Sobolev Spaces on Lie Manifolds and Polyhedral Domains

We study Sobolev spaces on Lie manifolds, which we define as a class of manifolds described by vector fields (see Definition 1.2). The class of Lie manifolds includes the Euclidean spaces Rn, asymptotically flat manifolds, conformally compact manifolds, and manifolds with cylindrical and polycylindrical ends. As in the classical case of Rn, we define Sobolev spaces using derivatives, powers of the Laplacian, or a suitable class of partitions of unity. We extend the basic results about Sobolev spaces on Euclidean spaces to the setting of Lie manifolds. These results include the definition of the trace map, a characterization of its range, the extension theorem, the density of smooth functions, and interpolation properties. One of the main motivations is that, in the examples we have studied so far, the totally-characteristic Sobolev spaces on polyhedral domains identify with Sobolev spaces on suitable Lie manifolds with boundary. The analysis we develop may be useful for solving certain types of non-linear partial differential equations on non-compact manifolds that appear, for instance, in Einstein’s constraint equations. We also sketch two applications, one to the Yamabe functional and one to the regularity of boundary value problems on polyhedral domains.