报告人：蒋美跃  教授

北京大学数学科学学院

报告题目：DIFFERENTIAL INCLUSIONS AND VARIATIONAL PROBLEMS IN BV SPACE IN DIMENSION 1

报告时间：2018年8月13日下午4：00-5:00

报告地点：实验楼105

内容摘要： 

Let $X$ be a Banach space, $E(u): X\to \mathbb R\cup \{+\infty\} $ be  a convex functional, may not be differentiable,  $F(u):X\to \mathbb R$ be a $C^1$ functional. we consider the critical points  of the functional $E(u)+F(u)$. As the functional may not be differentiable, the Euler-Lagrange equation $d(E(u)+F(u))=0$ does not make sense in general. One way to overcome this difficulty is to use the subdifferential $\partial E(u)$ to replace the differential $dE(u)$, and consider the differential inclusion $0\in \partial E(u) +dF(u)$ as  the Euler-Lagrange equation.

In this talk we will discuss the following variational problems in BV space, the space of functions with  bounded variation in dimension 1:

$$E(u)=\int \sqrt{1+|Du|^2}dx, \int |Du| dx,\quad F(u)=\int f(x,u) dx. $$

These functionals are related 1-dimensional prescribed mean curvature problem and 1-Laplacian equation. We will illustrate some  properties of the critical points in the above sense and show that one can add more restrictions to the critical points of the problem via the variation of the domain.

With some conditions on $f(x,u)$, existence of  new type critical points of $E(u)+F(u)$ are established.

报告人简介：

蒋美跃，北京大学数学科学学院教授，博士生导师，主要从事非线性分析的研究，在Hamilton系统以及临界群理论等方面，都有重要的研究成果。

学院联系人：张文教授