Global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature and Abreu's equations

$W^{4, p}$ solution to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem in $W^{4,p}$ under some restrictions on the sign or integrability of the affine mean curvature. We remove these restrictions in this paper and obtain $W^{4,p}$ solution to the second boundary value problem when the affine mean curv…

An obstacle problem for a class of Monge-Ampère type functionals

In this paper we study an obstacle problem for Monge-Ampère type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation.

Global Hölder estimates for 2D linearized Monge-Ampère equations with right-hand side in divergence form

We establish global Hölder estimates for solutions to inhomogeneous linearized Monge-Ampère equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Ampère measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global Hölder est…

The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Ampère equation

These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Ampère and linearized Monge-Ampère equations. The prescribed affine mean curvature equation is a fully nonlinear, fourth order, geometric partial differential equation of the following form $$\sum_{i, j=1}^n U^{ij}\frac{\partial^2}{\partial {x_i}\partial{x_j}}\left[(\det D^2 u)^{-\frac{n+1}{n+2}}\right]=f$$ where $(U^{ij})$ is the cofactor matrix of the Hessian matrix $D^2 u$ of a locally uniformly convex functio…

Two slope functions minimizing fractional seminorms and applications to misfit dislocations

We consider periodic piecewise affine functions, defined on the real line, with two given slopes and prescribed length scale of the regions where the slope is negative. We prove that, in such a class, the minimizers of $s$-fractional Gagliardo seminorm densities, with $0<s<1$, are in fact periodic with the minimal possible period determined by the prescribed slopes and length scale. Then, we determine the asymptotic behavior of the energy density as the ratio between the length of the two intervals where the slope is constant vanishes. Our results, for $s=\frac 1 2$, have relevant applications…

仿射变分泛函、等周问题与稳定性

Affine variational functionals, isoperimetric problems and stability

本文研究三个仿射变分量的等周问题，证明椭球是唯一极值，并借助L_p Busemann--Petty质心不等式的稳定性获得定量等周不等式及仿射Kohler-Jobin不等式的稳定性结果。