2 problems

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with $\mathcal C^2$ boundary. Let $D\subset\mathbb{R}^2$ be a disk with the same area as $\Omega$. For $a>0$, let…

Let $D$ be the unit disc, let $\mathcal{A}_0$ be the space appearing above, and let $R(f,\Omega)$ denote the associated quotient for a domain $\Omega\subset D$. Domains with radial…