论文标题
具有非平滑汉密尔顿和标量保护法的汉密尔顿 - 雅各比方程的可及
Reachable set for Hamilton-Jacobi equations with non-smooth Hamiltonian and scalar conservation laws
论文作者
论文摘要
我们对操作员的范围进行了完整的表征，该操作员范围与汉密尔顿 - 雅各比方程的时间$ t $ t $ t a Hamiltonian相关联的粘度解决方案。我们的主要动机是能够在没有其他规律性假设的情况下对待Hamiltonians的案例。我们特别注意$ h（p）= | p | $，为此，我们通过内部球条件在Sublevel套件上通过内部球条件提供了相当几何的粘度操作员范围。根据我们对可及的集合的表征，我们能够推断出有关可触及功能的尖锐规律性估计以及可触及集合的结构属性的进一步结果。结果最终适应了维度一号标量保护法的情况。
We give a full characterization of the range of the operator which associates, to any initial condition, the viscosity solution at time $T$ of a Hamilton-Jacobi equation with convex Hamiltonian. Our main motivation is to be able to treat the case of convex Hamiltonians with no further regularity assumptions. We give special attention to the case $H(p) = |p|$, for which we provide a rather geometrical description of the range of the viscosity operator by means of an interior ball condition on the sublevel sets. From our characterization of the reachable set, we are able to deduce further results concerning, for instance, sharp regularity estimates for the reachable functions, as well as structural properties of the reachable set. The results are finally adapted to the case of scalar conservation laws in dimension one.
