The equation: \begin{equation} -\Delta u=f\ \ \mathrm{in}\ \Omega,\ f\in L^2(\Omega)\\ \frac{\partial u}{\partial n}+u=g\ \ \mathrm{on}\ \partial\Omega,\ g\in L^2(\Omega) \end{equation}

Now to the weak formulation: $$\int_\Omega\nabla u\nabla v+\int_{\partial\Omega}uv=\int_\Omega fv+\int_{\partial\Omega}gv\ \ \forall v\in H^1(\Omega)$$ So I need to prove following:

  • functional $F(v)=\int_\Omega fv+\int_{\partial\Omega}gv$ is bounded - trivial
  • bilinear form $a(u,v)=\int_\Omega\nabla u\nabla v+\int_{\partial\Omega}uv$ is bounded (this I believe follows directly from the trace theorem) and coercive

But how do I prove it is coercive? i.e.: $$a(v,v)\geq\alpha\|v\|_1^2$$

  • 1
    $\begingroup$ The coercivity of $a$ is proved here. $\endgroup$ – Pedro Jan 2 '17 at 23:44

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