相关论文: Partial Regularity of a minimizer of the relaxed e…

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the strategy…

We examine the Casimir energy of 5D electro-magnetism in the recent standpoint. Z$_2$ symmetry is taken into account. After confirming the consistency with the past result, we do new things based on a new regularization. The regularization…

For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow N$ from a sequence of non-collapsed degenerating closed Einstein 4-manifolds $(M_j,g_j)$ with bounded Einstein constants, bounded diameters and bounded $L^2$…

We study minimizers $\boldsymbol{m}\colon \mathbb R^2\to\mathbb S^2$ of the energy functional \begin{align*} E_\sigma(\boldsymbol{m}) = \int_{\mathbb R^2} \bigg(\frac 12 |\nabla\boldsymbol{m}|^2 +\sigma^2 \boldsymbol{ m} \cdot \nabla…

We prove the semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$. First we show that damping stabilizes the system when the energy is…

In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that…

In this note we study the boundary regularity of minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions $n\geq 3 .$ In dimension $n=3,$ this…

We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous…

In this article, we investigate the regularity for certain elliptic systems without a $L^2$-antisymmetric structure. As applications, we prove some $\epsilon$-regularity theorems for weakly harmonic maps from the unit ball $B= B(m) \subset…

For a negatively curved manifold $M$ and a continuous map $\psi:\Sigma\to M$ from a closed surface $\Sigma$, we study complex submanifolds of Teichm\"uller space $\mathcal{S}\subset\mathcal{T}(\Sigma)$ such that the harmonic maps…

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 establish small energy H\"{o}lder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_\Omega W(\nabla u)+ \frac{1}{\varepsilon^2} \int_\Omega f(u),\] where $W$ is a positive definite quadratic form and the potential $f$…

We examine the Casimir energy of 5D electromagnetism in the recent standpoint. The bulk geometry is flat. Z$_2$ symmetry and the periodic property, for the extra coordinate, are taken into account. After confirming the consistency with the…

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperK\"ahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the…

In this paper, we consider the solutions of the relaxed Q-tensor flow in $\R^3$ with small parameter $\epsilon$. Firstly, we show that the limiting map is the so called harmonic map flow; Secondly, we also present a new proof for the global…

For $n\ge 3$, let $\Omega$ be a bounded domain in $R^n$ and $N$ be a compact Riemannian manifold in $R^L$ without boundary. Suppose that $u_n\in W^{1,n}(\Omega,N)$ are the Palais-Smale sequences of the Dirichlet $n$-energy functional and…

We study $p$--harmonic maps with Dirichlet boundary conditions from a planar domain into a general compact Riemannian manifold. We show that as $p$ approaches $2$ from below, they converge up to a subsequence to a minimizing singular…

We consider the energy spectrum $\Xi_E(N)$ of harmonic maps from the sphere into a closed Riemannian manifold $N$. While a well known conjecture asserts that $\Xi_E(N)$ is discrete whenever $N$ is analytic, for most analytic targets it is…

We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by…

We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $\Omega\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth…