Characterize the contact set of limiting ground-state profiles

Determine whether the contact set \(\Lambda_0(u_\infty)=\{x\in\Omega:|\nabla u_\infty(x)|=1\}\) of a limiting ground-state profile for the indefinite-weight Minkowski mean curvature Neumann problem can have full measure in \(\Omega\setminus\{a=0\}\), and characterize the size and geometry of this contact set.

Background

For the model nonlinearity g(u)=∣u∣p−2ug(u)=|u|^{p-2}u, ground-state solutions converge as λ→+∞\lambda\to+\infty to a maximizer u∞u_\infty of a constrained variational problem over 1-Lipschitz functions satisfying the compatibility constraint ∫Ωa∣u∣p−2u=0\int_\Omega a|u|^{p-2}u=0. The limiting profile saturates the global gradient constraint, ∥∇u∞∥L∞(Ω)=1\|\nabla u_\infty\|_{L^\infty(\Omega)}=1.

On connected open regions where the gradient constraint is uniformly inactive and a≠0a\neq0 almost everywhere, the profile is constant, taking only the values zero and a common positive plateau height. However, the paper does not rule out the possibility that the contact set where ∣∇u∞∣=1|\nabla u_\infty|=1 has full measure, in which case the plateau description is vacuous. The one-dimensional example later in the paper demonstrates that full saturation is possible for admissible competitors, but leaves unresolved whether it can occur for an actual maximizer or limiting ground-state profile.

References

We stress, however, that Theorem~\ref{thm:intro} does not exclude the possibility that $\Lambda_0(u_\infty)$ has full measure in $\Omega\setminus{a=0}$. In that case, the above plateau description becomes vacuous. Determining whether this can actually occur for a limiting ground state, as well as obtaining a finer description of the size and geometry of the contact set $\Lambda_0(u_\infty)$, remains an open problem, and is expected to depend heavily on the geometry of the indefinite weight $a(x)$ (cf. Example \ref{ex:final}).

— Positive Solutions for an Indefinite-Weight Minkowski Mean Curvature Neumann Problem: Multiplicity and Asymptotic Behaviour  (2609.09894 - Ziegele, 9 Sep 2026) in Section 1, immediately following the discussion of Theorem 1.1; see also the final paragraph before Example 4.1

The relevant question is instead whether such a fully saturated configuration can actually maximize $\mathcal J$ over $K_a$. Determining whether this can occur, and more generally understanding how the geometry of $a$ influences the size and structure of $\Lambda_0(u_\infty)$, remains an interesting open question.

— Positive Solutions for an Indefinite-Weight Minkowski Mean Curvature Neumann Problem: Multiplicity and Asymptotic Behaviour  (2609.09894 - Ziegele, 9 Sep 2026) in Final paragraph of Section 4, following Example 4.1

In the same borderline case, and in the regime (H2), we do not know whether the minimizer can saturate the constraint on a set of positive measure; by Corollary \ref{cor:H2nosat} this is excluded as soon as $L<\sigma$, and it is precisely the regime in which the maximal surfaces of are allowed to contain light rays.

— Gradient constraints, Born-Infeld, and maximal surfaces via superposition of infinitely many $p$-Laplacians  (2609.28288 - Hamid et al., 23 Sep 2026) in Section 6, subsection “The borderline datum L=σ”