= Solution
Outside a core of radius $r_0$, set $\lambda\simeq\lambda_0$ to leading order. The uniform-state bulk energy cancels on subtraction, while the divergence elastic density is $K\lambda_0^2/(2r^2)$. Its two-dimensional area integral is
$$
\Delta F_{\rm outer}=2\pi\int_{r_0}^L\frac{K\lambda_0^2}{2r^2}r\,dr=\pi K\lambda_0^2\log(L/r_0).
$$
Thus the <logarithmic energy of a planar nematic disclination> gives
$$
\boxed{\Delta F=\pi K\lambda_0^2\log(L/r_0)+O(K\lambda_0^2),\qquad L\gg r_0.}
$$
The core suppresses the nematic amplitude to regularize the singular director texture. Balancing lost bulk ordering energy against elasticity gives $r_0$ of order $\sqrt{K/|a|}$. The exact core radius and additive core energy depend on the amplitude profile.
The “local” amplitude disturbance need not be literally compactly supported. The radial <Euler-Lagrange equation> following from the PDF density is
$$
a\lambda+b\lambda^3-\frac K4\left(\lambda''+\frac{\lambda'}r-\frac{4\lambda}{r^2}\right)=0.
$$
Its far-field expansion is $\lambda=\lambda_0[1-K/(2|a|r^2)+O(r^{-4})]$. These algebraic tails change only finite contributions, not the logarithm. A regular core has $\lambda=O(r^2)$ for this integer winding.
The large cutoff $L$ is set by the sample size for an isolated defect with corresponding imposed far-field winding, or by the distance to compensating defects or other texture-screening boundaries. An isolated nonzero winding in an infinite ordered plane has a logarithmically divergent excess energy; in a neutral pair, far-field distortions cancel beyond the separation. If $L$ is not much larger than $r_0$, the leading logarithm is not a dominant approximation.
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