poisson's equation with robin's boundary, boundary value problem

Question

Consider the Poisson’s equation with Robin’s boundary conditions as follows \begin{array}{ll} −\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} \end{array} where $U$ is an open bounded subset of $\mathbb R^n$ and $\nu$ is the outward unit normal on $U$. Show that this boundary-value problem has at most one solution $u \in C^2(U) \cap C(\overline U)$.

I think I have to use the energy method but I'm really not sure how to go about this. Any help would be greatly appreciated.

Answer 1

In this answer of @ShuhaoCao, you can learn how to formulate this problem in a variation setting.

In this link, you can learn how to regularize it.

If you want to learn more about it, I suggest you to read section 4.11 of of this book of Maz'ja. There you will discover that a interesting space to work with, is the completion of $$H^1(U)\cap C^{\infty}(U)\cap C(\overline{U})$$

with respect to the norm $$\|u\|_{1,2}+\|u\|_{L^{2}(\partial U)}$$

Then, if $W$ is such completion, you can easily see that the functional $I:W\to\mathbb{R}$ defined by $$I(u)=\frac{1}{2}\int_U |\nabla u|^2+\frac{1}{2}\int_{\partial U} u^2-\int_U fu$$

is strictly convex, coercive and weakly sequentially lower semicontinous, w.s.l.s.c. for short. Indeed the first item is evident. The coerciveness will follow because $W$ is continuously embedded in $L^2(U)$. The third item (w.s.l.s.c.) will follow because $W$ is compactly embedded in $L^2(U)$. Therefore, $I$ has a unique minimum in $W$ which is a solution of your problem.