Additional sobolev regularity from laplace

Question

Given a bounded Lipschitz domain $U\subset \mathbb{R}^3$ and a function $u\in W^{2,2}(U)$ with $\Delta u\in L^p(U)$ for some $p>2$, does $u\in W^{1,p}(U)$ hold?

Answer 1

Yes, at least for $\,p=6\,$. This is according to the Sobolev Embedding Theorem which states

\begin{align} \begin{cases} k>l \\ 1\le q<p<\infty \\ \big(k-l\big)\,q<n \\ \dfrac{1}{p} = \dfrac{1}{q} - \dfrac{k-l}{n} \end{cases} \implies W^{k,q}\big(\mathbb R^n\big) \subseteq W^{l,p}\big(\mathbb R^n\big) \end{align}

In your case $\, n=3,\, k=2,\, l = 1,\, q = 2,\,$ and thus $\,p=6$.