Approximation problems for Sobolev homeomorphisms

本项目的主要目的是得到双Sobolov平面同胚的逼近结果。这将对非线性弹性力学的数学研究产生重大影响，实际上这是该领域一个长期悬而未决的问题。更准确地说，设u是平面的两个开子集之间的W^{1，p}同胚，具有W^{1，p}逆；这样的函数模拟了平面弹性物体在外力作用下的变形。我们要找出在W^{1，p}中收敛到u的两个集之间的微分同胚序列，同时它们的逆在W^{1，p}中收敛到u^{-1}。这个问题之所以不是平凡的，是因为对于通常的具有光滑核的卷积，通常失去了内射的性质，因此需要一种不同的和原创的方法。在过去的几年里，不同的作者致力于解决用W^{1，p}意义下的微分同胚逼近W^{1，p}同胚的问题：这是上述问题的简化版本，其中不需要关于逆映射的性质。特别是，该项目的申请者在一些不同合著者的论文中开发了最成功的方法之一。这种方法不仅解决了“简化”问题，而且还解决了特殊情况p=1的一般问题；我们希望进一步发展同样的方法可以解决任何p&gt；1的一般问题。这不是本项目的唯一目标：有几个不同的逼近问题，它们与主要问题相联系，应该通过同一主要策略的不同应用来解决；其中一些看起来很容易，另一些应该更复杂。这种具有不同难度的多重问题确保了项目成功的低风险。这项研究项目的一个关键因素将是致力于此的劳动力。事实上，几乎所有申请的拨款都将用于资助一个博士和一个博士后职位，我们希望将这些职位提供给数学水平非常高的年轻学生。

The main goal of this Project is to achieve approximation results for bi-Sobolev planar homeomorphisms. This would have significant consequences for the mathematical study of the nonlinear elasticity, and in fact this is a long-standing open problem in the field. More precisely, let u be a W^{1,p} homeomorphism, with W^{1,p} inverse, between two open subsets of the plane; such a function models the deformation of a planar, elastic object subject to some external force. One wants to find a sequence of diffeomorphisms between the two sets which converge to u in W^{1,p}, and contemporarily their inverses converge to u^{-1} in W^{1,p}. The reason why the problem is non-trivial is that with the usual convolution with a smoothing kernel the property of being injective is generally lost, hence a different and original method is required. In the last years, different authors have contributed to solve the problem of approximating a W^{1,p} homeomorphism with diffeomorphisms in the W^{1,p} sense: this is a simplified version of the problem described above, where no property about the inverse maps is required. In particular, one of the most successful methods has been developed by the applicant of this project, in some papers with different coauthors. This method not only solves the ``simplified'' problem, but also the general one for the special case p=1; our hope is that a further development of the same method can solve the general problem for any p>1. This is not the sole goal of the project: there are several different approximation problems, which are connected with the main one, and which should be solved by different applications of the same main strategy; some of them appear to be quite easy, others should be more complicate. This multiplicity of questions with different levels of difficulty ensures a low risk about the success of the project. A key ingredient for this research project will be the workforce working on that. In fact, almost the whole requested grant will be devoted to fund one Ph.D. and one postdoc position, which we hope to give to young students of very high mathematical level.