Approximation of variational problems with a convexity constraint by PDEs of Abreu type
Approximation of variational problems with a convexity constraint by PDEs of Abreu type
复制标题
用 Abreu 型偏微分方程逼近具有凸性约束的变分问题
DOI：
10.1007/s00526-019-1613-1
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发表时间：
2019
影响因子：
2.1
通讯作者：
T. Radice
中科院分区：
文献类型：
--
作者：
G. Carlier;T. Radice
受一些具有凸性约束的变分问题的启发，我们考虑了一种使用Hessian行列式的对数作为约束的障碍的近似。我们证明了这种惩罚的最小化可以通过解Abreu方程的第二边值问题来逼近，该边值问题是一个适定的四阶非线性椭圆型问题。更有趣的是，类似的近似结果也适用于初始约束变分问题。
Motivated by some variational problems subject to a convexity constraint, we consider an approximation using the logarithm of the Hessian determinant as a barrier for the constraint. We show that the minimizer of this penalization can be approached by solving a second boundary value problem for Abreu’s equation which is a well-posed nonlinear fourth-order elliptic problem. More interestingly, a similar approximation result holds for the initial constrained variational problem.