Free-Discontinuity Problems and Perimeter Inequalities under Symmetrisation

该项目将探索自由不连续问题，该问题是由Ennio De Giorgi和Luigi Ambrosio在1988年提出的。这些是变分问题，其中要最小化的能量涉及体积和表面项。“自由不连续”是指表面能集中的集合不是先验的固定集合，可以描述为一个函数的不连续集合。自由不连续问题理论的应用范围从图像分割到断裂力学，再到液晶的研究。这类问题的自然框架是SBV函数之一，即有界变差的特殊函数。粗略地说，这些函数在一个表面上是不连续的，但它们仍然允许一个分配导数，这是一个“规则”度量。在某些应用中，可以考虑更一般的SBD函数（有界变形的特殊函数）。能量仅由表面项组成的情况也很有趣，包括（各向异性）周长泛函。这个项目可能的主题是：-自由不连续问题的确定性和/或随机均匀化-研究对称对周长泛函的影响

The project will explore free-discontinuity problems, which were introduced by Ennio De Giorgi and Luigi Ambrosio in 1988. These are variational problems where the energy to be minimised involves both volume and surface terms. The expression "free-discontinuity" refers to the fact that the set where the surface energy is concentrated is not a priori fixed, and can be described as the discontinuity set of a function. Applications for the theory of free-discontinuity problems range from image segmentation, to fracture mechanics, to the study of liquid crystals.The natural framework for such problems is that one of SBV functions, i.e. Special functions of Bounded Variation. Roughly speaking, these functions are allowed to be discontinuous on a surface, but they still admit a distributional derivative, which is a "regular" measure.In some applications, the more general class of SBD functions can be considered (Special functions of Bounded Deformation).The case when the energy only consists of a surface term is also very interesting and includes, for instance, the (anisotropic) perimeter functional. Possible topics for this project are:- deterministic and/or stochastic homogenisation of free-discontinuity problems - studying the effect of symmetrisation on perimeter functionals