DOI: 10.68381/jca27041 ISSN: 0944-6532
Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions
Julius Ross, David Witt Nyström
Suppose
f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2
f
(
x
,
y
)
+
κ
2
∥
x
∥
2
−
σ
2
∥
y
∥
2
is convex where
\kappa\ge 0, \sigma>0
κ
≥
0
,
σ
>
0
, and the argmin function
\gamma(x) = \{ \gamma: \inf_y f(x,y) = f(x,\gamma)\}
γ
(
x
)
=
{
γ
:
inf
y
f
(
x
,
y
)
=
f
(
x
,
γ
)
}
exists and is single valued. We will prove
\gamma
γ
is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.