11
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • $\begingroup$ The integral at the LHS is on $\partial \Omega$, not on $\Omega$, right? $\endgroup$
    – Piero D'Ancona
    Commented Aug 19, 2016 at 20:22

1 Answer 1

Reset to default
1
$\begingroup$

Given a concrete bounded domain, the best constant can be evaluated by using FEM to solve the Stokes eigenvalue problem along with mathematically correct precision.

I am not sure whether this information is useful to you.

This is a new result in press (to appear soon). If you are interested, please contact me.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .