On Friedrichs inequalities for a disk

We consider the Friedrichs inequality for functions defined on a disk of unit radius $\Omega$ and equal to zero on almost all boundary except for an arc $\gamma_\varepsilon$ of length $\varepsilon$, where $\varepsilon$ is a small parameter. Using the method of matched asymptotic expansions, we construct a two-term asymptotics for the Friedrichs constant $C(\Omega,\partial\Omega\backslash\overline\gamma_\varepsilon)$ for such functions and present a strict proof of its validity. We show that $C(\Omega,\partial\Omega\backslash\overline\gamma_\varepsilon)=C(\Omega,\partial\Omega)+\varepsilon^2C(\Omega,\partial\Omega)(1+O(\varepsilon^{2/7}))$. The calculation of the asymptotics for the Friedrichs constant is reduced to constructing an asymptotics for the minimum value of the operator $-\Delta$ in the disk with Neumann boundary condition on $\gamma_\varepsilon$ and Dirichlet boundary condition on the remaining part of the boundary.