Global uniqueness for degenerate-weight Dirichlet problems

Establish global uniqueness for the Dirichlet problem associated with the infinity-Laplacian equation with a nonnegative degenerate weight, such as \(\Delta_\infty u-\lambda(x)u_+^m=h(x)\), from the prescribed outer boundary data alone, without prescribing the solution on the degeneracy set.

Background

For strictly positive weights, comparison principles can yield uniqueness for strong-absorption infinity-Laplacian equations. When the weight vanishes, the paper proves a degeneracy-set comparison principle: two solutions with the same boundary values are uniquely determined once they agree on the degeneracy set. For the prototype Hénon-type weight $\lambda(x)=|x|^\alpha$, the degeneracy set is $F=\{0\}$, so the result gives uniqueness only within each slice indexed by the value $u(0)$.

The unresolved issue is whether the value on the degeneracy set is itself determined by the outer boundary data. Resolving this would upgrade slice uniqueness to global uniqueness and clarify whether the degeneracy set functions as an additional interior boundary.