In the original definition of \(U_2\) and \(U_3\) the function \(z\) is defined differently. The authors assumed, that \(U_2\) and \(U_3\) are bounded from below as shown in the picture below. However, \(U_2\) and \(U_3\) are not bounded from below, as was pointed out in the follow-up paper [1] where the authors use this result.
In [1] the authors suggest to add the condition \(y>-\frac{\alpha_- \delta}{\beta}\) to \(U_2\) and \(U_3\) without mentioning how exactly the proof in [2] changes. We were not able to fix the proof by adding the condition. That is because, in our opinion, adding this condition requires changing the cut-off function in the proof. As a consequence, we obtain integrals on the right-hand side where \(u_-\) is involved.
We suggest to change the definition of \(z\).
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References Link to heading
- C. Cârstea and J. Wang,
Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients,
Journal of Differential Equations, vol. 268, no. 12, p. 7609–7628, 2020. doi:10.1016/j.jde.2019.11.088 - V. Franceschi, K. Naderi, and K. Pankrashkin,
Embedded trace operator for infinite metric trees,
Mathematische Nachrichten, vol. 298, no. 1, p. 190–243, 2025. doi:10.1002/mana.202300574