On minimax theorems for lower semicontinuous functions in Hilbert spaces

We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\mathrm{\Phi }$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\mathrm{\Phi }$-convex functions. These conditions are expressed in terms of abstract $\mathrm{\Phi }$-subgradients.