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?
