  In this paper we study the evolution problem \[ \left\lbrace\begin{array}{ll}
u_t (x,t)- \lambda_j(D^2 u(x,t)) = 0, & \text{in } \Omega\times (0,+\infty), \\
u(x,t) = g(x,t), & \text{on } \partial \Omega \times (0,+\infty), \\ u(x,0) =
u_0(x), & \text{in } \Omega, \end{array}\right. \] where $\Omega$ is a bounded
domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its
boundary) and $\lambda_j(D^2 u)$ stands for the $j-$st eigenvalue of the
Hessian matrix $D^2u$. We assume that $u_0 $ and $g$ are continuous functions
with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial \Omega$.
  We show that the (unique) solution to this problem exists in the viscosity
sense and can be approximated by the value function of a two-player zero-sum
game as the parameter measuring the size of the step that we move in each round
of the game goes to zero.
  In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$,
we show that viscosity solutions to this evolution problem stabilize and
converge exponentially fast to the unique stationary solution as $t\to \infty$.
For $j=1$ the limit profile is just the convex envelope inside $\Omega$ of the
boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this
result with two different techniques: with PDE tools and and with game
theoretical arguments. Moreover, in some special cases (for affine boundary
data) we can show that solutions coincide with the stationary solution in
finite time (that depends only on $\Omega$ and not on the initial condition
$u_0$).
