A sufficient condition for the differentiable optimal solution for a problem of Bellman (Q1191858)

The paper gives a new result concerning the solution of a problem of Bellman: Minimize \(\int_ I(f+g)\) over all pairs \((f,g)\) satisfying \(f(x)+g(y)\geq k(x,y)\), the latter being a continuous kernel. Specifically, it is proved that the problem above has a unique and differentiable solution provided \(k\in C^ 2\) and \(k_{xy}\) does not change its sign on \(I\times I\). The methods used rely on earlier results of the second author.