I want t maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^2(\mu)}^2:w_1+w_2=1\;\mu\text{-almost surely}\right\},$$ where $(E,\mathcal E,\lambda)$ is a measure space, $\mu\ll\lambda$ is a probability measure on $(E,\mathcal E)$ and $f_i,g_{ij}:E^2\to[0,\infty)$ are $\mathcal E$-measurable.

By Theorem 10.47 in the book of Clark, if $w^\ast$ is a minimizer of $F$ over $S$, then $$0\in\partial F(w^\ast)+N_S(w^\ast)\tag1,$$ where $\partial F(w^\ast)$ denotes Clarke's generalized gradient of $F$ at $w^\ast$ (see Definition 2 on page 53 in the paper of Clarke) and $$N_S(w^\ast):=\left\{\varphi\in\left({L^2(\mu}^2\right)':\langle\varphi,v-w^\ast\rangle\le0\text{ for all }v\in S\right\}$$ denotes the normal cone to $S$ at $w^\ast$.

Now, denoting the integrand in the definition of $F(w)$ by $G((x,y),w)$, by Theorem 1 on page 59 of Clarke's paper, $$\partial F(w)\subseteq\int\lambda^{\otimes2}({\rm d}(x,y))\partial_wG((x,y),w)\tag2$$ for all $w\in{\mathcal L^2(\mu)}^2$.

For fixed $(x,y)\in E^2$, $\partial_wG((x,y),w)$ can be computed as in this answer. (The generalized gradient is only subadditive in general, but in the particular case, it should be additive.)

At this point I'm stuck. What are we able to infer on the form of $w^\ast$ from what we know?