Some regularity properties of viscosity solution defined by Hopf formula

Some properties of characteristic curves in connection with viscosity solution of Hamilton-Jacobi equations $(H,\sigma)$ defined by Hopf formula $u(t,x)=\max_{q\in\R^n}\{ \langle x,q\rangle -\sigma^*(q)-tH(q)\}$ are studied. We are concerned with the points where the solution $u(t,x)$ is differentiable, and the strip of the form $\mathcal R=(0,t_0)\times \R^n$ of the domain $\Omega$ where $u(t,x)$ is of class $C^1(\mathcal R).$ Moreover, we investigate the propagation of singularities in forward of this solution.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1208.3288, arXiv:1309.254

A Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem

In this paper, we introduce and study the following problem and its further generalizations: given two finite collections of sets in a normed space, find a ball whose center lies in a given constraint set with the smallest radius that encloses all the sets in the first collection and intersects all the sets in the second one. This problem can be considered as a generalized version of the Sylvester smallest enclosing circle problem introduced in the 19th century by Sylvester which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We also consider a generalized version of the Fermat-Torricelli problem: given two finite collections of sets in a normed space, find a point in a given constraint set that minimizes the sum of the farthest distances to the sets in the first collection and shortest distances (distances) to the sets in the second collection

Modified defect relations of the Gauss map of complete minimal surfaces on annular ends

In this article, we study the modified defect relations of the Gauss map of complete minimal surfaces in $\mathbb R^3$ and $\mathbb R^4$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto~[J. Differential Geometry \textbf{29} (1989), 245-262] for (the whole) complete minimal surfaces. We thus give some improvements of the previous results for the Gauss maps of complete minimal surfaces restricted on annular ends.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1304.706

INKA: An Ink-based Model of Graph Visualization

Common quality metrics of graph drawing have been about the readability criteria, such as small number of edge crossings, small drawing area and small total edge length. Bold graph drawing considers more realistic drawings consisting of vertices as disks of some radius and edges as rectangles of some width. However, the relationship that links these readability criteria with the rendering criteria in node-link diagrams has still not been well-established. This paper introduces a model, so-called INKA (Ink-Active), that encapsulates mathematically the relationship between all common drawing factors. Consequently, we investigate our INKA model on several common drawing algorithms and real-world graphs

Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Pr\'ekopa's theorem

We give a new approach, inspired by H\"ormander's $L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the Brunn-Minkowski inequalities. We also present several applications of these variance inequalities, including reverse H\"older inequalities for convex functions, weighted Brascamp-Lieb inequalities and sharp weighted Poincar\'{e} inequalities for generalized Cauchy measures.Comment: 24 pages, some minor corrections. To appear in Journal of Functional Analysi

On the infinite loop space structure of the cobordism category

We show an equivalence of infinite loop spaces between the classify- ing space of the cobordism category, with infinite loop space structure induced by taking disjoint union of manifolds, and the infinite loop space associated to the Madsen-Tillmann spectrum.Comment: 14 pages, 1 figure, rewritten introduction and corrected some errors in section

Improved Moser--Trudinger inequality for functions with mean value zero in Rn\mathbb R^n and its extremal functions

Let $\Omega$ be a bounded smooth domain in $\mathbb R^n$, $W^{1,n}(\Omega)$ be the Sobolev space on $\Omega$, and $\lambda(\Omega) = \inf\{\|\nabla u\|_n^n: \int_\Omega u dx =0, \|u\|_n =1\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-\Delta_n$ on $\Omega$. For $0 \leq \alpha < \lambda(\Omega)$, let us define $\|u\|_{1,\alpha}^n =\|\nabla u\|_n^n -\alpha \|u\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $\Omega$, $$\sup_{u\in W^{1,n}(\Omega), \int_\Omega u dx =0, \|u\|_{1,\alpha} =1} \int_{\Omega} e^{\beta_n |u|^{\frac n{n-1}}} dx < \infty,$$ where $\beta_n = n (\omega_{n-1}/2)^{1/(n-1)}$, and $\omega_{n-1}$ denotes the surface area of unit sphere in $\mathbb R^n$. We also show that this supremum is attained by some function $u^*\in W^{1,n}(\Omega)$ such that $\int_\Omega u^* dx =0$ and $\|u^*\|_{1,\alpha} =1$. This generalizes a result of Ngo and Nguyen \cite{NN17} in dimension two and a result of Yang \cite{Yang07} for $\alpha=0$, and improves a result of Cianchi \cite{Cianchi05}.Comment: 24 pages, to appear in Nonlinear Analysi

The sharp higher order Hardy--Rellich type inequalities on the homogeneous groups

We prove several interesting equalities for the integrals of higher order derivatives on the homogeneous groups. As consequences, we obtain the sharp Hardy--Rellich type inequalities for higher order derivatives including both the subcritical and critical inequalities on the homogeneous groups. We also prove several uncertainty principles on the homogeneous groups. Our results seem to be new even in the case of Euclidean space $\mathbb R^n$ and give a simple proof of several classical Hardy--Rellich type inequalities in $\mathbb R^n$.Comment: 47 pages, comments are welcom

Extremal functions for the sharp Moser--Trudinger type inequalities in whole space RN\mathbb R^N

This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\mathbb R^N$, $N \geq 2$ in more general case. We first compute explicitly the \emph{normalized vanishing limit} and the \emph{normalized concentrating limit} of the Moser-Trudinger type functional associated with our inequalities over all the \emph{normalized vanishing sequences} and the \emph{normalized concentrating sequences}, respectively. Exploiting these limits together with the concentration-compactness principle of Lions type, we give a proof of the exitence of maximizers for these Moser-Trudinger type inequalities. Our approach gives an alternative proof of the existence of maximizers for the Moser-Trudinger inequality and singular Moser-Trudinger inequality in whole space $\mathbb R^N$ due to Li and Ruf \cite{LiRuf2008} and Li and Yang \cite{LiYang}.Comment: 27 page

Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds: the influence of curvature

We first establish a family of sharp Caffarelli-Kohn-Nirenberg type inequalities on the Euclidean spaces and then extend them to the setting of Cartan-Hadamard manifolds with the same best constant. The quantitative version of these inequalities also are proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli-Kohn-Nirenberg type inequalities with the same sharp constant as in $\mathbb R^n$ (shortly, sharp CKN inequalities). Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Krist\'aly to a larger class of the sharp CKN inequalities.Comment: 21 pages, comment are welcome. This is the second part of my previous preprint arXiv:1708.09306v