Let $1<p<\infty$ and $\lambda>0$. When $\Omega$ is a bounded $C^1$ or a bounded Lipschitz domain with small Lipschitz constant in $\mathbb{R}^d$, then for every $f\in L_p(\Omega)$ and $\mathbf{F}\in L_{p}(\Omega;\mathbb{R}^d)$, there exists a unique weak solution $u\in \mathring{W}^1_p(\Omega)$ of $$ -\triangle u +\lambda u = f+\mathrm{div}\, \mathbf{F}\quad \text{in } \Omega,\quad u=0\quad \text{on } \partial\Omega.$$ Moreover, we have $$ \Vert \nabla u \Vert_{L_p(\Omega)}+\lambda^{1/2} \Vert u \Vert_{L_p(\Omega)}\leq C\left(\lambda^{-1/2} \Vert f \Vert_{L_p(\Omega)}+\Vert \mathbf{F}\Vert_{L_p(\Omega)} \right) $$ for some constant $C=C(n,p,\Omega)>0$.
I tried to find a reference the same $W_{p}^1$-estimate when $\Omega$ is an arbitrary bounded Lipschitz domain in $\mathbb{R}^d$. It seems to be standard result, but I failed to find an exact reference for this. When $p=2$, the estimate holds obviously. I want to find such estimate for $p\neq 2$.
The question is motivated by the classical result of Jerison-Kenig (1995). In the case of $d=2$, Jerison-Kenig proved the following result:
Theorem. Let $\Omega$ be bounded Lipschitz domain in $\mathbb{R}^2$. Then there exists $\varepsilon$, $0<\varepsilon\leq 1/2$ depending only on the Lipschitz constant of $\Omega$ such that the following hold for every $p_0<p<p_0'$, where $1/p_0=1/2+\varepsilon$: for every $f\in L_{p}(\Omega)$ and $\mathbf{F} \in L_{p}(\Omega;\mathbb{R}^2)$, there exists a unique $u\in \mathring{W}^{1}_{p}(\Omega)$ satisfying $$ -\triangle u = f+\mathrm{div}\, \mathbf{F}\quad \text{in } \Omega.$$ Moreover, there exists a constant $C=C(p,\Omega)>0$ such that $$ \Vert \nabla u \Vert_{L_p(\Omega)}+ \Vert u \Vert_{L_p(\Omega)}\leq C\left( \Vert f \Vert_{L_p(\Omega)}+\Vert \mathbf{F}\Vert_{L_p(\Omega)} \right) $$ for some constant $C=C(p,\Omega)>0$. If $\Omega$ is a bounded $C^1$-domain in $\mathbb{R}^2$, then $p_0$ can be taken to be $1$.
Thank you for time.
