ETD

• calculus of variations
• elliptic regularity
• linear PDOs
• quasiconvexity

We investigate the relaxation, in the $L^1$ topology, of the functional
$$
{\mathcal{F}}[u]=\begin{cases}\int_{\Omega} f(\mathscr{B} u(x))\, dx & \text{ if }u\in W^{1,1}_g(\Omega,\mathbb{R}^m);\\
+\infty &\text{otherwise.}\end{cases}
$$
Where $\mathscr{B}$ is a first order, homogeneous, vector-valued, elliptic and canceling differential operator, $g$ is some boundary datum and $f$ is a strongly $\mathscr{B}$-quasiconvex lagrangian that admits a strong recession function. We provide existence of minimizer $u$ and derive an explicit local minimality condition. Then we show that this condition entails an $\varepsilon$-regularity result for $\mathscr{B} u$, which \textit{a priori} in just a finite measure. The same result was previously known only in the cases $\mathscr{B}=\nabla$ and $\mathscr{B}=\nabla+\nabla^T$. We also give a brief overview of the space of maps with bounded $\mathscr{B}$-variation.