Geometric operators on singular domains

边值问题由于其在物理、几何和数值分析中的应用而得到广泛的研究。如果边界是光滑的，我们对标准狄利克雷和诺依曼边界条件有很好的理解。这里，解的正则性仅被所考虑的算子下的边界值之一和图像所阻碍。对于具有奇异性的边界，特别是在更高的维度上，这个问题还没有完全得到很好的理解。本文研究了奇异区域上Laplacian算子在混合边界条件下的适定性问题。我们强调，正则性和适定性的结果也很重要的数值应用。例如，通常的Galerkin格式的收敛阶由Sobolev尺度决定。用于我们的程序的基本思想是，通过适当的权重函数对度量的共形改变可以通过将边界的奇点发送到无穷大来将具有分层边界的域变换为具有边界的有界几何的流形。 当奇异域上的适定性结果总是在某个加权Sobolev空间中时，我们可以用保形爆破来处理将标准Sobolev空间转化为非紧爆破所得到的边值问题。这将允许一个统一的处理更一般的奇性也在更高的dimensions.In我们的项目的第二部分，我们研究的非相对论薛定谔算子的N电子在库仑型势。特别是，我们感兴趣的是本征函数的正则性。 这与物理和化学中的应用高度相关，因为它有助于建立用于本征函数的数值计算的改进的自适应算法。 虽然这与上面的适定性问题不同，但类似的想法将被应用。我们把势的奇点，即多电子或电子与核碰撞的点，作为我们再次爆破的边值问题的奇点。这将与现有的工具相结合，如自然紧化。我们还将Kustaanheimo-Stiefel变换纳入我们的图像中，这种方法以前曾成功地应用于经典力学中的库仑势和两粒子碰撞中薛定谔算子本征函数的强正则性结果。此外，薛定谔本征函数，我们的结果可以作为一个起点，在未来的数值算法。

Boundary value problems are and were extensively studied due to their applications to physics, geometry, and numerical analysis. If the boundary is smooth, we have a very good understanding of the standard Dirichlet and Neumann boundary conditions. Here, the regularity of the solutions is only obstructed by the one of the boundary values and the image under the operator under consideration. For boundaries with singularities, in particular in higher dimensions, the problem is not completely well-understood. In this project we study the question of well-posedness for the Laplacian with mixed boundary conditions on singular domains. We stress that regularity and well-posedness results are also important for numerical applications. For example the order of convergence of usual Galerkin schemes are determined by the Sobolev scale. The underlying idea used for our programme is that a conformal change to a metric by an appropriate weight function can transform domains with stratified boundary into a manifold of bounded geometry with boundary by sending the singularities of the boundary to infinity. While the well-posedness results on the singular domains are always in some weighted Sobolev space we can treat with a conformal blow-up the boundary value problem obtained by translation to the noncompact blow-up with standard Sobolev spaces. This will allow for a uniform treatment of more general singularities also in higher dimensions.In the second part of our project we study the nonrelativistic Schrödinger operator for N electrons in a Coulomb-type potential. In particular we are interested in the regularity of the eigenfunctions. This is of high relevance to applications in physics and chemistry as it helps to establish improved adaptive algorithms for numerical calculations of the eigenfunctions. Although this is different from the well-posedness question from above, similar ideas will be applied. We treat the singularities of the potential, i.e. points of multi-electron or electron-nucleus collisions, as singularities of the boundary value problems which we blow-up again. This will be combined with existing tools like natural compactifications. We also incorporate the Kustaanheimo-Stiefel transform in our picture, a method which was previously successfully applied for Coulomb potentials in classical mechanics and for strong regularity results for eigenfunctions of the Schrödinger operator in two-particle collisions. Also for Schrödinger eigenfunctions, our results may serve as a starting point for numerical algorithms in the future.