I'm looking for sufficient conditions that guarantee that partial subgradients of a convex, lower-semicontinuous functional $f:X_1\times X_2\rightarrow\overline{\mathbb{R}}$ form a subgradient of $f$. Here, $X_1$ and $X_2$ should be (reflexive) Banach spaces or even locally convex spaces...
More specifically, for fixed $(\bar x_1,\bar x_2)$ let us consider the functionals $f_1:X_1\rightarrow\overline{\mathbb{R}}$ and $f_2:X_2\rightarrow\overline{\mathbb{R}}$ given by $$ f_1(x_1):=f(x_1,\bar x_2),\hskip1cm f_2(x_2):=f(\bar x_1,x_2) $$ and let us assume that $\bar x_1^*\in\partial f_1(\bar x_1)$ and $\bar x_2^*\in\partial f_2(\bar x_2)$ meaning that $\bar x_1^*$ and $\bar x_2^*$ belong to the set of subgradients of $f_1$ in $\bar x_1$, respectively of $f_2$ in $\bar x_1$. Under which conditions on $f$ can we conclude that $(\bar x_1^*,\bar x_2^*)\in X_1^*\times X_2^*\cong(X_1\times X_2)^*$ is a subgradient of $f$ in $(\bar x_1,\bar x_2)$? This would be equivalent to $$ f(x_1,x_2)\ge f(\bar x_1,\bar x_2) + \langle \bar x_1^*,x_1-\bar x_1\rangle + \langle \bar x_2^*,x_2-\bar x_2\rangle \hskip8mm\forall (x_1,x_2)\in X_1\times X_2. $$ If you now any conditions that are sufficient (or, that are necessary), please let me know. Thanks a lot!