Related papers: Lipschitz regularity of energy-minimal mappings be…

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups.…

Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an…

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the…

We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we…

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ of finite uniton number, from a Riemann surface $M,$ compact or non-compact, into inner symmetric spaces of compact or non-compact type. As…

Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus…

A set of locally finite perimeter $E \subset \mathbb{R}^{n}$ is called an anisotropic minimal surface in an open set $A$ if $\Phi(E;A) \le \Phi(F;A)$ for some surface energy $\Phi(E;A) = \int_{\partial^{*}E \cap A} \| \nu_{E}\| d…

We consider an energy functional motivated by the celebrated $K_{13}$ problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional…

Let $M$ be a $C^{2}$-smooth Riemannian surface. A classical theorem in differential geometry states that the Gauss curvature function $K : M \to \mathbb{R}$ vanishes everywhere if and only if the surface is locally isometric to the…

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics…

Motivated by establishing Neumann Talenti type comparison results, we concern the minimization of the following shape functional under volume constraint: \begin{align*} T(\Omega):=\inf\left\{\frac12 \int_{\Omega} |\nabla u|^2\,dx…

We consider, in a smooth bounded multiply connected domain $\dom\subset\R^2$, the Ginzburg-Landau energy $\d E_\v(u)=1/2\int_\dom{|\n u|^2}+\frac{1}{4\v^2}\int_\dom{(1-|u|^2)^2}$ subject to prescribed degree conditions on each component of…

We show that for any purely 2-unrectifiable metric space $M$, for example the Heisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carath\'{e}odory metric, every homotopy class $[\gamma]$ of Lipschitz paths contains a length minimizing…

We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m}, N]$, provided that the homotopy group $\pi_{2m}(N)$ is…

We show that if $\{M_t\}_{t\in \Delta}$ is a polarized family of compact K\"ahler manifolds over the open unit disk $\Delta$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if $\{\phi_t:M_t\to N\}_{t\in…

We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a…

In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $\mathbb{S}^2$. We show that every boundary map can be modified to a boundary map that admits multiple minimizers of the Dirichlet energy by a small…

Let $\Sigma$ a closed $n$-dimensional manifold, $\mathcal{N} \subset \mathbb{R}^M$ be a closed manifold, and $u \in W^{s,\frac ns}(\Sigma,\mathcal{N})$ for $s\in(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if…

We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;\Omega)= \int_{\Omega} f (x, Dv) dx \end{equation*} under $(p,q)$-growth conditions. The main novelty…

In this paper, we first extend the finite distortion problem from the bounded domains in $\mathbb{R}^2$ to the closed genus-zero surfaces in $\mathbb{R}^3$ by the stereographic projection. Then we derive a theoretical foundation for…