Geometric measure theory and hyperbolic operators ant its numerical calculations

在本研究工作中，我们处理了双曲自由边界问题。基本方程表达了从平面上剥离胶带的模型。在此模型的基础上，建立了一种分析气泡在水面上或小液滴在障碍物上的动态接触角运动的新方法。对于多个附着气泡的情况，我们开发了一种高效的算法，可以自动处理包括拓扑变化在内的运动连接。另一方面，我们用离散Morse流方法构造了弹性壳弹性体问题的数值求解器。使用该算法，我们能够结合内部结构并分析外壳与其内容物之间的相互作用。

In this research work, hyperbolic free boundary problems have been treated. The basic equation expresses a model for peeling off a tape from a plane. Based on this model, we established a new method analyzing bubble motion on water surface or small droplet motion with dynamic contact angle on obstacle. In the case of everal attached bubbles, we developed an efficient algorithm which can automatically deal with moving junctions including topological changes. On the ther hand, we have constructed a numerical solver for the problem of bouncing elastic shell via the discrete Morse flow method. Using this algorithm, we are able to incorporate inner structure and analyze the interactionbetween the shell and its contents.