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let $\Omega$ an open bounded connexe and regular on $\mathbb{R}^n$, and let the problem $$ \begin{aligned} -&\Delta u = f(x),&& x\in \Omega\\ &\nabla u \cdot n + u=g(x),&& x\in \Gamma \end{aligned} $$ where $f\in L^2$ and $g\in L^2$ and $n$ is the unitary vector on $\Gamma.$ Prove that this problem admits a unique solution in the adequate Hilbert space $V.$

I take $V=H^1(\Omega)$, then the variational problem is $$a(u,v)=l(v),\quad \forall v\in V$$ with $$a(u,v)=\int_{\Omega}\nabla u \cdot \nabla v dx +\int_{\Gamma} uvdx\quad\text{and}\quad l(v)=\int_{\Omega} (f(x)-g(x))v dx$$ How we prove the continuity and the coercivity of $a$? (The difficulty is the integral on $\Gamma$ I don't know how we can estimate it). Thanks for your help.

user1729
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jijii
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  • I think that it has already been answered here: http://math.stackexchange.com/questions/361423/variational-formulation-of-robin-boundary-value-problem-for-poisson-equation-in – Dr_Sam Aug 16 '13 at 11:51
  • why the continuity of $a$ is natural? How we can prouve it and how we choose the adequate space $V$? I wan't to understand how we choose $V$ and how we prouve the continuity? – jijii Aug 16 '13 at 11:57
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    Note that your right hand side $l(v)$ is wrong: the function $g(x)$ is defined on the boundary but you integrate over the whole domain: you should split that integral in two, as you do in the definition of $a(u,v)$. – Miguel Aug 16 '13 at 12:10
  • The continuity of the first term of $a(.,.)$ is natural: $\int_{\Omega} \nabla u \cdot \nabla v \leq |u|{H^1} |v|{H^1} \leq ||u||{H^1} ||v||{H^1}$ where the first inequality is Cauchy-Schwarz, the second holds from the definition of the $H^1$ norm. – Dr_Sam Aug 16 '13 at 12:15
  • @pluton It is best not to remove "Thank you" comments from posts. See here. – user1729 Aug 16 '13 at 13:27

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In order to prove continuity recall that the trace operator

$$ \gamma :H^{1} ( \Omega ) \rightarrow H^{1/2} ( \Gamma ) $$

is continuous:

$$ \| \gamma ( u ) \|_{H^{1/2} ( \Gamma )} \leqslant c \| u \|_{H^{1} ( \Omega ) } . $$

In order to prove coercivity what you need is a different version of the Friedrichs-Poincaré inequality which includes the integral over the boundary. Under sufficient regularity assumptions on the domain, I think the proof should be the same (i.e. consider a bounding box, integrate in one dimension using the absolute continuity of functions in $H^1(\mathbb{R})$, then square, integrate in the other direction and use Cauchy-Schwarz (don't have the precise steps in mind, sorry). It should be something like

$$ \int_{\Omega} | u |^{2} \leqslant C \left( \int_{\Omega} | \nabla u |^{2} + \int_{\Gamma} | u |^{2} \right). $$

Miguel
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