A property of strictly convex functions which differ from each other by a constant on the boundary of their domain

In this paper, in particular, we prove the following result: Let $E$ be a reflexive real Banach space and let $C\subset E$ be a closed convex set, with non-empty interior, whose boundary is sequentially weakly closed and non-convex. Then, for every function $\phi :\mathrm{\partial }C\to \mathbf{R}$ and for every convex set $S\subseteq E^{\ast }$ dense in $E^{\ast }$, there exists $\tilde{\gamma }\in S$ having the following property: for every strictly convex lower semicontinuous function $J:C\to \mathbf{R}$, Gâteaux differentiable in $\text{int}(C)$, such that $J_{\mathrm{\mid }\mathrm{\partial }C}-\phi$ is constant in $\mathrm{\partial }C$ and ${\lim }_{\mathrm{\parallel }x\mathrm{\parallel }\to +\mathrm{\infty }}\frac{J(x)}{\mathrm{\parallel }x\mathrm{\parallel }}=+\mathrm{\infty }$ if $C$ is unbounded, $\tilde{\gamma }$ is an algebraically interior point of $J^{\mathrm{\prime }}(\text{int}(C))$ (with respect to $E^{\ast }$).