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The present paper provides a generalization of the previous authors' work onBellman functions for integral functionals on $\mathrm{BMO}$. Those Bellmanfunctions are the minimal locally concave functions on parabolic strips in theplane. Now we describe the algorithm for constructing minimal locally concavefunctions on a planar domain that is a difference of two unbounded convexdomains. This leads to many sharp estimates for functions in the classes like$\mathrm{BMO}$, $A_p$, or the Gehring classes.

Bellman functions on simple non-convex domains in the plane