Let $n\in\mathbb N$, let $\Omega\subset \mathbb R^n$ be a bounded Lipschitz domain, let $p>1$, and let $\Gamma\subset \partial\Omega$ be a relatively measurable boundary part with positive $\mathcal H^{n-1}$-measure. Let $\mathcal A\subset W^{1,p}(\Omega;\mathbb R^n)$ be nonempty and suppose that all elements of $\mathcal A$ have the same trace on $\Gamma$. Assume the corresponding trace-Poincare estimate holds on $\mathcal A$: there exist constants $K>0$ and $D\ge 0$ such that
Assume $\inf_{v\in\mathcal A} I[v]<\infty$. Then every minimizing sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A$ with $I[u_k]<\infty$ for all sufficiently large $k$ is bounded in $W^{1,p}(\Omega;\mathbb R^n)$.