Let $U\subset\mathbb R^n$ be bounded, open, and Lipschitz, let $1<p<\infty$, and let $f:\mathbb R^n\to[0,\infty)$ be continuous. Suppose there exist constants $c>0$ and $C>0$ such that, for every $\xi\in\mathbb R^n$,
Let $f^{**}:\mathbb R^n\to\mathbb R$ denote the convex lower semicontinuous envelope of $f$, equivalently the Fenchel biconjugate of $f$. For $u\in W^{1,p}(U)$, define the weak $W^{1,p}$ sequential relaxation of $F$ by
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\begin{align*}
\overline F[u]=\inf\left\{\liminf_{k\to\infty}F[u_k]:(u_k)_{k=1}^{\infty}\subset W^{1,p}(U),\ u_k\rightharpoonup u\text{ in }W^{1,p}(U)\right\}.
\end{align*}