Abstract
Motivated by the vanishing contact problem, we study in the present paper
the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let
H
(
x
,
p
,
u
)
{H(x,p,u)}
be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter
λ
>
0
{\lambda>0}
, we denote
by
u
λ
{u^{\lambda}}
the unique viscosity solution of the Hamilton–Jacobi equation
H
(
x
,
D
u
(
x
)
,
λ
u
(
x
)
)
=
c
.
H\big{(}x,Du(x),\lambda u(x)\big{)}=c.
Under quite general assumptions, we prove that
u
λ
{u^{\lambda}}
converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation
H
(
x
,
D
u
(
x
)
,
0
)
=
c
{H(x,Du(x),0)=c}
. We also characterize the limit solution in terms of Peierls barrier and Mather measures.
Geometric Conditions for Regularity of Viscosity Solution to the Simplest Hamilton-Jacobi Equation
