So far I understood that in one dimension Laplace equation gives out a simple equation $V(x)=mx+b$ which is because if you differentiate this equation two times you will get 0 which is Laplace identity.

After this it says(Griffiths Electrodynamics) thus we need two boundary conditions What does this mean  ?

In my understanding $V1$ and $V2$ are two solution two the Laplace equation for potential.And thus$\bigtriangledown^2 V=0$ is satisfied by thee.Then it takes potential at a third point which is difference of $V1$ and $V2$ and thus taking Laplacian of $V3=V1-V2$ gives us zero and thus $V3$ also satisfies the Laplacian and after than whatever he has written I am sorry I don't understand  ?

Whats the sole purpose of Uniqueness Theorem and Boundary conditions where and how do we use it  ?

I went through Rochester and Ph Texas online notes got nothing

Then it provides this proof

He says there may be islands inside what does that mean  ?

So far I understood that in one dimension Laplace equation gives out a simple equation $V(x)=mx+b$ which is because if you differentiate this equation two times you will get 0 which is Laplace identity.

After this it says(Griffiths Electrodynamics) thus we need two boundary conditions What does this mean?

In my understanding, $V_1$ and $V_2$ are two solutions two the Laplace equation for potential. And thus$\nabla^2 V=0$ is satisfied by thee. Then it takes potential at a third point which is the difference of $V_1$ and $V_2$ and thus taking Laplacian of $V_3=V_1-V_2$ gives us zero and thus $V_3$ also satisfies the Laplacian and after than whatever he has written I am sorry I don't understand?

Whats the sole purpose of Uniqueness Theorem and Boundary conditions, where and how do we use it?

I went through Rochester and Ph Texas online notes got nothing

Then it provides this proof

He says there may be islands inside what does that mean?