We prove a lower bound for the value of the L1-relaxed area of the graph of the map u:Bl(0)∖{0}⊂R2→R2, u(x):=x/|x|, x≠0, for all values of the radius l>0. In the computation of the singular part of the relaxed area, for l in a certain range, in particular l not too large, a nonparametric Plateau-type problem with partial free boundary has to be solved. Our lower bound turns out to be optimal, in view of an upper bound proven in a companion paper.

We consider the question of existence of weak solutions for the fully inhomogeneous, stationary generalized Navier–Stokes equations for homogeneous, shear-thinning fluids. For a shear rate exponent p∈(2dd+1,2), previous results require either smallness of the norm or vanishing of the normal component of the boundary data. In this work, combining previous methods, we propose a new, more general smallness condition.

We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding–Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit core metrics, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.

Consider a uniformly distributed random linear subspace L and a stochastically independent random affine subspace E in Rn, both of fixed dimension. For a natural class of distributions for E we show that the intersection L∩E admits a density with respect to the invariant measure. This density depends only on the distance d(o,E∩L) of L∩E to the origin and is derived explicitly. It can be written as the product of a power of d(o,E∩L) and a part involving an incomplete beta integral. Choosing E uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable d(o,E∩L) and study the behavior of the probability that E∩L hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where E is tangent to the unit sphere, in which case we again derive the density for d(o,E∩L). Our probabilistic results are derived by means of a new integral–geometric transformation formula of Blaschke–Petkantschin type.