3 problems

Let $\Lambda_{\epsilon}$ be a bounded connected planar annulus with two boundary components, and let $\lambda_{1}(\Lambda_{\epsilon})$ be its first nontrivial Steklov eigenvalue, g…

Let $\Lambda_{\epsilon}$ denote the planar annulus with outer radius $1$ and inner radius $\epsilon\in(0,1)$, let $|\partial\Lambda_{\epsilon}|$ be its perimeter, and let…

Let $\Omega$ be a bounded planar domain with one hole, meaning that its complement has two connected components, one bounded and one unbounded, and assume that $\Omega$ has Lipschi…