Related papers: Dirichlet-type energy of mappings between two conc…

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space. Under the setting, we prove that the Korevaar-Schoen energy admits a unique minimizer.

Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*}…

We study the regularity of minimizers of the functional $\mathcal E(u):= [u]_{H^s(\Omega)}^2 +\int_\Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $\Omega\subset\mathbb R^N$. More precisely,…

In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, with $n\geq 4$, let $a, b,…

We study identifiability of finite mixtures of Dirichlet distributions on the interior of the simplex. We first prove a shift identity showing that every Dirichlet density can be written as a mixture of $J$ shifted Dirichlet densities,…

Given a family of critical points $u_{\epsilon}:M^n\to\mathbb{C}$ for the complex Ginzburg--Landau energies \begin{align*} &E_\epsilon(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4\epsilon^2}\right), \end{align*} on a manifold $M$,…

Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with…

We consider the varifold associated to the Allen--Cahn phase transition problem in $\mathbb R^{n+1}$(or $n+1$-dimensional Riemannian manifolds with bounded curvature) with integral $L^{q_0}$ bounds on the Allen--Cahn mean curvature (first…

We prove a ${\Gamma}$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $\Omega \subseteq \mathbb{R}^{n+k}$ to $\mathscr{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold…

We study the constrained minimum energy problem with an external field relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant…

Let $n\geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in…

This work provides an extension of parts of the classical finite dimensional sub-elliptic theory in the context of infinite dimensional compact connected metrizable groups. Given a well understood and well behaved bi-invariant Laplacian,…

We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies $\Gamma$-converge in the weak-$*$ topology to the Riesz functional…

We consider the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree $1$ maps from closed…

We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schr\"odinger system \begin{align*} - \Delta u + \lambda_1 u &= \frac{\alpha u(\alpha…

In this paper we deal with the existence, regularity and Beltrami field property of magnetic energy minimisers under a helicity constraint. We in particular tackle the problem of characterising local as well as global minimisers of the…

In this paper we prove that the space $\cM(n,\rv,D,\Lambda):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv>0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le \Lambda\}$ has at most $C(n,\rv,D,\Lambda)$ many diffeomorphism…

Let $W:R^m\rightarrow R$ be a nonnegative potential with exactly two nondegenerate zeros $a_-\neq a_+\in R^m$. We assume that there are$ N\geq 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $ u_1,\ldots,u_N$…

For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For $H^2$-regular graphs we show that bounds for the…

The notion of horizontal energy minimizers between C-C spaces is introduced. We prove existence of such energy minimizers when the domain is a $C^{2}$, noncharacteristic bounded open set in a C-C space and the target is a C-C space of…