Approximation problems for Sobolev homeomorphisms

本课题的主要目标是获得双sobolev平面同胚的逼近结果。这将对非线性弹性的数学研究产生重大影响，实际上这是该领域一个长期存在的开放性问题。更准确地说，设u是平面上两个开放子集之间的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。我们想要找到两个集合之间的微分同态序列，它们在W^{1，p}中收敛于u，同时它们的逆在W^{1，p}中收敛于u^{-1}。问题之所以不平凡，是因为通常的带平滑核的卷积通常失去了内射的性质，因此需要一种不同的、新颖的方法。在过去的几年里，不同的作者致力于解决用W^{1，p}意义上的微分同态逼近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，该项目的申请人在与不同合作者的一些论文中开发了一种最成功的方法。该方法既解决了“简化”问题，又解决了特殊情况p=1时的一般问题；我们的希望是，同样的方法的进一步发展，可以解决任何p bbb1的一般问题。这不是项目的唯一目标：有几个不同的近似问题，这些问题与主要问题有关，并且应该通过相同主要策略的不同应用来解决；其中一些看起来很简单，而另一些则要复杂得多。不同难度的问题的多样性确保了项目成功的低风险。这个研究项目的一个关键因素将是从事这项工作的劳动力。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，我们希望给数学水平很高的年轻学生。

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.