The generalized (nonlinear) Poisson problem: A dual variational approach

A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity, isotropy, and material homogeneity. In this paper a generalization with respect to nonliearity, anisotropy, and inhomogeneity is made. Starting from the canonical basic equations in the primal and dual formulation respectively we derive systematically the corresponding generalized variational principles; under certain conditions they can be extended to so calle complementary extremum principles allowing for global bounds. For simplicity a restriction to two dimensional problems is made, including twice-connected domains.