3 problems

Let $s\in(0,1)$, and consider stable $s$-minimal hypercones in $\mathbb{R}^n\setminus\{0\}$. Flatness conjecture. For every integer $n$ with $3\leqslant n\leqslant 7$, there exists…

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 $\|\nabla u\|_{L^\infty(\mathbb{R}^n)}<\infty$…