Hamilton-Jacobi equations on metric measure spaces

第一个项目研究了度量测度空间中的程函方程，其中非齐次项在区域中是不连续的、无界的且仅p-可积。 推广了度量空间中Monge解的概念，建立了相应Dirichlet边值问题的唯一性和存在性结果。第二个项目是关于Heisenberg群中给定连续函数的水平拟凸包络的PDE方法。我们得到了非局部Hamilton-Jacobi方程Dirichlet边值问题粘性解的存在唯一性。

The first project studies the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely p-integrable in the domain. Generalizing the notion of Monge solutions in metric space, we establish uniqueness and existence results for the associated Dirichlet boundary problem.The second project is concerned with a PDE-based approach to the horizontally quasiconvex envelope of a given continuous function in the Heisenberg group. We obtain the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton-Jacobi equation.