Linear and Nonlinear Laplacians on Riemannian Manifolds

这个建议包含三个独立的项目：高阶椭圆算子的特征函数节点集；Bergman度规的L2上同调，以及秩一对称空间间调和映射在无穷远处的Dirichlet问题。对于节点集方案，研究者提出寻找其Hausdorff测度的下界。对于L2上同调工程，他打算将严格伪凸域的消失定理推广到包括弱伪凸域。对于Dirichlet问题，研究者有兴趣提供调和映射存在的反例。这个项目的主要推力是研究黎曼流形上的拉普拉斯算子。给定一个黎曼流形，即一个具有距离函数的弯曲空间，它的拉普拉斯决定了空间上定义的一类非常重要的函数的行为，即调和函数。调和函数出现在许多不同的环境中：在数学中，它们作为复解析函数的纯部和虚部出现；在应用中，它们描述了各种热和波现象。

9622709 Donnelly This proposal contains three separate projects: nodal sets of eigenfunctions for higher order elliptic operators; L2 cohomology of the Bergman metric, and the Dirichlet problem at infinity for harmonic maps between rank one symmetric spaces. For the nodal set project, the investigator proposes to find a lower bound for its Hausdorff measure. For the L2 cohomology project, he intends to generalize the vanishing theorem for strictly pseudoconvex domains to include weakly pseudoconvex domains. For the Dirichlet problem, the investigator is interested in providing a counterexample to existence of harmonic maps. The main thrust of this project is the study of Laplacians on Riemannian manifolds. Given a Riemannian manifold, that is, a curved space possessing a distance function, its Laplacian determines the behavior of a very important class of functions defined on the space, namely, harmonic functions. Harmonic functions arise in many different contexts: in mathematics, they arise as pure and imaginary parts of complex-analytic functions; in applications, they describe various heat and wave phenomena.