4 problems

Lundholm–Nam–Portmann conjecture. In the strong-coupling limit,

Extremal uniqueness conjecture. Equality holds in the sharp inequality if and only if $f=h_p$ up to translations, dilations and multiplicative constants.

Let $a\geqslant n>p>1$, and consider the Gagliardo–Nirenberg trace inequality whose optimal constant is attained by … Here, dilatations and translations act on functions in the usu…

Ilyin's torus extremal conjecture. For every multidimensional torus $\Omega=\mathbb{T}^d$ with $d>1$ and $l=0$, this analogous effect holds.