Apply Banach's Fixed Point Theorem to a nonlinear boundary-value problem I am attempting Exercise 5 in Chapter 9 of PDE Evans, 2nd edition:


  
*Consider the nonlinear boundary-value problem $$\begin{cases}-\Delta u + b(Du)=f & \text{in }U \\ \qquad \qquad \quad \, \, \, u=0 & \text{on }\partial U. \end{cases}$$ Use Banach's Fixed Point Theorem to show there exists a unique weak solution $u \in H^2(U) \cap H_0^1(U)$ provided $b : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz continuous, with $\text{Lip}(b)$ small enough.
  

Here are my thoughts so far on this problem:
I am tempted to follow Theorem 2 on page 536 in the textbook, which says (and Theorem 2 in the textbook refers to $(2)$ on page 535 which I wrote down as well):

THEOREM 2 (Existence). There exists a unique weak solution of the initial-boundary value problem for the reaction-diffusion system $$\begin{cases}\mathbf{u}_t-\Delta\mathbf{u}=\mathbf{f}(\mathbf{u}) &\text{in }U_T \\ \qquad \quad \mathbf{u}=\mathbf{0} & \text{on }\partial U \times [0,T] \\\qquad \quad \mathbf{u}=\mathbf{g} & \text{on }\partial U \times \{t=0\}.\end{cases}\tag{2}$$ Here $\mathbf{u}=(u^1,\ldots,u^m)$, $\mathbf{g}=(g^1,\ldots,g^m)$, and as usual $U_T = U\times (0,T]$, where $U \subset \mathbb{R}^n$ is open and bounded, with smooth boundary. The time $T > 0$ is fixed. We assume that the initial function $\mathbf{g}$ belongs to $H_0^1(U;\mathbb{R}^m)$.

It seems that Exercise 5 calls for the reader to construct a proof similar to that of Theorem 2. (Please comment if you need me to reproduce the latter proof here.) Anyway, the proof of Theorem 2 applies Banach's Fixed Point Theorem, just like what Exercise 5 asks to. 
Somewhere in the beginning of the proof of Theorem 2, one sentence reads "Given a function $\mathbf{u} \in X$, set $\mathbf{h}(t):=\mathbf{f}(\mathbf{u}(t))$ ($0 \le t \le T)$." For Exercise 5, however, $f$ does not depend on $u$, but $b(Du)$ does. 
 A: I think you have the right notion, and in Exercise 5, things are 'nicer' for us in that there is no time variable to deal with. 
In the following, I am assuming that the given $f \in L^2(U)$.


*

*Given a function $u \in H^1_0(U)$, set $g := f - b(Du)$, and consider the linear PDE


\begin{align}
-\Delta w &= g \quad \text{in } U\\
w &= 0 \quad \text{on } \partial U.
\end{align}


*By appealing to existence and regularity results (see Chapter 6 in Evans), we know that there exists a unique weak solution $w \in H^1_0(U)\cap H^2(U)$ to the above linear PDE.

*Define an operator $A: H^1_0(U) \to H^1_0(U)$ by
\begin{equation}
A[u] = w
\end{equation}
Let $u,\tilde{u} \in H^1_0(U)$, with $A[u] = w, A[\tilde{u}] = \tilde{w}$. To show: $A$ is a strict contraction. (This is a good place to try yourself first without reading further). Then

\begin{align} \left\|A[u] - A[\tilde{u}]\right\|_{H^1_0(U)} &= \|w-\tilde{w}\|_{H^1_0(U)}\\ &\leq C\|D(w - \tilde{w})\|_{L^2(U)}\quad(*).\end{align}
In the above, $(*)$ follows by a Sobolev estimate. Now
\begin{align}
\|D(w - \tilde{w})\|_{L^2(U)}^2 &= \int_{U} D(w - \tilde{w})\cdot D(w - \tilde{w}) dx\\
&= -\int_{U}[\Delta(w - \tilde{w})] (w - \tilde{w}) dx\\
&= \int_{U} (b(D\tilde{u}) - b(Du))(w - \tilde{w}) dx\\
&\leq \|b(D\tilde{u}) - b(Du)\|_{L^2(U)}\|w - \tilde{w}\|_{L^2(U)}\quad (**)\\
&\leq C\|b(D\tilde{u}) - b(Du)\|_{L^2(U)}\|D(w - \tilde{w})\|_{L^2(U)} \quad (***)
\end{align}
In $(**)$, I have used the fact that $b$ is Lipschitz so $b(Du), b(D\tilde{u}) \in L^2(U)$. In $(***)$, I have used a Sobolev estimate. 
 Our conclusion, having reached $(***)$, is that
\begin{equation}
\|D(w - \tilde{w})\|_{L^2(U)} \leq C\|b(D\tilde{u}) - b(Du)\|_{L^2(U)}.
\end{equation}
Hence returning to $(*)$, we see that
\begin{align}
\left\|A[u] - A[\tilde{u}]\right\|_{H^1_0(U)} &\leq C\|b(D\tilde{u}) - b(Du)\|_{L^2(U)}\\
&\leq C\text{Lip}(b)\|D\tilde{u} - Du\|_{L^2(U)}\\
&\leq C\text{Lip}(b)\|u - \tilde{u}\|_{H^1_0(U)}.
\end{align}
Provided $\text{Lip}(b)$ is small enough, $A$ is a strict contraction (here the constant $C$ is from the Sobolev estimates, so only depends on things like the dimension $n$, $U$, etc.). 
Thus by the Banach fixed point theorem, we deduce the existence of a unique fixed point $u_* \in H^1_0(U)$, such that $A[u_*] = u_*$. The regularity theory for the PDE in 1. gives $u_* \in H^2(U)\cap H^1_0(U)$.
