| Abstract: |
| In this talk, we will discuss the intrinsic approach of the propagation of singularities of viscosity solutions of the first order Hamilton-Jacobi equations, mainly based on the joint work with Piermarco Cannarsa and Albert Fathi.
Let $H$ be a Tonelli Hamiltonian. We consider the propagation of singularities along generalized characteristics by an intrinsic method. We will show that, for a prescribed solution u which has the representation in the form of inf-convolution, the relevant precess of sup-convolution determines the propagation of singulars and generalized characteristics for singular initial data. This method leads to the global result under mild Tonelli conditions.
Based on this global result, we can discuss the associated singular dynamics in both topological and differential sense. We obtained the homotopy equivalence result between the complement of Aubry set and the cut locus of u, and the local path-connected result of the cut locus. We also studied the $\omega$-limit set of the semi-flows defined by generalized characteristics, and their connections to the regular dynamics.
We shall also discuss this problem under different type of initial/boundary conditions. |
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