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For nonnegative even kernels $K$, we consider the $K$-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated $K$-nonlocal mean curvature equation in an open set $\Omega\subset\mathbb{R}^n$, we built a calibration for the nonlocal perimeter inside $\Omega\subset\mathbb{R}^n$. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in $\Omega$ of each leaf of the foliation. As an application, we prove the minimality of $K$-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions., Comment: To appear in Annali di Matematica Pura ed Applicata

We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of $k$-contact structure and $k$-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the $k$-symplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers' equation., 26 pp. Minor corrections. The bibliography has been updated