On Some Boundary Value Problems for the Equation nabla \cdot (F(|nabla w|)nabla w) = 0

A method based on the use of a reciprocal theorem and dual functions in the strain plane (γ _x, γ _y), where γ _x = partial w/partial x, γ _y = partial w/partial y is used to deduce solutions of the equation nabla \cdot (F(|nabla w|)nabla w) = 0 for various boundary value problems. The method is applied, in principle, to general (F(|nabla w|) and specific examples, previously treated by other methods (e.g. F(|nabla w|) = |nabla w|^N), are worked out in detail as a check on the method.