I am interested in the relation between the definition of an 'elliptic boundary value problem' and an 'elliptic partial differential equation'. From the wiki entries it seems that 'elliptic boundary value problem' is a stable state boundary value problem which does not have a time variable. An elliptic partial differential equation is a category of partial differential equations categorized by the discriminant $B^2-4AC$. Are these two concepts related in any way? Would I be right in stating that an elliptic boundary value problem always has an underlying elliptic partial differential equation?
Thanks.
Yes, it would be correct to say that an elliptic boundary value problem always has an underlying elliptic PDE. However, I should warn you that the Wikipedia article Elliptic partial differential equation considers only second-order linear equations in nondivergence form. Ellipticity is defined differently for divergence and non-divergence type linear equations, yet differently for quasilinear equations, and yet differently for fully nonlinear equations. I find it unfruitful to discuss the meaning of "ellipticity", either for operators or for boundary value problems, without having in mind a particular class of equations.
By the way: not every boundary condition, when imposed on an elliptic PDE, results in good behavior, as Hadamard's example shows.