Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry
Publications Mathématiques de l'Institut des Hautes Études Scientifiques - Tập 133 - Trang 327-366 - 2021
Tóm tắt
If
$U:[0,+\infty [\times M$
is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation
$$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$
where
$M$
is a not necessarily compact manifold, and
$H$
is a Tonelli Hamiltonian, we prove the set
$\Sigma (U)$
, of points in
$]0,+\infty [\times M$
where
$U$
is not differentiable, is locally contractible. Moreover, we study the homotopy type of
$\Sigma (U)$
. We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.
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