Approximation problems for Sobolev homeomorphisms

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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.