Differentiability properties of the minimal average action

Given a Z n+l-periodic variational principle on R n+l we look for solutions u : R" ---+ R minimizing the variational integral with respect to compactly supported variations. To every vector c~ E R" we consider a subset ~/g~ of solutions which have an average slope c~ when averaging over R n. The minimal average action A(c 0 is defined by the average value of the variational integral given by a solution with average slope c~. Our main result is: A is differentiable at c~ if and only if the set ~/~a is totally ordered (in the natural sense). In case that J / / ~ is not totally ordered, A is differentiable at c~ in some direction/3 E R ~ \ {0} if and only if fl is orthogonal to the subspace defined by the rational dependency of c~. Assuming that the i th component of c~ is rational with denominator s i c N in lowest terms, we show: The difference of rightand left-sided derivative in the i th standard unit direction is bounded by const, s-!r. Mathematics Subject Classification (1991): 58C20; 46G05; 26B25