On Friedrichs inequalities for a disk (Q2447023)

The classical Friedrichs inequality \(\|u\|_{L_2(\Omega)}\leq C(\Omega, \Gamma)\|\operatorname{grad} u\|_{L_2(\Omega)}\) holds in the case where \(u\equiv 0\) on the boundary \(\Gamma\) of the unit disk \(\Omega\). In this paper, by allowing \(u\) not to be zero on a small piece of \(\Gamma\) with length \(\epsilon\), the authors show that the coefficient function in the Friedrichs inequality becomes \[ C(\Omega, \Gamma_{\epsilon})=C(\Omega, \Gamma)[1+\epsilon^2(1+O(\epsilon^{2/7}))]. \]