3 problems

Fernández-Real–Ros-Oton conjecture. Suppose that $u:\mathbb{R}^n\to(0,\infty)$ minimizes $I_1$ locally. If $n\leq n_e^*-1$, then $u$ is one-dimensional: in a suitable Euclidean coo…

Assume (A1): $\beta$ is non-negative and $C^1$, with $\textrm{supp}\,\beta=[0,1]$ and $\int_0^1\beta=1$. Let … where $\Phi'=\beta$. Dimension-seven non-one-dimensional minimizer co…

Assume (A1)–(A2): $\beta$ is non-negative and $C^1$ with $\textrm{supp}\,\beta=[0,1]$ and $\int_0^1\beta=1$, while $u\geq0$ satisfies $\|\nabla u\|_{L^\infty(\mathbb{R}^n)}<\infty$…