Mathematics Department Colloquium: Singularities of solutions of the Hamilton-Jacobi equation. A toy model: distance to a closed subset.

Colloquium | October 18 | 4:10-5 p.m. | 60 Evans Hall

 Albert Fathi, Georgia Institute of Technology

 Department of Mathematics

This is a joint work with Piermarco Cannarsa and Wei Cheng.

If A is a closed subset of the Euclidean space $R^k$, the Euclidean distance function $d_A : R^k \to [0, + \infty[$ is defined by

$$d_A(x) = \mathrm{min}_{a \in A} ||x − a||.$$

This function is Lipschitz, therefore differentiable almost everywhere. We will give topological properties of the set Sing(F) of points in $R^k \setminus M$ where F is not differentiable. For example it is locally connected. We will also discuss the homotopy type of Sing(F).

Although, we will concentrate on $d_A$, we will explain that it is a particular case of a more general result on the singularities of a viscosity solution $F:R^k × ]0, +\infty[ \to R$ of the evolution Hamilton-Jacobi equation

$$ \partial_t F + H(x, \partial_x F) = 0,$$

where $H : R^k × R^k \to R$, $(x, p) \mapsto H(x, p)$ is a $C^2$ Tonelli Hamiltonian, i.e. convex and superlinear in the momentum p. If time permits we will explain the methods of proof for the case of $d_A$.