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{\bf HW \#4, 269C} Due during the week of May 27 - May 31st (note May 27 is a
holiday).
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{\bf [1]} Recall the ``coercive'' inhomogeneous Neumann problem:
$$(D) \ \ \Big\{-\triangle u +u=f \mbox{ in }\Omega, \ \ \frac{\partial u}{\partial \vec{n}}=g \mbox{ on }\partial \Omega\Big\}.$$
We know that the corresponding weak variational problem (with $V=H^1(\Omega)$) is: find $u\in H^1(\Omega)$ such that
$$(V) \ \ \int_{\Omega}(\nabla u\cdot \nabla v +uv)dx =\int_{\Omega}fv dx +\int_{\partial \Omega}gv ds,$$
for all $v\in H^1(\Omega)$. We also know that $(D)\Rightarrow (V)$.
Assume now that $u,f,g$ are sufficiently smooth. Show that, if $u$ is a smooth solution of $(V)$ (for example $u\in C^2(\overline{\Omega})$, then $u$ satisfies $(D)$; in other words, $(V)\Rightarrow (D)$.
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{\bf [2]} Let $K$ be a tetrahedron with vertices $a^i$, $i=1,...,4$, and let
$a^{ij}$ denote the midpoint on the straight line $a^ia^j$, $i