Hamilton-Jacobi equations on metric measure spaces

第一个课题研究了度量测度空间中的对角方程，其中非齐次项在域内允许不连续、无界且仅为p可积。推广度量空间中Monge解的概念，建立了相关Dirichlet边界问题的唯一性和存在性结果。第二个项目是关于一个基于pde的方法来水平拟凸包络在海森堡群中给定的连续函数。得到了非局部Hamilton-Jacobi方程Dirichlet边界问题黏性解的唯一性和存在性。

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.