Consider nonlinear transmission systems,
$$
\hbox{div} (A(\nabla u)\chi_{D^c} + B(\nabla u)\chi_D) = 0,\tag{1}
$$
where $u:B_1\subset {\Bbb R}^n \to {\Bbb R}^m$, and
both $A$ and $B$ are strongly elliptic, nonlinear operators.
It is well-known that nonlinear systems do not have Lipschitz solutions, in general, even if $A = B$ and the dependence on $\nabla u$ is smooth. This remains true even for minimisers of a nonlinear functional, see [2]. It is also known that the boundary regularity fails for nonlinear systems, even if the boundary data is smooth, see e.g., [3].
However, if we assume that $u$ is Lipschitz up to $\partial D$, then the Lipschitz regularity may have some chances of propagating to the other side, in some small neighborhood, depending on the geometry of $\partial D$. This is because the governing system yields a matching condition of the normal derivatives of $u$ on $\partial D$: formally,
$$
A_i^\alpha (\nabla u|_{D^c})\nu_\alpha + B_i^\alpha (\nabla u|_D)\nu_\alpha = 0,
$$
whenever the outward normal $\nu$ is defined on $\partial D$. This may leave us in a better situation than a Dirichlet boundary problem, since for the latter problem the normal derivatives of the solution does not need to match those of the boundary data. ...
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