Let $\Omega$ be a "nice (say $C^1$)" open domain in $\mathbb{R}^n$. Consider the equation $$ \Delta u = 0 \text{ in } \Omega$$ $$\alpha u + \frac{du}{dn} = f(x) \text{ on the boundary} .$$
I noticed that the energy method fails when $\alpha <0$ and $|\alpha|$ large, since in this case, the energy is not bounded below and thus doesn't yield a minimizer.
I am thinking of alternative ways to show existence. If $\Omega$ is bounded, I could try using Fredholm alternative, since $-\Delta$ has a compact inverse.
Another alternative is to try using Greens functions, without knowing what it explicitly is.
However, I can only find reference on the Dirichlet boundary problem for the above approaches. I wonder how could we apply these methods to this Robin boundary condition?
I am also looking for other alternatives of dealing with this problem.
Thank you very much.
Let $G(x,x')$ be the Green's function of the laplacian operator satisfying homogeneous Robin boundary conditions: $$ \Delta G(x,x')=-\delta(x-x')\,\,\text{for}\,\,x\in\Omega, \tag{1.a} $$ $$ \alpha G(x,x')+\frac{\partial}{\partial n}G(x,x')=0\,\,\text{for}\,\,x\in \partial\Omega, \tag{1.b} $$ and let $u$ be the function defined in the question. Applying Green's identity to $u$ and $G$, one obtains $$ \int_{\Omega}(u\Delta G-G\Delta u)\,dV =\int_{\partial\Omega}(u\partial_nG-G\partial_n u)\,dS. \tag{2} $$ The LHS of $(2)$ yields $$ -\int_{\Omega}u(x)\delta(x-x')\,dV=-u(x'), \tag{3} $$ whereas the RHS gives $$ \int_{\partial\Omega}(-u\alpha G-G\partial_n u)\,dS=-\int_{\partial\Omega}Gf\,dS, \tag{4} $$ hence $$ u(x')=\int_{\partial\Omega}G(x,x')f(x)\,dS. \tag{5} $$