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被广泛研究。如果边界是光滑的，我们就很好地理解了标准的狄利克雷和诺伊曼边界条件。在这里，解的正则性仅被其中一个边界值和所考虑的算子下的图像所阻碍。对于具有奇点的边界，特别是在高维中，这个问题还没有完全被理解。本文研究奇异域上具有混合边界条件的拉普拉斯算子的适定性问题。我们强调正则性和适定性结果对数值应用也很重要。例如，通常的伽辽金格式的收敛阶是由Sobolev尺度决定的。我们的程序使用的基本思想是，通过适当的权函数对度规进行保角改变，可以通过将边界的奇点发送到无穷远，将具有分层边界的域转换为具有边界的有界几何流形。由于奇异域上的适定性结果总是在某些加权Sobolev空间中，我们可以用保角爆破来处理由转化为标准Sobolev空间的非紧化爆破得到的边值问题。这将允许在更高维度上对更一般的奇点进行统一处理。在我们项目的第二部分，我们研究了库仑型势中N电子的非相对论性Schrödinger算符。我们特别感兴趣的是特征函数的规律性。这与物理和化学的应用有很大的相关性，因为它有助于建立改进的自适应算法来计算特征函数的数值。虽然这与上面的适位性问题不同，但类似的想法将被应用。我们把势的奇点，即多电子或电子-核碰撞的点，作为我们再次放大的边值问题的奇点。这将与现有的工具如自然压实相结合。我们还在我们的图中加入了Kustaanheimo-Stiefel变换，这种方法以前成功地应用于经典力学中的库伦势和两粒子碰撞中Schrödinger算子的本征函数的强规则性结果。同样对于Schrödinger特征函数，我们的结果可以作为未来数值算法的起点。

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.