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The most important open problem in Monotone Operator Theory concerns themaximal monotonicity of the sum of two maximal monotone operators provided thatRockafellar's constraint qualification holds.  In this paper, we prove the maximal monotonicity of $A+B$ provided that $A$and $B$ are maximal monotone operators such that $\dom A\cap\inte\domB\neq\varnothing$, $A+N_{\overline{\dom B}}$ is of type (FPV), and $\domA\cap\overline{\dom B}\subseteq\dom B$. The proof utilizes the Fitzpatrickfunction in an essential way.

The sum of a maximal monotone operator of type (FPV) and a maximal monotone operator with full domain is maximal monotone