A counterexample to regular boundary points for minimizers of variational integrals under subquadratic growth

Question

Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\rightarrow\mathbb{R}$ such that \begin{equation} \lambda|z|^p\leq f(x, y, z)\leq L(1+|z|^2)^{\frac{p}{2}} \end{equation}for some $\lambda,L>0$ and for all $(x, y, z)\in\overline{\Omega}\times\mathbb{R}^N\times\mathbb{R}^{Nn}$. Now consider the integral: \begin{equation} I[u]=\int_{\Omega}f(x, u, Du) \ \mathrm{d}x\quad (u\in W^{1, p}_0(\Omega;\mathbb{R}^N)). \end{equation}Does there exist an example of an $I$ and a (local) minimiser $u$ of $I$ such that we can split the boundary $\partial\Omega$ into a set of zero surface measure and a set of non-zero surface measure (via usual Hausdorff measure) with $Du$ being irregular (not continuous) on the set of positive surface measure?