Conceptual Doubt regarding principle of superposition for Linear PDE(s) This is in regards to a problem I have been trying to solve. I have already posted a question regarding it previously, but the part of the problem I describe here leans more towards conceptual understanding, so I post this as a separate question. I have the Laplacian:
$$\nabla^2 T(x,y,z)=0 \tag{A}$$
on a domain where $x \in [0,a], y \in [0,b], z \in [0,c]$ with BC(s) as:
$$T(0,y,z)=T_{w1}$$
$$T(x,0,z)=T_{w2}$$
$$\frac{\partial T}{\partial x}\bigg \vert_{a,y,z}=\frac{\partial T}{\partial y}\bigg \vert_{x,b,z}=0$$
$$T(x,y,0)=T(x,y,c)=0$$
Since there ae two non-homogeneous BC(s), I decided to split the problem in two parts to later add up their individual solutions. 
P1 $$T(0,y,z)=T_{w1}$$
$$\frac{\partial T}{\partial x}\bigg \vert_{a,y,z}=0$$ $$T(x,0,z)=T(x,b,z)=T(x,y,0)=T(x,y,c)=0$$
P2 $$T(x,0,z)=T_{w2}$$ $$\frac{\partial T}{\partial y}\bigg \vert_{x,b,z}=0$$
$$T(0,y,z)=T(a,y,z)=T(x,y,0)=T(x,y,c)=0$$
When I divide the problem as explained above, it's evident that each sub-problem has a homogeneous Neumann condition on a particular face. 
This same face behaves as a homogeneous Dirichlet in the conjugate/other problem.
I am unable to understand that when we write the final solution by adding up individual answers, whether the final answer will reflect the Neumann (insulated condition) or the Dirichlet condition? (For ex. for the face a $x=a$, will the final solution reflect $\frac{\partial T}{\partial x}\bigg \vert_{a,y,z}=0$ or will it reflect $T(a,y,z)=0$ ?)
My doubt stems from the fact that both are homogeneous in nature. If there is some serious conceptual flaw in the way I am dealing with things here? Any input is appreciated.
 A: Your problem is
$$
                              \nabla^2 T(x,y,z) = 0, \\
 0 \le x \le a,\; 0 \le y \le b,\; 0 \le z \le c. \\
      \mbox{Conditions on X faces}:       T(0,y,z)= T_{w1},\;\;\; T_x(a,y,z)=0,\\
      \mbox{Conditions on Y faces}:       T(x,0,z)= T_{w2},\;\;\; T_y(x,b,z)=0,\\
      \mbox{Conditions on Z faces}:\;\;\;T(x,y,0)= 0,\;\;\; T(x,y,c)=0.
$$
This is not a Dirichlet or Neumann problem; this is a mixed problem. So you're questioning whether or not you have uniqueness of such a solution, provided it exists, and that seems like a reasonable thing to question. The difference $Q=T_1-T_2$ of two such solutions will satisify the same mixed condition, but where $Q_{w1}=Q_{w2}=0$. Then
$$
      0=\int_{\Omega} Q(\nabla^2Q) dV=\int_{\Omega}\nabla\cdot(Q\nabla Q)-\nabla Q\cdot\nabla QdV \\
   = \int_{\partial\Omega}Q\frac{\partial Q}{\partial n}dS-\int_{\Omega}|\nabla Q|^2dV.
$$
Because either $Q$ or $\frac{\partial Q}{\partial n}$ is $0$ on a given outer face of the cubic domain, then $\int_{\Omega}|\nabla Q|^2 dS=0$, which forces $Q$ to be constant throughout the domain. And that constant must be $0$ because of the boundary conditions. So $Q\equiv 0$, which means $T_1=T_2$ and this establishes uniqueness of the solution $T$. With uniqueness, you can focus on existence. Existence can be exhibited by solving for $T$ using separation of variables, which you have already considered.
