The Lp dual Minkowski problem for p > 1 and q > 0

General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang ref="#br0240" class="workspace-trigger">[24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. $L_p$ dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang ref="#br0240" class="workspace-trigger">[24] formulated the $L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual curvature measures. In this paper, we solve the existence part of the $L_p$ dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity of the solution.     

Caractère de Chern bivariant

We construct a bivariant Chern character with values in Jones-Kassel's bivariant cyclic cohomology. This is done forK-theoretic objects such as idempotents, bimodules, quasi-homomorphisms à la Cuntz and extensions of algebras.

     

Curvature Estimates for the Level Sets of Solutions to the Monge-Amp`ere Equation detD2u = 1

For the Monge-Ampere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.     

On certain degenerate and singular elliptic PDEs I: Nondivergence form operators with unbounded drifts and applications to subelliptic equations

We prove a Harnack inequality for nonnegative strong solutions to degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts. Our main theorem extends the Harnack inequality for the linearized Monge–Ampère equation due to Caffarelli and Gutiérrez and it is related, although under different hypotheses, to a recent work by N.Q. Le.Since our results are shown to apply to the convex functions with $p\ge 2$ and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge–Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot–Carathéodory metric in the study of certain subelliptic PDEs.     

A generalized rotationally symmetric case of the centroaffine Minkowski problem

In this paper the centroaffine Minkowski problem, a critical case of the $L_p$-Minkowski problem in the $n+1$ dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.     

Congruences of lines in {{\mathbb P}^5}, quadratic normality, and completely exceptional Monge–Ampère equations

The existence is proved of two new families of sextic threefolds in , which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge–Ampère equations. One of these families comes from a smooth congruence of multidegree (1, 3, 3) which is a smooth Fano fourfold of index two and genus 9.      

Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation

Given a strictly convex, smooth, and bounded domain Ω in rect.com/cache/MiamiImageURL/B6WK2-4PPF6GY-4-2/0?wchp=dGLbVzz-zSkWz" alt="View the MathML source" title="View the MathML source" align="absbottom" border="0" height=12 width="20"/> we establish the existence of a negative convex solution in rect.com/cache/MiamiImageURL/B6WK2-4PPF6GY-4-3/0?wchp=dGLbVzz-zSkWz" alt="View the MathML source" title="View the MathML source" align="absbottom" border="0" height=17 width="108"/> with zero boundary value to the singular Monge–Ampère equation det(D²u)=p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates.     

Interior gradient estimates for solutions to the linearized Monge–Ampère equation

Let ? be a convex function on a convex domain Ω⊂Rn, n?1. The corresponding linearized Monge–Ampère equation istrace(ΦD²u)=f, where rect.com/content/image/1-s2.0-S000187081100226X-si4.gif"> is the matrix of cofactors of D²?. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to Lp(Ω) for some p>n. The function ? is assumed to be such that rect.com/content/image/1-s2.0-S000187081100226X-si8.gif"> with ?=0 on ∂Ω and the Monge–Ampère measure rect.com/content/image/1-s2.0-S000187081100226X-si10.gif"> is given by a density g∈C(Ω) which is bounded away from zero and infinity.     

Monge-Ampère Equation with Bounded Periodic Data

We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named author.     

关于精确结构静力重分析法的注记

结构重分析是与结构优化设计紧密相关的分析环节。位移约束条件、载荷与单元刚度矩阵可修改的精确结构静力重分析是较新的一种方法，文献[1,2]提出其结构静力重分析的列式，这一重分析方法被多篇综述文章引用。本文从另一角度推导结构修改的基本方程，并给出广义柔度矩阵的简便可行算法，推导过程简单明了，力学意义明显。