Let $U\subset\mathbb R^n$ be a bounded Lipschitz domain, let $1<p<\infty$, and let $g\in W^{1,p}(U)$. Let $\mathbb N:=\{1,2,3,\dots\}$. Define the affine Dirichlet class
\begin{align*}
f:U\times\mathbb R\times\mathbb R^n\to\mathbb R
\end{align*}
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be Borel measurable. Suppose that there exist $\alpha>0$ and $a\in L^1(U)$ such that, for $\mathcal L^n$-a.e. $x\in U$ and every $(s,\xi)\in\mathbb R\times\mathbb R^n$,
is bounded in $W^{1,p}(U)$. Equivalently, for every sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A_g$, if $\|u_k\|_{W^{1,p}(U)}\to\infty$, then $I[u_k]\to+\infty$.