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


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

  • $\begingroup$ +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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.