each compact subset of $\mathbb{C}^{n}$, where a defining function of a smooth bounded domain $D$ is defined as a smooth real-valued function say $\psi$ on $\mathbb{C}^{n}$ such that $\psi$ satisfies the following: $$ \begin{align} D =\{z\in\mathbb{C}^{n}:\psi(z)<0\}&,\\ \partial D =\{z\in\mathbb{C}^{n}:\psi(z)=0\}&,\text{ and}\\ \operatorname{grad}(\rho)\neq0\text{ on }\partial D.& \end{align}$$

This notion of convergence implies the Hausdorff convergence of open domains. Precisely if I say: For any compact set $K\subset\Omega$, then $K\subset\Omega_{j}$ for sufficiently large $j$. Also, for any compact set $L\subset\mathbb{C}^{n}\setminus\overline{\Omega}$, then $L\subset\mathbb{C}^{n}\setminus\overline{\Omega_{j}}$. My question is: for fixed $x_{0}\in \Omega$ can we imply that there exists a positive constant $c$ such that $\min\{P_{j}(x_{0},y):y\in\partial\Omega_{j}\}\geq c$ for large $j$, where $P_{j}$ denotes the Poisson kernel of $\Omega_{j}$. It would be a great help if any suggestions I get.

Answer 1

No. Take $\Omega$ to be, say, the unit disc $B_1$ in $\mathbb{C}$ with the defining function $\psi(z)=|z|^2-1$. In fact, any sequence of smooth domains $\Omega_j$ such $B_{1-\frac{1}{j}}\subset\Omega_j\subset B_1$ converges to $\Omega$ in your sense: if $\psi_j$ are defining functions for $\Omega_j$, then, using a partition of unity, you can modify them so that they coincide with $\psi$ on $B_{1-\frac{2}{j}}$.

Now, if $\Omega_j$ have long fjords, then the Poisson kernel inside those fjords can be arbitrarily small. Indeed, imagine the fjord is a strip divided into equal squares by segments $l_1,l_2,\dots $, with $l_1$ closest to the dead-end. Then, $$\max_{y\in l_n} G(x_0,y)\leq \max_{y\in l_n} \mathrm{hm}(y,l_{n+1})\cdot\max_{y\in l_{n+1}} G(x_0,y)\leq c\cdot \max_{y\in l_{n+1}} G(x_0,y),$$ where the harmonic measure is taken in the part of the fjord cut out by $l_{n+1}$, and $c<1$ is the corresponding maximum when one replaces that part by a half-infinite strip. Consequently, the Green's function inside the fjord is exponentially small in the aspect ratio of the fjord, and by Harnack estimate so is its normal derivative.