A regularity theory for multiple-valued Dirichlet minimizing maps

A Theorem on Frequency Function for Multiple-Valued Dirichlet Minimizing Functions

This paper discusses the frequency function of multiple-valued Dirichlet minimizing functions in the special case when the domain and range are both two dimensional. It shows that the frequency function must be of value k/2 for some nonnegative integer k. Futhermore, by looking at the blowing-up functions, we characterize the local behavior of the original Dirichlet minimizing function.

Regularity of the optimal sets for the second Dirichlet eigenvalue

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $Ω$ minimizes the functional \[ \mathcal F_Λ(Ω)=λ_2(Ω)+Λ|Ω|, \] among all subsets of a smooth bounded open set $D\subset \mathbb{R}^d$, where $λ_2(Ω)$ is the second eigenvalue of the Dirichlet Laplacian on $Ω$ and $Λ>0$ is a fixed constant, then $Ω$ is equivalent to the union of two disjoint open sets $Ω_+$ and $Ω_-$, which are $C^{1,α}$-regular up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in the one-phase…

多边形域中狄利克雷最优控制问题解的正则性研究

On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains

本文研究了非凸多边形域上线性二次狄利克雷控制问题的解正则性，证明在存在控制约束时，最优控制连续性。

About Hölder-regularity of the convex shape minimizing λ2

In this paper, we consider the well-known following shape optimization problem: $$λ_2(Ω^*)=\min_{\stackrel{|Ω|=V_0} {Ω\textrm{ convex}}} λ_2(Ω),$$ where $λ_2(\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Om\subset\R^2$, and $|\Om|$ is the area of $\Om$. We prove, under some technical assumptions, that any optimal shape $Ω^*$ is $\mathcal{C}^{1,\frac{1}{2}}$ and is not $\C^{1,α}$ for any $α>\frac{1}{2}$. We also derive from our strategy some more general regularity results, in the framework of partially overdetermined boundary va…

Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. In fact, provided that the lengths of the singular curves under consideration are bounded with respect to a given complete Riemannian metric, we demonstrate that such a set is a semianalytic cu…

亚里叶米安几何中长度最短曲线的正则性

Sharp regularity of sub-Riemannian length-minimizing curves

本文研究了亚里叶米安结构中长度最短曲线的正则性，证明了此类曲线至少为C²类，并指出[6]中的定理1.1是尖锐的。