Local boundary Hölder estimate for three-dimensional mixed Lamé systems

Establish whether a scale-invariant local boundary Hölder estimate holds for homogeneous Lamé systems with mixed Dirichlet–traction boundary conditions on three-dimensional Lipschitz domains.

Background

The two-dimensional Lamé result in the paper relies on a boundary Hölder estimate for homogeneous solutions. In three dimensions, the mixed Dirichlet–traction interface produces singular behavior that prevents the pure-boundary non-tangential estimates from directly supplying the required regularity.

The requested estimate is the missing ingredient for obtaining the annular decay estimate in the atomic argument and, consequently, corresponding global W2,pW^{2,p} solvability for data with pp close to 1. The paper states that such an estimate is not known even for arbitrary three-dimensional Lipschitz polyhedra, although it is available under additional geometric assumptions.

References

Does a scale-invariant local boundary H\"older estimate hold for homogeneous Lam\'e systems with mixed Dirichlet--traction boundary conditions on three dimensional Lipschitz domains?

— 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”

Nicaise further conjectured that the latter vertex condition could be removed Remark~4.4.

— 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”