  We propose a notion of a pluri-Lagrangian problem, which should be understood
as an analog of multi-dimensional consistency for variational systems. This is
a development along the line of research of discrete integrable Lagrangian
systems initiated in 2009 by Lobb and Nijhoff, however having its more remote
roots in the theory of pluriharmonic functions, in the Z-invariant models of
statistical mechanics and their quasiclassical limit, as well as in the theory
of variational symmetries going back to Noether. A d-dimensional
pluri-Lagrangian problem can be described as follows: given a d-form L on an
m-dimensional space (called multi-time, m>d), whose coefficients depend on a
sought-after function x of m independent variables (called field), find those
fields x which deliver critical points to the action functionals
$S_{\Sigma}=\int_{\Sigma} L$ for any d-dimensional manifold $\Sigma$ in the
multi-time. We derive the main building blocks of the multi-time Euler-Lagrange
equations for a discrete pluri-Lagrangian problem with d=2, the so called
corner equations, and discuss the notion of consistency of the system of corner
equations. We analyze the system of corner equations for a special class of
three-point 2-forms, corresponding to integrable quad-equations of the ABS
list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff
by showing that the corresponding 2-forms are closed not only on solutions of
(non-variational) quad-equations, but also on general solutions of the
corresponding corner equations. We also find an example of a pluri-Lagrangian
system not coming from a multidimensionally consistent system of
quad-equations.
