# Green Solution to Laplace Equation with Robin Boundary Conditions

Let's say that I know a solution for the Laplace equation in the whole plane: $$\nabla^2u(\mathbf{x})=0\quad \mathbf{x}\in\mathbb{R}^2$$ And I need a solution for the laplace equation in the semiplane $$x>0$$. I know that if there are Dirichlet homogeneous conditions on the boundary: $$u(\mathbf{x})=0\quad\text{on }\mathbf{x}=(0,y)$$ I can construct a solution $$u_\mathrm{D}$$ in that semiplane satisfiying the condition by setting:

$$u_\mathrm{D}(x)=u(x)-u(-x) \quad \forall\mathbf{x}=(x,y);x>0$$

I know I can do the same for homogeneous Neumann conditions: $$u'(\mathbf{x})=0\quad \text{on }\mathbf{x}=(0,y)$$ then the function $$u_\mathrm{N}=u(x)+u(-x)$$ will also solve the problem on the on the semiplane with the Neumann boundary condition.

My question is: Can I do the same for homogeneous Robin boundary conditions i.e.: $$u(\mathbf{x})+Au'(\mathbf{x})=0 \quad \text{on }\mathbf{x}=(0,y)$$

and if is possible, which would be the formula?

Thank you