In the 1960s, Almgren [3, 4] initiated a program to find minimal hypersurfaces in Riemannian manifolds using min-max method. This program was largely advanced by Pitts  and Schoen–Simon  in the 1980s when the manifold is compact without boundary. In this paper, we finish this program for general compact manifold with non-empty boundary. As a corollary, we establish the existence of a smooth embedded minimal hypersurface with non-empty free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves  shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many properly embedded minimal hypersurfaces with non-empty free boundary.
"Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory." J. Differential Geom. 118 (3) 487 - 553, July 2021. https://doi.org/10.4310/jdg/1625860624