Fractional Sobolev Regularity for the Brouwer Degree

We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and $n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega, \mathbb R^{n}\right)$ belongs to the Sobolev space $W^{\beta, p} (\mathbb R^n)$ for every $0\leq \beta < \frac{n}{p} - \frac{d}{\alpha}$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $\beta\geq 0$ and $p\geq 1$ with $\beta > \frac{n}{p} - \frac{n-1}{\alpha}$ there is a vector field $v\in C^{0, \alpha} (B_1, \mathbb R^n)$ with \mbox{deg}\, (v, \Omega, \cdot)\notin W^{\beta, p}, where $B_1 \subset \mathbb R^n$ is the unit ball.Comment: 12 pages, 1 figur

Rigidity and Flexibility of Isometric Extensions

In this paper we consider the rigidity and flexibility of $C^{1, \theta}$ isometric extensions and we show that the H\"older exponent $\theta_0=\frac12$ is critical in the following sense: if $u\in C^{1,\theta}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $\Sigma$ and $\theta> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $\Sigma$. On the other hand, for any $\theta<\frac12$ we can construct $C^{1,\theta}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, \theta}$ isometric embeddings, $\theta<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.Comment: 36 page

C1,13−C^{1,\frac{1}{3}-} very weak solutions to the two dimensional Monge-Amp\'ere equation

For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one

Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood

In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of "A selection principle for the sharp quantitative isoperimetric inequality" by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands

C1,α isometric embeddings of polar caps

We study isometric embeddings of C2 Riemannian manifolds in the Euclidean space and we establish that the Hölder space C1,12is critical in a suitable sense: in particular we prove that for α >12 the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any α <12 we construct C1,α isometric embeddings of portions of the standard 2-dimensional sphere for which such property fails

A Nash–Kuiper theorem for C1,15−δC^{1,\frac{1}{5}-\delta} immersions of surfaces in 3 dimensions

We prove that, given a $C^2$ Riemannian metric $g$ on the 2-dimensional disk $D_2$, any short $C^1$ immersion of $(D_2,g)$ into $\mathbb{R}^3$ can be uniformly approximated with $C^{1,α}$ isometric immersions for any $α<\frac{1}{5}$. This statement improves previous results by Yu. F. Borisov and of a joint paper of the first and third author with S. Conti