Peskin and Schroeder give a brief outline of Lagrangian field theory on page fifteen in their Quantum Field Theory book, where they write:
Lagrangian Field Theory
The fundamental quantity of classical mechanics is the action, $S$, the time integral of the Lagrangian, $L$. In a local field theory the Lagrangian can be written as the spatial integral of a Lagrangian density, denoted by $\mathcal{L}$, which is a function of one or more fields $\phi(x)$ and their derivatives $\partial_\mu\phi$. Thus we have
$$ S = \int Ldt = \int\mathcal{L}(\phi,\partial_\mu\phi)\ d^4x.$$
Since this is a book on field theory, we will refer to $\mathcal{L}$ simply as the Lagrangian. The principle of least action states that when a system evolves from one given configuration to another between times $t_1$ and $t_2$, it does so along the “path” in configuration space for which $S$ is an extremum (normally a minimum).
In relativistic classical field theory, I'd expect the Lagrangian density to be integrated over all space and time so that no point or boundary is more privileged than any other. So what do we assume about the fields, if at all, that enables the Lagrangian density to be integrated between two fixed times, while maintaining a classical field theory that's still relativistic?