Weak maximum principle for the Dirichlet Lamé problem in high-dimensional Lipschitz domains

Prove or disprove the weak maximum principle for the Dirichlet Lamé problem on arbitrary Lipschitz domains in dimensions n≥4.

Background

The paper discusses extending its mixed-boundary regularity and solvability results for Lamé systems from two-dimensional polygons to higher-dimensional domains. Such an extension requires local boundary Hölder estimates for homogeneous solutions. The authors note that even the corresponding pure Dirichlet problem presents unresolved difficulties in dimensions n≥4.

The weak maximum principle is identified as an especially fundamental unresolved issue for the Dirichlet Lamé system on arbitrary Lipschitz domains. Resolving it would contribute to the boundary regularity theory needed for higher-dimensional mixed problems.

References

In fact, for $n\geq4$, even the weak maximum principle for the Dirichlet Lam\'e problem on an arbitrary Lipschitz domain remains open; see, for example, Problem~3.2.38 and .

— On the $W^{2,p}$ solvability for mixed boundary value problems  (2609.30090 - Khandelwal et al., 24 Sep 2026) in Section “Discussion for higher dimensions”