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{ \large \textbf{\textsc{Math 8385 -- Homework 2b (Due November 22)}}}
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\begin{enumerate}
\item Find $L=L(x,z,p)$ so that the PDE
\[-\Delta u + \nabla \varphi \cdot \nabla u = f \ \ \text{ in }U\]
is the Euler-Lagrange equation associated with the functional $I(u) = \int_U L(x,u,\nabla u)\, dx$. [Hint: Look for a Lagrangian with an exponential term.]
\item Let $u,v\in H^1_0(U)$ be minimizers of the Dirichlet energy
\[I(w) = \int_U |\nabla w|^2,\, dx.\]
Suppose also that $u,v>0$ within $U$. Use the hints below to give a new proof that $u=v$ almost everywhere in $U$.
[Hint: Define $w:=\left( \frac{u^2+v^2}{2} \right)^{1/2}$, $s:=\frac{u^2}{u^2+v^2}$ and $\eta = \frac{u^2 + v^2}{2}$. Show that 
\[|\nabla w|^2 = \eta\left| s \frac{\nabla u}{u} + (1-s)\frac{\nabla v}{v}\right|^2.\]
Deduce that
\[|\nabla w|^2 \leq \eta\left( s \left|\frac{\nabla u}{u}\right|^2 + (1-s)\left|\frac{\nabla v}{v}\right|^2\right) = \frac{1}{2}|\nabla u|^2 + \frac{1}{2}|\nabla v|^2,\]
and therefore $\frac{\nabla u}{u} = \frac{\nabla v}{v}$ a.e.~in $U$.]
\item (Pointwise gradient constraint)
\begin{enumerate}
\item Show there exists a unique minimizer $u\in \A$ of 
\[I(w):=\int_U \frac{1}{2}|\nabla w|^2 - fw \, dx\]
where
\[\A = \{w \in H^1_0(U) \, | \, |\nabla w|\leq 1 \text{a.e.}\}.\]
\item Prove that
\[\int_U \nabla u \cdot \nabla (w-u)\, dx \geq \int_U f(w-u)\, dx\]
for all $w\in \A$.
\end{enumerate}
\end{enumerate}

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