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For a class of formally hypoelliptic differential operators in divergence form we prove a generalized Girding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.doi:10.1017/s1446788700032134 fatcat:bwgyfidkevh4vcmbant7w25hte