Strong convergence of extended conormal derivatives on the full domain

Determine whether the extended conormal derivatives of the truncated-domain solutions converge strongly to the conormal derivative of the degenerate-domain solution in L^2(0,T;L^2(Γ)) as δ→0^+, rather than only on fixed nondegenerate boundary portions.

Background

The paper approximates the boundary-degenerate hyperbolic equation on Ω by uniformly elliptic problems posed on truncated domains Ω_δ={x∈Ω:x_2>δ}. For each fixed boundary portion separated from the degenerate boundary, the authors establish strong convergence of the approximate conormal derivatives and use this convergence to transfer hidden-regularity estimates to the original problem.

The authors explicitly caution that the corresponding strong convergence statement over the entire nondegenerate boundary Γ may fail. Thus, the behavior of the extended approximate conormal derivatives on the full boundary as the truncation parameter tends to zero remains unresolved, even though a uniform L2 estimate for the limiting conormal derivative is ultimately obtained.

References

From the proof of Theorem \ref{03.13.T2}, we note that this may not hold:

\frac{\partial Ey_\delta}{\partial \nu}\to \frac{\partial y}{\partial \nu}\quad\mbox{strongly in } L2(0,T; L2(\Gamma)) \mbox{ as } \delta\to 0+.

— Application of the Shape Design Method to Hidden Regularity of Degenerate Hyperbolic Equations with Degenerate Boundary  (2608.17767 - Yang et al., 18 Aug 2026) in Remark 3.15.R1, Section 3