http://www.cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap6.pdf 6.1 "Classification of PDE's" clarifies which boundary conditions are needed for existence and uniqueness of each <classification of second order partial differential equations into elliptic, parabolic and hyperbolic>[type of second order of PDE]:
* <elliptic partial differential equation> and <parabolic partial differential equation>: <Dirichlet boundary condition> or <Neumann boundary condition>
* <hyperbolic partial differential equation>: <Cauchy boundary condition>


 Which boundary conditions lead to existence and uniqueness of a second order PDE