8 problems

Consider a positive function $u$ homogeneous of degree one in a domain $E gsubseteqmathbb{R}^n$, where $partial E$ is a cone, solving the one-phase free boundary problem … Assume t…

Let $\varepsilon_*$ be the maximal small-angle threshold for which $\mathbf{C}_{7,1,\theta}$ is strictly minimizing, and let $\varepsilon^*$ be the near-$\frac{\pi}{2}$ threshold f…

Let $n$ and $k$ be in the ranges considered in the theorem establishing minimization for small contact angles, excluding $(n,k)\in\{(7,1),(7,5)\}$. Let…

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…

For $s\in(0,1)$, let $n_s^*$ be the minimum dimension $n\geq 3$ for which there exists a smooth, stable $s$-minimal cone in $\mathbb{R}^n\setminus\{0\}$ that is not a hyperplane. C…

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

Let $n\geq 1$, and let $\Sigma\subset\mathbb{R}^{n+2}$ be a nonplanar smooth proper minimal hypersurface. For $1\leq p\leq n-1$, define the minimal hypersurface … of…