8 problems

Let $\widehat{\Sigma}_i\in\mathcal{K}_{\omega_0}$ for $1\leqslant i\leqslant n+1$, where $\mathcal{K}_{\omega_0}$ is the class of anisotropic $\omega_0$-capillary convex bodies, an…

Maggi's conjecture. Wulff shapes are the unique such sets.

Let $\phi$ be a smooth, strictly convex norm on $\mathbf{R}^{n+1}$, with Wulff shape $\mathcal{W}^\phi=\{x\in\mathbf{R}^{n+1}:\phi^*(x)=1\}$. Let $X_0:M\to\mathbf{R}^{n+1}$ be a cl…

Anisotropic Serrin-type conjecture for rough domains. Wulff shapes are the unique such sets satisfying the overdetermined system.

Let $B$ be the Euclidean unit disk, and for each even integer $n\geq 4$ let $P_n^*$ be the regular polygon used as the anisotropy in the source. Write … where $h_{P_n^*}(B)$ is the…

Let $\phi$ be the norm defining the anisotropic curvature measures $\mathcal{C}_m^\phi(C,\cdot)$ of a convex body $C\subseteq\mathbf{R}^{n+1}$. Anisotropic curvature-measure charac…

Let $\gamma:S^n\to {\mathbb R}_{>0}$ be a $C^2$ function, and let $X:M\to {\mathbb R}^{n+1}$ be a closed CAMC hypersurface. For each $r=1,\ldots,n$, assume that the $r$-th anisotro…

Alexandrov's conjecture. Alexandrov's theorem should hold, in some proper formulation, for every anisotropic energy.