DOI: 10.68381/jca18069 ISSN: 0944-6532
An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations
Sandro Zagatti
We devote the integro-extremization method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations
\begin{cases} F(Du)=0 & \quad \textrm{in} \quad{\Omega}\cr u(x)=\varphi(x) & \quad \textrm{for} \quad x\in \partial {\Omega}, \end{cases}
{
F
(
D
u
)
=
0
in
Ω
u
(
x
)
=
φ
(
x
)
for
x
∈
∂
Ω
,
with a particular interest for non coercive hamiltonians
F
F
, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations
\begin{cases} \frac{\partial u}{\partial t}+ F(\nabla u)=0 & \quad \textrm{in} \quad ]0,T[\times {\Omega}\cr u(0,x)=\eta(x) & \quad \textrm{for} \quad x\in{\Omega}\cr u(t,x)=\psi(x) & \quad \textrm{for} \quad (t,x)\in[0,T]\times \partial {\Omega}. \end{cases}
{
∂
u
∂
t
+
F
(
∇
u
)
=
0
in
]
0
,
T
[
×
Ω
u
(
0
,
x
)
=
η
(
x
)
for
x
∈
Ω
u
(
t
,
x
)
=
ψ
(
x
)
for
(
t
,
x
)
∈
[
0
,
T
]
×
∂
Ω
.
We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian
F
F
, on the domain
\Omega
Ω
and on the boundary datum, without any growth assumptions on
F
F