Erik Lindgren: Uniqueness of extremals for Morrey's inequality

A celebrated result in the theory of Sobolev spaces is Morrey's
inequality, which establishes in particular that for a bounded domain
$\Omega\subset \mathbb{R}^n$ and $p>n$, there is $c>0$ such that
$$
c\|u\|^p_{L^\infty(\Omega)}
\le
\int_\Omega|Du|^pdx, \quad u\in W^{1,p}_0(\Omega).
$$
Interestingly enough the equality case of this inequality has not been
thoroughly investigated (unless the underlying domain is
$\mathbb{R}^n$ or a ball).

I will discuss uniqueness properties of extremals of this inequality.
These extremals are minimizers of the nonlinear Rayleigh quotient
$$
\inf\left\{\frac{\int_\Omega|Du|^pdx}{\|
u\|_{L^\infty(\Omega)}^p}:u\in W_0^{1,p}(\Omega)\setminus\{0\}\right\}.
$$

In particular, I will present the result that in convex domains,
extremals are determined up to a multiplicative factor. I will also
explain why convexity is not necessary and why stareshapedness is not
sufficient for this result to hold.

The talk is based on recent results obtained with Ryan Hynd.