SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2014-2015
Jeudi 13 novembre 2014 :
Nao Hamamuki
(Waseda University, Tokyo)
Hamilton-Jacobi equations with discontinuous source terms.
We study the initial-value problem for a Hamilton-Jacobi equation
whose Hamiltonian is discontinuous with respect to space variables.
Our motivation comes from the two dimensional nucleation
in crystal growth phenomena. The associated equation is nonlinear
and has a semicontinuous source term, but, because of the discontinuity,
solutions are not unique even in Ishii's sense of viscosity solutions.
To overcome this issue, we introduce a new notion of viscosity solutions
and prove that the initial-value problem admits a unique global-in-time
solution.
We also give a representation formula of the solution as a value
function of the optimal control problem with a semicontinuous running cost
function.
Moreover, we discuss the large time behavior of the solution via the
scaling method.
