Peakedness conjecture for extremal Sobolev functions

Let $n \geq 2$ and let $\Omega \subset \mathbf{R}^n$ be a bounded domain with piecewise Lipschitz boundary satisfying a uniform cone condition. For each allowable exponent $p$, let $u_p^*$ be the corresponding positive extremal function, normalized by

$$\sup_{x \in \Omega} u_p^*(x)=1.$$

Define its distribution function by

$$\mu_p(t)=\left|\left\{x \in \Omega:u_p^*(x)>t\right\}\right|.$$

Peakedness conjecture. Within the allowable range of exponents,

$$1\leq p<q\quad\Rightarrow\quad \mu_p(t)>\mu_q(t) \quad\text{for almost every }t\in(0,1).$$

If $n=2$, the allowable range is $1\leq p<q$; if $n\geq3$, it is $1\leq p<q<\frac{2n}{n-2}$. This formalizes the expectation that extremal functions become more peaked as $p$ increases; the paper presents numerical evidence, but the inequality remains conjectural.