We show that a nef line bundle on a proper scheme over an excellent base is semiample if and only if it is semiample after restricting to characteristic zero and to positive characteristics. In the process of the proof, we provide a generalization to mixed characteristic of the fact that the perfection of the Picard functor in positive characteristic is a stack in groupoids for the h-topology.

Let $(M,g)$ be a Riemannian three-manifold that is asymptotic to Schwarzschild. We study the existence of large area-constrained Willmore spheres $\mathrm{\Sigma }\subset M$ with nonnegative Hawking mass and inner radius ρ dominated by the area radius λ. If the scalar curvature of $(M,g)$ is nonnegative, we show that no such surfaces with $log\mathit{\lambda }\ll \mathit{\rho }$ exist. This answers a question of Huisken.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$, $n\ge 2$ be a bounded open and connected set satisfying the corkscrew condition with uniformly n-rectifiable boundary. In this paper we study the connection among the solvability of $(D_{p^{\prime }})$, the Dirichlet problem for the Laplacian with boundary data in $L^{p^{\prime }}(\partial \mathrm{\Omega })$, and $(R_p)$ (resp., $({\tilde{R}}_p)$), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space $W^{1,p}(\partial \mathrm{\Omega })$ (resp., ${\tilde{W}}^{1,p}(\partial \mathrm{\Omega })$, the usual Sobolev space in terms of the tangential derivative), where $p\in (1,2+\mathit{\epsilon })$ and $1∕p+1∕p^{\prime }=1$. Our main result shows that $(D_{p^{\prime }})$ is solvable if and only if $(R_p)$ also is. Under additional geometric assumptions (two-sided local John condition or weak Poincaré inequality on the boundary), we prove that $(D_{p^{\prime }})\Rightarrow ({\tilde{R}}_p)$. In particular, we deduce that in bounded chord-arc domains (resp., two-sided chord-arc domains), there exists $p_0\in (1,2+\mathit{\epsilon })$ so that $(R_{p_0})$ (resp., $({\tilde{R}}_{p_0})$) is solvable. We also extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with n-Ahlfors–David regular boundaries, the single-layer potential operator is invertible from $L^p(\partial \mathrm{\Omega })$ to the inhomogeneous Sobolev space $W^{1,p}(\partial \mathrm{\Omega })$. Finally, we provide a counterexample of a chord-arc domain ${\mathrm{\Omega }}_0\subset {\mathbb{R}}^{n+1}$, $n\ge 3$, so that $({\tilde{R}}_p)$ is not solvable for any $p\in [1,\mathrm{\infty })$.