Let's say that I know a fundamental solution for the Laplace equation in the whole plane: $$\nabla^2u(\mathbf{x})=\delta\quad \text{in the sense of distributions}$$ 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: $$u(\mathbf{x})=0\quad\mathbf{x}=(0,y)$$ 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: $$u'(\mathbf{x})=0\quad \mathbf{x}=(0,y)$$ then the function $$u_\mathrm{N}=u(x)+u(-x) $$ will also solve the problem on the on the semi plane with the Neumann boundary condition.

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

and if it is possible, who would be de expression?

Thank you


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