Related papers: $C^{1+\alpha}$-Regularity for Two-Dimensional Almo…

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by…

We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. Based on the properties of…

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the $n$-dimensional Heisenberg group into $CAT(0)$ spaces. Our main theorem establishes that these maps have the…

In analogy with Almgren's Theorem for area minimizing currents of general dimension and codimension, we prove that an $m$-dimensional semicalibrated current in a $(n+m)$-dimensional $C^{3,\varepsilon_0}$ manifold, semicalibrated by a…

We study a natural biharmonic analogue of the classical Alt-Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and $C^{1,\alpha}$-regularity of minimisers. For the Navier…

We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of…

In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying H\"ormander's bracket generating condition, where the target is a nonempty closed set. We show that,…

Ahlfors and Gehring asked for the Riemann Mapping Theorem for quasiconformal mappings (QC) of R^3. We summarise our solution: (a) QC reflections are tame (b) T is the fixed set of a QC reflection iff T is a uniform sphere (i.e. the limits…

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into…

We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing…

We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension.

We prove that minimizers of the $L^{d}$-norm of the Hessian in the unit ball of $\mathbb{R}^d$ are locally of class $C^{1,\alpha}$. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate…

A semigroup generated by two dimensional $C^{1+\alpha}$ contracting maps is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives of…

Let $(M,g)$ be a smooth connected Riemannian manifold. We show an improvement of flatness theorem for hypersurfaces of $M$ of bounded nonlocal mean curvature in the viscosity sense. It implies local $ C^{1,\alpha}$ regularity of these…

We study the limiting behavior of minimizing $p$-harmonic maps from a bounded Lipschitz domain $\Omega \subset \mathbb{R}^{3}$ to a compact connected Riemannian manifold without boundary and with finite fundamental group as $p \nearrow 2$.…

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a…

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to…

We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the…

We show $C^{1,\alpha}$-regularity for energy minimizing maps from a 2-dimensional Riemannian manifold into a Finsler space $(\R^n, F)$ with a Finsler structure $F(u,X)$.

These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some…