Lipschitz regularity in vectorial nonlinear transmission problems

Problem's Description

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.

For instance, let $\partial D$ be a hyperplane. Then from the assumptions that $u$ is Lipschitz up to $\partial D$ from $D$, and that the equation yields a matching condition of the normal derivative of $u$ on $\partial D$, it is reasonable to expect the propagation of the Lipschitz regularity to the other side.

On the other hand, if $\partial D$ has a cusp so that $D$ does not have positive density at a point on $\partial D$, then the nice information from $D$ may lose its effect, and the nonlinearity of the operators in Question 2 does not supplement the loss of information. More precisely, in the blowup regime the limit solution of $u$ will solve $\hbox{div} A(\nabla u_0) = 0$ everywhere (recall that $A$ is the governing operator in the region $D^c$). Unlike the case of linear systems, the blowup limit $u_0$ may fail to be Lipschitz, so this strategy cannot give any regularity improvement for the original solution $u$.

Question: Suppose the solution to equation (1) is uniformly Lipschitz in $D \cap B_1$. Does it follow that $u$ is Lipschitz in $B_{1/2}$? Prove or give a counterexample to this.

1. ArticleIs an originLipschitz regularity in vectorial linear transmission problems

   • Alessio Figalli,
   • sunghan kim,
   • Henrik Shahgholian

   Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 221, Paper No. 112911, 2022fulltext

2. Article Non-Lipschitz minimizers of smooth uniformly convex functionals

   • Vladim\'\ir Sverák,
   • Xiaodong Yan

   Proceedings of the National Academy of Sciences of the United States of America 99, 15269-15276, 2002fulltext

3. Article A counter-example to the boundary regularity of solutions to elliptic quasilinear systems

   • Mariano Giaquinta

   Manuscripta Mathematica 24, 217-220, 1978fulltext