Geodesic
Non-uniqueness of solutions to boundary value problems with Wentzell condition
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 14 (2021) no. 4, pp. 102-105 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Recently, in the mathematical literature, the Wentzell boundary condition is considered from two points of view. In the first case, let us call it classical one, this condition is an equation containing a linear combination of the values of the function and its derivatives on the boundary of the domain. Moreover, the function itself also satisfies the equation with an elliptic operator defined in the domain. In the second case, which we call neoclassical one, the Wentzell condition is an equation with the Laplace–Beltrami operator defined on the boundary of the domain understood as a smooth compact Riemannian manifold without boundary, and the external action is represented by the normal derivative of a function defined in the domain. The paper shows the non-uniqueness of solutions to boundary value problems with the Wentzell condition in the neoclassical sense both for the equation with the Laplacian and for the equation with the Bi-Laplacian given in the domain.
Keywords: Wentzell condition. 
@article{VYURU_2021_14_4_a7,
     author = {N. S. Goncharov and S. A. Zagrebina and G. A. Sviridyuk},
     title = {Non-uniqueness of solutions to boundary value problems with {Wentzell} condition},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {102--105},
     year = {2021},
     volume = {14},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2021_14_4_a7/}
}
TY  - JOUR
AU  - N. S. Goncharov
AU  - S. A. Zagrebina
AU  - G. A. Sviridyuk
TI  - Non-uniqueness of solutions to boundary value problems with Wentzell condition
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2021
SP  - 102
EP  - 105
VL  - 14
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VYURU_2021_14_4_a7/
LA  - en
ID  - VYURU_2021_14_4_a7
ER  - 
%0 Journal Article
%A N. S. Goncharov
%A S. A. Zagrebina
%A G. A. Sviridyuk
%T Non-uniqueness of solutions to boundary value problems with Wentzell condition
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2021
%P 102-105
%V 14
%N 4
%U http://geodesic.mathdoc.fr/item/VYURU_2021_14_4_a7/
%G en
%F VYURU_2021_14_4_a7
N. S. Goncharov; S. A. Zagrebina; G. A. Sviridyuk. Non-uniqueness of solutions to boundary value problems with Wentzell condition. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 14 (2021) no. 4, pp. 102-105. http://geodesic.mathdoc.fr/item/VYURU_2021_14_4_a7/

[1] Ventcel' A.D., “On Boundary Conditions for Multidimensional Diffusion Processes”, Theory of Probability and Its Applications, 4:2 (1960), 164–177 | MR | Zbl

[2] Luo Y., Trudinger N.S., “Linear Second Order Elliptic Equations with Venttsel Boundary Conditions”, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, 118:3–4 (1991), 193–207 | DOI | MR | Zbl

[3] Apushkinskaya D.E., Nazarov A.I., “An Initial-Boundary Value Problem with a Venttsel' Boundary Condition for Parabolic Equations not in Divergence Form”, St. Petersburg Mathematical Journal, 6:6 (1995), 1127–1149 | MR

[4] Lukyanov V.V., Nazarov A.I., “Solving of Vent'sel Boundary-Value Problem for Laplace and Helmholtz Equations by Iterated Potentials”, Journal of Mathematical Sciences, 102:4 (2000), 4265–4274 | DOI | MR | Zbl

[5] Favini A., Goldstein G.R., Goldstein J.A., Romanelli S., “$C_0$-Semigroups Generated by Second Order Differential Operators with General Wentzell Boundary Conditions”, Proceedings of the American Mathematical Society, 128:7 (2000), 1981–1989 | DOI | MR | Zbl

[6] Favini A., Goldstein G.R., Goldstein J.A., Romanelli S., “The Heat Equation with Generalized Wentzell Boundary Condition”, Journal of Evolution Equations, 2:1 (2002), 1–19 | DOI | MR | Zbl

[7] Goldstein G.R., “Derivation and Physimathcal Interpretation of General Boundary Conditions”, Advances in Differential Equations, 4:11 (2006), 419–456 | MR

[8] Denk R., Kunze M., Ploss D., “The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains”, Integral Equations and Operator Theory, 93:2 (2021), 13 pp. | DOI | MR | Zbl

[9] Triebel H., Interpolation Theory. Function Spaces. Differential Operators, Veb Deutscher Verlag der Wissenschaften, Berlin, 1978 | MR

[10] Warner F.W., Foundations of Differentiable Manifold and Lie Groups, Springer, New York–Berlin–Heidelberg–Tokyo, 1983 | MR

[11] Goncharov N.S., Zagrebina S.A., Sviridyuk G.A., “Showalter–Sidorov and Cauchy Problems for the Linear Dzektser Equation with Wentzel and Robin Boundary Conditions in a Bounded Domain”, Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2022 | Zbl