18.42.30 Corrections from the smooth weightBDE9EA5F5F58

The boundary $\Gamma _ t$ can be expanded in powers of $R$, and the leading terms of this expansion give explicit corrections to the uncorrected law. Write

where $P_{2r+1}$ is a real homogeneous polynomial of degree $2r+1$ in $(z,\bar z)$. Let $\Lambda $ be the Dirichlet-to-Neumann operator of the unit circle, $\Lambda \mathrm{e}^{\mathrm{i} n\theta }=|n|\mathrm{e}^{\mathrm{i} n\theta }$. The first boundary coefficient is given by

To derive this, write the interior obstacle, to first order in $R$ and in polar coordinates $(s,\theta )$ for $\zeta $, as $1+2r\log s+R H$ with $H$ harmonic. Matching values on the boundary $s=1+Rb_1$ gives $H|_{s=1}=P$; the terms involving $b_1$ cancel there, since the radial model already matches first derivatives. Matching radial derivatives then gives $4r^2b_1+(2r+1)P-\Lambda P=0$. If $P_{2r+1}$ is harmonic, then $\Lambda P=(2r+1)P$ and $b_1=0$, as it must be, since such a term can be removed by a change of frame.

For $q=O(m^{1/(r+1)})$, the argument of $\Phi $ at the uncorrected observation point of (18.42.28) is

uniformly in $\theta $, and the normalized kernel differs from $\Phi $ of this argument by $o(1)$. The remainder is small because $\sqrt q R^2\to 0$ in this range. For $w=z+cz^2$ we have $P=2r\operatorname {Re}(c\mathrm{e}^{\mathrm{i}\theta })$, and the shift is $2\sqrt{rq}R\operatorname {Re}(c\mathrm{e}^{\mathrm{i}\theta })$, in agreement with the compact example above.

For every fixed integer $J\geq 1$, the parameter construction of lemma 18.42.10 gives

in every fixed norm needed. The coefficients satisfy an explicit recursion, obtained by expanding the interior harmonic obstacle in powers of $R$. At order $j$, matching values determines the boundary data of the new harmonic term from the Taylor term of degree $2r+j$ of $\phi $ and all preceding coefficients, and harmonic extension determines the term itself. Matching derivatives then gives an equation $4r^2b_ j+\text{(known terms)}=0$; the new coefficient $b_ j$ does not enter the value equation of order $j$. For $j\geq 2$, nonlinear contributions of lower-order terms also appear.

Replacing $\Gamma _ t$ by the curve given by the truncated expansion moves the center of the transition by $O(R^{J+2})$. This curve can therefore be used in place of $\Gamma _ t$ whenever

If $b_1=\cdots =b_{j-1}=0$ but $b_ j\not\equiv 0$, the criterion (18.42.29) shows that, for this fixed weight, the uncorrected law holds exactly in the range $q=o(m^{j/(r+j)})$, and this range is sharp when all directions of observation are allowed. Indeed, the higher terms are relatively $O(R)$ at an angle where $b_ j\ne 0$.

In the radial case $\phi =f(u)$ with $u=|z|^2$, the hole is a disk by uniqueness and rotational invariance. By (18.42.11), its boundary circle $\{ u=u_*\} $ is determined by

With $\rho (u)=f'(u)+uf''(u)$, the asymptotic variance of the dominant monomial degree is $m u\rho (u)$, and the corresponding Gaussian argument is

For bounded arguments, Taylor expansion at $u_*$ identifies it with $2\sqrt{m\rho (u_*)}(\sqrt u-\sqrt{u_*})+o(1)$. Hence theorem 18.42.15 proves this radial formula for all sublinear orders. For $f=u^ r+c u^{r+1}+\cdots $, the first shift at the uncorrected observation point is $c(r+1)r^{-3/2}\sqrt q R^2$, so the critical growth is $m^{2/(r+2)}$. This larger radial threshold does not apply in arbitrary coordinates.

No fixed finite Taylor jet of the weight determines the correction for every $q=o(m)$. For example, near zero take

with the second term extended by zero at $u=0$, and interpolated globally as above. It has the same Taylor jet as $u^ r$ at zero to every finite order, but at $u=R^2$ its relative contribution to $uf'(u)$ is $c \mathrm{e}^{-1/R}(1+1/(2rR))$. For $q=\lfloor m/\log m\rfloor $, we have $\sqrt q \mathrm{e}^{-1/R}/R\to \infty $. Hence the uncorrected observation point with $\tau =0$ lies outside the true boundary circle by many transition widths, and the normalized kernel tends to one. The quadrature-domain correction includes this flat contribution.