Regularity of Solutions of Hamilton-Jacobi Equation on a Domain

PDE Seminar
Tuesday, October 28, 2014 - 15:05
1 hour (actually 50 minutes)
Skiles 006
École Normale Supérieure de Lyon, France
In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type.  The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p.  Let D be a compact smooth domain with boundary $\partial D$ contained in $T^N \times  ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then this is also the case on the interior of D.  There are several variants of this result in different settings.  To make the result accessible to the layman, we will explain the method on the function distance to a closed subset of an Euclidean space. This example contains all the ideas of the general case.