Existence and uniqueness for a two-dimensional Ventcel problem modeling the equilibrium of a prestressed membrane

Antonio Greco; Giuseppe Viglialoro

Applications of Mathematics (2023)

  • Volume: 68, Issue: 2, page 123-142
  • ISSN: 0862-7940

Abstract

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This paper deals with a mixed boundary-value problem of Ventcel type in two variables. The peculiarity of the Ventcel problem lies in the fact that one of the boundary conditions involves second order differentiation along the boundary. Under suitable assumptions on the data, we first give the definition of a weak solution, and then we prove that the problem is uniquely solvable. We also consider a particular case arising in real-world applications and discuss the resulting model.

How to cite

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Greco, Antonio, and Viglialoro, Giuseppe. "Existence and uniqueness for a two-dimensional Ventcel problem modeling the equilibrium of a prestressed membrane." Applications of Mathematics 68.2 (2023): 123-142. <http://eudml.org/doc/299365>.

@article{Greco2023,
abstract = {This paper deals with a mixed boundary-value problem of Ventcel type in two variables. The peculiarity of the Ventcel problem lies in the fact that one of the boundary conditions involves second order differentiation along the boundary. Under suitable assumptions on the data, we first give the definition of a weak solution, and then we prove that the problem is uniquely solvable. We also consider a particular case arising in real-world applications and discuss the resulting model.},
author = {Greco, Antonio, Viglialoro, Giuseppe},
journal = {Applications of Mathematics},
keywords = {Ventcel boundary condition; Laplace-Beltrami operator; composite Sobolev space; well-posedness},
language = {eng},
number = {2},
pages = {123-142},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence and uniqueness for a two-dimensional Ventcel problem modeling the equilibrium of a prestressed membrane},
url = {http://eudml.org/doc/299365},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Greco, Antonio
AU - Viglialoro, Giuseppe
TI - Existence and uniqueness for a two-dimensional Ventcel problem modeling the equilibrium of a prestressed membrane
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 2
SP - 123
EP - 142
AB - This paper deals with a mixed boundary-value problem of Ventcel type in two variables. The peculiarity of the Ventcel problem lies in the fact that one of the boundary conditions involves second order differentiation along the boundary. Under suitable assumptions on the data, we first give the definition of a weak solution, and then we prove that the problem is uniquely solvable. We also consider a particular case arising in real-world applications and discuss the resulting model.
LA - eng
KW - Ventcel boundary condition; Laplace-Beltrami operator; composite Sobolev space; well-posedness
UR - http://eudml.org/doc/299365
ER -

References

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  1. Apushkinskaya, D. E., Nazarov, A. I., 10.1023/A:1022288717033, Appl. Math., Praha 45 (2010), 69-80. (2010) Zbl1058.35118MR1738896DOI10.1023/A:1022288717033
  2. Apushkinskaya, D. E., Nazarov, A. I., Palagachev, D. K., Softova, L. G., 10.1137/19M1286839, SIAM J. Math. Anal. 53 (2021), 221-252. (2021) Zbl1458.35182MR4198569DOI10.1137/19M1286839
  3. Bonnaillie-Noël, V., Dambrine, M., Hérau, F., Vial, G., 10.1137/090756521, SIAM J. Math. Anal. 42 (2010), 931-945. (2010) Zbl1209.35035MR2644364DOI10.1137/090756521
  4. Brezis, H., 10.1007/978-0-387-70914-7, Universitext. Springer, New York (2011). (2011) Zbl1220.46002MR2759829DOI10.1007/978-0-387-70914-7
  5. Caffarelli, L., Silvestre, L., 10.1080/03605300600987306, Commun. Partial Differ. Equations 32 (2007), 1245-1260. (2007) Zbl1143.26002MR2354493DOI10.1080/03605300600987306
  6. Creo, S., Lancia, M. R., Nazarov, A., Vernole, P., 10.3934/dcdss.2019004, Discrete Contin. Dyn. Syst., Ser. S. 12 (2019), 57-64. (2019) Zbl1416.35098MR3836592DOI10.3934/dcdss.2019004
  7. Carmo, M. P. do, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs (1976). (1976) Zbl0326.53001MR0394451
  8. Evans, L. C., 10.1090/gsm/019, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (2010). (2010) Zbl1194.35001MR2597943DOI10.1090/gsm/019
  9. Favini, A., Goldstein, G. Ruiz, Goldstein, J. A., Romanelli, S., 10.1007/s00028-002-8077-y, J. Evol. Equ. 2 (2002), 1-19. (2002) Zbl1043.35062MR1890879DOI10.1007/s00028-002-8077-y
  10. Favini, A., Goldstein, G. Ruiz, Goldstein, J. A., Romanelli, S., The heat equation with nonlinear general Wentzell boundary condition, Adv. Differ. Equ. 11 (2006), 481-510. (2006) Zbl1149.35051MR2237438
  11. Gilbarg, D., Trudinger, N. S., 10.1007/978-3-642-61798-0, Grundlehren der Mathematischen Wissenschaften 224. Springer, Berlin (1983). (1983) Zbl0562.35001MR0737190DOI10.1007/978-3-642-61798-0
  12. Grisvard, P., 10.1137/1.9781611972030, Monographs and Studies in Mathematics 24. Pitman, Boston (1985). (1985) Zbl0695.35060MR0775683DOI10.1137/1.9781611972030
  13. Kashiwabara, T., Colciago, C. M., Dedè, L., Quarteroni, A., 10.1137/140954477, SIAM J. Numer. Anal. 53 (2015), 105-126. (2015) Zbl1326.65161MR3296617DOI10.1137/140954477
  14. Luo, Y., Trudinger, N. S., 10.1007/BF01053434, Potential Anal. 3 (1994), 219-243. (1994) Zbl0823.35059MR1269282DOI10.1007/BF01053434
  15. McShane, E. J., 10.1090/S0002-9904-1934-05978-0, Bull. Am. Math. Soc. 40 (1934), 837-842. (1934) Zbl0010.34606MR1562984DOI10.1090/S0002-9904-1934-05978-0
  16. Nicaise, S., Li, H., Mazzucato, A., 10.1002/mma.4083, Math. Methods Appl. Sci. 40 (2017), 1625-1636. (2017) Zbl1375.35145MR3622421DOI10.1002/mma.4083
  17. Pucci, P., Serrin, J., 10.1007/978-3-7643-8145-5, Progress in Nonlinear Differential Equations and Their Applications 73. Birkhäuser, Basel (2007). (2007) Zbl1134.35001MR2356201DOI10.1007/978-3-7643-8145-5
  18. Quarteroni, A., Valli, A., Domain Decomposition Methods for Partial Differential Equations, Numerical Mathematics and Scientific Computation. Clarendon Press, New York (1999). (1999) Zbl0931.65118MR1857663
  19. Salsa, S., 10.1007/978-3-319-31238-5, Unitext 99. Springer, Cham (2016). (2016) Zbl1383.35003MR3497072DOI10.1007/978-3-319-31238-5
  20. Venttsel', A. D., 10.1137/1104014, Theor. Probab. Appl. 4 (1959), 164-177. (1959) Zbl0089.13404MR0121855DOI10.1137/1104014
  21. Viglialoro, G., González, Á., Murcia, J., 10.1080/00207160.2016.1154950, Int. J. Comput. Math. 94 (2017), 933-945. (2017) Zbl1371.35070MR3625207DOI10.1080/00207160.2016.1154950

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