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 and related inequalities. Extremals of the above inequality 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 results obtained with Ryan Hynd.