I have some issues with the following problem.
Let ${\Omega \subset \Bbb{R}^n}$ be a bounded open set with smooth boundary $\Gamma$, and consider the following problem
$$(\text{R})\displaystyle \begin{cases} -\Delta u(x)+c(x)u(x) =f(x), &\text{ }x\in\Omega \\ \dfrac{\partial u}{\partial \nu}(x)+\alpha u(x) =g(x), & x\in\Gamma \end{cases}, $$
where ${\alpha>0}$ is a constant. This is a problem with Robin boundary conditions.
If $f\in L^2(\Omega), g\in L^2(\Gamma)$ and $c\in L^\infty(\Omega)$ satisfy $c(x)\geq c_0>0$,
I must prove that (R) is a well posed problem which has a unique weak solution.
As usual, one wants to use the Lax-Milgram theorem, so I must seek for a bilinear form $B\colon V\times V\to \mathbb R$ which is continuous and coercive. I don't have issues with proving that such $B$ is continous and coercive, but I get confused at the moment of proposing it (as well as chosing the appropiate space, I think that $V$ must be $H^1(\Omega )$). Is there some compatibility condition ?
I'm thinking in proposing $B$ as
$$B(u, v):=\int_\Omega \nabla u\cdot \nabla v+\int_\Omega cuv+\alpha\int_{\Gamma }uv,$$
and so we want an unique $u\in V$ such that for all $v\in V$
$$B(u, v)=\int_\Omega fv+\int_{\Gamma }gv.$$
Is this right? Can you help me in reasoning the formulation of this problem?
What is the appropiate choice of $V$?
Thanks in advance, this is my first time solving these kind of problems
Best Answer
Indeed it follows from trace theorem that, the injection $$H^1(\Omega)\to H^{1/2}(\Gamma)\to L^{2}(\Gamma)$$ are continuous therefore, there is a constant k such that for all $u\in H^1(\Omega)$
$$\| u\|_{ L^{2}(\Gamma)}\le k\|u\|_{H^1(\Omega)}$$
from this, you easily get the continuity of the bilinear form $B$ on $H^1(\Omega)\times H^1(\Omega)$ whereas the coercivity is straightforward since, $c(x)>c_0$. By the same token you get the continuity on $H^1(\Omega)$ of the linear form
$$v\mapsto \int_\Omega fv+\int_{\Gamma }gv.$$
therefore the existence and uniqueness for the weak solution follow from Lax-Milgramm