6 problems

Let $f:\mathbb{R}\to\mathbb{R}$ be Lipschitz, and let $\Omega\subset\mathbb{R}^n$ be a smooth (in fact, Lipschitz) connected domain such that $\mathbb{R}^n\setminus\Omega$ is conne…

Let Omega\right?

Consider the overdetermined problem … where $\Omega\subset\mathbb{R}^d$ is a smooth domain, $f$ is Lipschitz, and $\nu$ is the outward unit normal to $\partial\Omega$. BCN conjectu…

First Schiffer conjecture D in $\mathbb{S}^2$. If such a solution exists, then $\Omega$ is a geodesic ball. The paper states that this first-eigenvalue version is true on…

Let $f:\mathbb{R}\to\mathbb{R}$ be Lipschitz, and let $\Omega\subset\mathbb{S}^2$ be a topological disk with $C^2$ boundary. Suppose the overdetermined elliptic problem … admits a…

Let $f$ be a Lipschitz function on $\mathbb{R}_{+}$, and let $\Omega\subset\mathbb{H}^{n}$ be a smooth domain such that $\mathbb{H}^{n}\backslash\overline{\Omega}$ is connected. As…