This note corrects an inaccuracy in the paper "Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition" by the author, which appeared in Commun. Anal. Mech. 15 (2023), no. 4,811–830. In particular, we point out that the bounded open set $ \Omega $ must be connected for some of the main results in the paper to hold. To motivate this claim, we give an example of a disconnected open set $ \Omega $ combined with a partition $ (\Gamma_0, \Gamma_1) $ of its boundary, for which the potential–well depth associated to the problem vanishes. When this occurs, the framework developed in the paper breaks down. We also show that, conversely, when $ \Omega $ is connected, this phenomenon does not show up and all assertions in the paper are correct.
Citation: Enzo Vitillaro. On the necessity of the connectedness condition of $ \Omega $ in : "Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition"[J]. Communications in Analysis and Mechanics, 2026, 18(2): 400-405. doi: 10.3934/cam.2026015
This note corrects an inaccuracy in the paper "Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition" by the author, which appeared in Commun. Anal. Mech. 15 (2023), no. 4,811–830. In particular, we point out that the bounded open set $ \Omega $ must be connected for some of the main results in the paper to hold. To motivate this claim, we give an example of a disconnected open set $ \Omega $ combined with a partition $ (\Gamma_0, \Gamma_1) $ of its boundary, for which the potential–well depth associated to the problem vanishes. When this occurs, the framework developed in the paper breaks down. We also show that, conversely, when $ \Omega $ is connected, this phenomenon does not show up and all assertions in the paper are correct.
| [1] |
E. Vitillaro, Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition, Commun. Anal. Mech., 15 (2023), 811–830, https://doi.org/10.3934/cam.2023039 doi: 10.3934/cam.2023039
|
| [2] | P. Grisvard, Elliptic problems in nonsmooth domains, vol. 69 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, Reprint of the 1985 original, with a foreword by Susanne C. Brenner. https://doi.org/10.1137/1.9781611972030.ch1 |
| [3] | W. P. Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded variation, In: Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. |
| [4] | G. Leoni, A first course in Sobolev spaces, vol. 181 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2017. |
| [5] | D. Mugnolo, E. Vitillaro, The wave equation with acoustic boundary conditions on non-locally reacting surfaces, Mem. Amer. Math. Soc., 2024. https://doi.org/10.1090/memo/1526 |
| [6] |
J. L. Vázquez, E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143–2161, https://doi.org/10.1016/j.jde.2010.12.012 doi: 10.1016/j.jde.2010.12.012
|
| [7] | E. Vitillaro, Strong solutions for the wave equation with a kinetic boundary condition, In: Recent trends in nonlinear partial differential equations. I. Evolution problems, vol. 594 of Contemp. Math, 295–307, Amer. Math. Soc., Providence, RI, 2013. https://doi.org/10.1090/conm/594/11793 |
| [8] |
E. Vitillaro, On the Wave Equation with Hyperbolic Dynamical Boundary Conditions, Interior and Boundary Damping and Source, Arch. Ration. Mech. Anal., 223 (2017), 1183–1237, https://doi.org/10.1007/s00205-016-1055-2 doi: 10.1007/s00205-016-1055-2
|
| [9] |
E. Vitillaro, On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, J. Differential Equations, 265 (2018), 4873–4941, https://doi.org/10.1016/j.jde.2018.06.022 doi: 10.1016/j.jde.2018.06.022
|
| [10] |
E. Vitillaro, Blow-up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 4575–4608, https://doi.org/10.3934/dcdss.2021130 doi: 10.3934/dcdss.2021130
|
| [11] |
G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457–480, https://doi.org/10.57262/ade/1355867704 doi: 10.57262/ade/1355867704
|
Enzo Vitillaro. On the necessity of the connectedness condition of $ \Omega $ in : "Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition"[J]. Communications in Analysis and Mechanics, 2026, 18(2): 400-405. doi: 10.3934/cam.2026015
DownLoad: