Let $ \Omega \subseteq \mathbb{R}^2 $ be some bounded and simply connected domain and $ \partial \Omega $ be its boundary of class $ C^3 $. Then there exist a $ 2\pi $ periodic representation $ \boldsymbol{\gamma} (t) $ of $ \partial \Omega $ and $ \boldsymbol{\gamma} (t):= (a(t),b(t)) \in \mathbb{R}^2 $ is thirdly continuously differentiable.
Set \begin{equation} Q(t) = \int^{2\pi}_0 (\ln \vert \boldsymbol{\gamma}(t) - \boldsymbol{\gamma} (s)\vert ) \vert \boldsymbol{\gamma}' (s)\vert ds, \ t \in [0,2\pi]. \end{equation} Is $ Q(t) $ at least twicely continuously differentiable?
Edit: The improper integral $ Q(t) $ exists for any $ t \in [0,2\pi]$ since we can rewrite the integral as \begin{equation*} \int_{\partial \Omega} \ln \vert x - y \vert ds(y), \ \text{where} \ ds(y) \ \text{means the arc element in } \ \partial \Omega \end{equation*} The well-definedness is guaranteed by Theorem 6.15 of [Linear integral equation, Third edition, Rainer Kress, Springer]. And the firstly continuously differentiability of $ Q(t) $ can be guaranteed by Corollary 7.31.
Thank you in advance!