Regularity properties of stationary and evolution free boundary problems

该建议侧重于固定和演化自由边界问题的正则性问题。这个项目的一部分是处理由一致椭圆算子驱动的分布源两相自由边界问题。我们计划开发新的技术来解决平坦或Lipschitz自由边界是否光滑的基本问题。我们还建议在海森堡群的自由边界问题的背景下，这不能在经典框架内处理类似的问题。 将制定适当的战略，以考虑到微妙的几何特征的问题。另一个正在调查的问题是在Stefan型的非齐次演化问题的移动自由边界的规律性。最后，我们建议通过研究其更高的规则性（解析性）问题来完成对薄的单相自由边界的分析。所研究的自由边界问题通常出现在不同的数学和科学领域：形状优化，燃烧理论，概率和统计等。具有分布源的两相自由边界问题的典型例子是流体动力学中的Prandtl-Bachelor模型或磁流体动力学中的特征值问题。Stefan型演化问题描述了具有固-液界面的材料的熔化（或固化）。最后，薄的自由边界是相关的介质中的远程（非局部）相互作用的经典物理模型，如在不连续表面的传播，如平面裂纹扩展。因此，更好地了解这些问题是广大科学界的极大兴趣。

This proposal focuses on regularity issues for stationary and evolution free boundary problems. Part of this project deals with two-phase free boundary problems with distributed sources, driven by uniformly elliptic operators. We plan to develop new techniques to settle the fundamental question of whether flat or Lipschitz free boundaries are smooth. We also propose to investigate similar issues in the context of free boundary problems in the Heisenberg group, which cannot be treated within the classical framework. Appropriate strategies will be developed to take into account the subtle geometric features of the problem. Another problem under investigation is the regularity of the moving free boundary originating in a non-homogeneous evolution problem of Stefan-type. Finally, we propose to complete the analysis of thin one phase free boundaries by investigating the question of their higher regularity (analyticity).The free boundary problems under investigation arise in different areas of mathematics and science in general: shape optimization, combustion theory, probability and statistics among others. Typical examples of two-phase free boundary problems with distributed sources are the Prandtl-Bachelor model in fluid dynamics or the eigenvalue problem in magneto hydrodynamics. Stefan-type evolution problems describe the melting (or solidification) of a material with a solid-liquid interphase. Finally, thin free boundaries are relevant in classical physical models in mediums where long range (non-local) interactions are present, as in the propagation of surfaces of discontinuities, like planar crack expansion. A better understanding of these problems is thus of great interest to a wide scientific community.