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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

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It suffices to use the space $V=H^1$ along with the trace theorem

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

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  • $\begingroup$ +1 Should be $v \mapsto \int_{\Omega} fv + \int_{\Gamma} gv$. $\endgroup$ – fourierwho Jan 15 '18 at 22:39
  • $\begingroup$ @fourierwho thanks for carefull reading $\endgroup$ – Guy Fsone Jan 15 '18 at 22:40
  • $\begingroup$ Thanks, let me check it our carefully! $\endgroup$ – EternalBlood Jan 15 '18 at 23:06

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