Regularity properties of stationary and evolution free boundary problems

本文主要讨论了平稳和无演化边界问题的正则性问题。本课题部分研究了均匀椭圆算子驱动的分布源两相自由边界问题。我们计划开发新技术来解决平坦或利普希茨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研究类似的问题，这些问题不能在经典框架内处理。将制定适当的策略，以考虑到这一问题的微妙几何特征。另一个研究的问题是源于非齐次stefan型演化问题的运动自由边界的规律性。最后，我们建议通过研究其高正则性（解析性）问题来完成薄的一相自由边界的分析。所研究的自由边界问题通常出现在数学和科学的不同领域：形状优化、燃烧理论、概率论和统计学等。具有分布源的两相自由边界问题的典型例子是流体力学中的Prandtl-Bachelor模型或磁流体力学中的本征值问题。stefan型演化问题描述了具有固-液界面的材料的熔化（或凝固）。最后，在经典物理模型中，在存在长程（非局部）相互作用的介质中，薄的自由边界是相关的，例如在不连续面的传播中，如平面裂纹扩展。因此，更好地理解这些问题对广大科学界具有极大的兴趣。

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.