Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla u|^2$$ for functions $u \in H^1(\Omega)$ such that $\int_{\partial \Omega} u =0$.

The best constant is, in fact, the first Steklov eigenvalue corresponding to $\Omega$. Do you know any references for results of this type? I am interested in the case of unbounded domains. For me $\Omega$ has fixed measure and bounded perimeter, but if any other necessary assumptions are needed, I'm still interested.

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla u|^2$$ for functions $u \in H^1(\Omega)$ such that $\int_{\partial \Omega} u =0$.

The best constant is related to the first Steklov eigenvalue corresponding to $\Omega$. Do you know any references for results of this type? I am interested in the case of unbounded domains. For me $\Omega$ has fixed measure and bounded perimeter, but if any other necessary assumptions are needed, I'm still interested.

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla u|^2$$ for functions $u \in H^1(\Omega)$ such that $\int_{\partial \Omega} u =0$.

The best constant is, in fact, the first Steklov eigenvalue corresponding to $\Omega$. Do you know any references for results of this type? I am interested in the case of unbounded domains.

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla u|^2$$ for functions $u \in H^1(\Omega)$ such that $\int_{\partial \Omega} u =0$.

The best constant is, in fact, the first Steklov eigenvalue corresponding to $\Omega$. Do you know any references for results of this type? I am interested in the case of unbounded domains. For me $\Omega$ has fixed measure and bounded perimeter, but if any other necessary assumptions are needed, I'm still interested.