I am reading this paper and would like to check how they derive the inequality in (5) on page 4.
Denotes $S^d$ the set of symmetric second order tensors on $\mathbb{R}^d$. Define the permeability tensor $$k(x)=\exp(u(x)), u\in S^d.$$ Let $D\subset\mathbb{R}^d$ be an open bounded set with regular boundary. Let $X=L^\infty(D;S^d)$.
Consider the elliptic equation \begin{equation} \begin{aligned} \nabla \cdot v & =f, \quad x \in D \\ p & =0, \quad x \in \partial D \\ v & =-k \nabla p\end{aligned} \label{eq1} \end{equation} where $f\in H^{-1}(D)$ is known. The unique solution $p\in H^1_0(D)$ satisfies
$$ \|\nabla p\|_{L^{2}} \leq c_{1} \exp \left(\|u\|_{X}\right)\|f\|_{H^{-1}} $$
Thus, we may define the map $G:X\rightarrow H_0^1(D)$ by $G(u)=p$.
The paper claimed the following: Using above inequality one may show that $G$ is Lipschitz. Indeed if $p_i$ denotes the solution to the PDE with $k_i(x)=\exp(u_i)$, then we have
$$ \left\|\nabla p_{1}-\nabla p_{2}\right\|_{L^{2}} \leq\left(c_{1}\right)^{2}\left\|u_{1}-u_{2}\right\|_{X} \exp \left(2\left(\left\|u_{1}\right\|_{X}+\left\|u_{2}\right\|_{X}\right)\right)\|f\|_{H^{-1}}. $$
I have no idea how they derive this inequality. Any helps are greatly appreciated.