FOR 3013: Vector- and Tensor-Valued Surface PDEs

曲面上偏微分方程一直是应用数学中一个活跃的研究领域。由于它们与几何结构的耦合，这种表面偏微分方程本质上是非线性的。这给建模和数值分析带来了新的挑战。这些挑战大多是针对标量值表面偏微分方程的。在标量情况下，表面几何形状与偏微分方程之间的耦合相对较弱，因此在平面空间中建立的数值方法在稍加修改后仍然适用。对于矢量值和张量值曲面偏微分方程，这些方法不再足够。表面向量场和张量场通常需要满足额外的约束条件。一个例子是这些场的切线性，因此它们需要被视为切线束的元素，导致表面几何形状和PDE之间的非线性耦合。应对这些新的挑战是这个研究单位的动力。在过去的三年中，随着新模型、新数值方法、新数值分析结果、新软件工具和新应用的发展，我们看到了研究活动的巨大增长。我们的研究单位对这些发展作出了重大贡献，并在若干领域发起了这些发展。在第二个资助期，我们将深化这方面的知识，以描述和理解揭示普遍原则，并使所考虑的应用程序之间的异同变得透明。与第一个资助期不同，重点将放在表面动力学上，包括可变形流体表面、预压板的速率无关演化、生长和膨胀现象，以及针对这些新挑战的数值方法的发展及其分析。解决这些挑战只能通过结合各种数学学科来探索，这是本研究单元非常适合的。我们加强了我们在数值分析和力学方面的专业知识，并在第一个资助期的成功基础上，期望我们的成果进一步促进数学和其他学科中这个快速增长的研究领域的发展，以实现突破性的发展。

PDEs on surfaces remain an active field of research in applied mathematics. Due to their coupling with the geometry such surface PDEs are intrinsically nonlinear. This leads to new challenges in modeling and numerical analysis. Most of these challenges are addressed for scalar-valued surface PDEs. In the scalar case the coupling between surface geometry and the PDE is relatively weak, and thus numerical approaches established in flat spaces are applicable after small modifications. For vector- and tensor-valued surface PDEs these approaches are no longer sufficient. The surface vector- and tensor-fields often need to fulfill additional constraints. One example is the tangentiality of these fields, for which they need to be considered as elements of the tangent bundle, leading to a nonlinear coupling between the surface geometry and the PDE. To deal with these new challenges was the motivation for this research unit. Within the last three years we have seen a tremendous growth in research activities, with development of new models, new numerical methods, new numerical analysis results, new software tools, and new applications. Our research unit has significantly contributed to these developments and in several fields also initiated them. In the second funding period we will deepen this knowledge for a description and understanding that uncovers universal principles and makes similarities and differences between the considered applications transparent. Different from the first funding period a focus will be on surface dynamics, including deformable fluid surfaces, rate-independent evolution of prestrained plates, growth and swelling phenomena and the development of numerical methods and their analysis for these new challenges. Addressing these challenges can only be explored by combining various mathematical disciplines for which this research unit is perfectly suited. We strengthen our expertise in numerical analysis and mechanics and based on the success of the first funding period expect our results to further foster this fast growing research field within mathematics and other disciplines to enable breakthrough developments.