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I'm trying to show the existence and uniqueness of weak solutions to the following boundary value problem: \begin{align} -\nabla \cdot ( k \nabla u) &= f \quad \text{in } \Omega \subset \mathbb{R}^n\\ -k \nabla u \cdot \boldsymbol{n} - c u &= g \quad \text{on } \partial \Omega. \end{align} The associated variational formulation is to find $u \in H^1(\Omega)$ such that $$ a(u,w) = \ell(w) \quad \forall w \in H^1(\Omega), $$ where \begin{align} a(u,w) &= \int_{\Omega} k \nabla u \cdot \nabla w ~\mathrm{d} \boldsymbol{x} + \int_{\partial \Omega} c u w ~\mathrm{d} \boldsymbol{l}\\ \ell(w) &= \int_{\Omega} f w ~\mathrm{d} \boldsymbol{x} - \int_{\partial \Omega} g w ~\mathrm{d} \boldsymbol{l}. \end{align} I'm assuming $f\in L^2(\Omega)$, that $g$ and $c$ are in $L^2(\partial \Omega)$, and there exists constants $k_{min}, k_{max} \in \mathbb{R}$ such that $0 < k_{min} < k < k_{max}$ for all $\boldsymbol{x} \in \Omega$. Additionally, it is assumed that $c$ is strictly positive.

I want to use the Lax-Milgrim Theorem to establish the existence and uniqueness of the weak solution. I can show that $a(\cdot, \cdot)$ and $\ell(\cdot)$ are bounded above in $H^1(\Omega)$ using the trace inequality. However, I am having some trouble establishing coercivity of $a(\cdot, \cdot)$.

I did find this really nice proof by contradiction to a similar question: Variational formulation of Robin boundary value problem for Poisson equation in finite element methods. However, I can use a constructive proof to show coercivity in the $1$-dimensional setting. It makes me think there might be a way to directly establish coercivity in the the $n$-dimensional setting. I was wondering if anyone knows a clever way to establish coercivity without the compactness argument?

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  • $\begingroup$ Does $c$ have a sign? $\endgroup$ – miles Feb 13 '16 at 15:40
  • $\begingroup$ $c$ is strictly positive. $\endgroup$ – Steve Feb 14 '16 at 22:40
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Since $c$ is strictly positive, there is a measurable portion $E\subset \partial\Omega$ such that $c>c_{min}>0$ on $E$. Therefore, coercivity follows from the Poincare type inequality $\|v\|_2^2 \leq C(\Omega,E)(\|\nabla v \|_2^2 + \int_E |v|^2~d\sigma)$ for all $v\in H^1(\Omega)$.

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  • $\begingroup$ Thanks for the hint. This inequality definitely gives the desired coercivity. I am wondering where you know it from? or how I can prove it? $\endgroup$ – Steve Jan 12 '17 at 22:48
  • $\begingroup$ Thanks for the help MathManiac. For completeness, a common name for this inequality is the Poincare-Friedrichs inequality. A nice proof of its validity can be found in the text "The mathematical theory of finite element methods" by Brenner and Scott. $\endgroup$ – Steve Jan 31 '18 at 16:58

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