Let \(U=\{x\in \mathbb {R}^2:\frac {1}{2}<|x|<1\}\), and denote \(\partial U=\Gamma _1\cup \Gamma _2\) where \[ \Gamma _1=\{|x|=\tfrac {1}{2}\},\quad \Gamma _2=\{|x|=1\}. \] Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ \alpha u+\frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _1,\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _2, \end {aligned} \] where \(\alpha >0\), \(\eta \) is the unit outward normal on \(\partial U\), and \(f\in L^2(U)\).
Prove that there exists a constant \(C>0\) such that \[ C\int _U u^2\,dx \le \int _U |Du|^2\,dx+\int _{\Gamma _1} u^2\,dS \] for all \(u\in C^1(\overline {U})\). Polar coordinates may be helpful. Explain how this estimate could be used in a proof of existence and uniqueness for solutions of the problem above.