The Obstacle Problem on the Stratified Set
DOI:
https://doi.org/10.52575/2687-0959-2025-57-2-111-116Keywords:
Stratified Set, Laplacian, Variational Problem, Obstacle ProblemAbstract
In this paper we consider an analog of the obstacle problem for a mechanical system composed of strings and membranes; as well as a generalization of this problem to multidimensional case. Small displacements of this system under external small loads may be modeled as an elliptic equation of second order (outside of a contact zone) on the stratified set, equipped with Dirichlet’s condition on the boundary. The main result of this job is a proof of solvability of a corresponding boundary value problem in sobolev-type space. The main assumption is so-called firmness of the stratified set.
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Pham F. Introduction a l’´etude topologique des singularit´es de Landau. Paris: Gauthier-Villars ´Editeur; 1967. 142 p.
Покорный Ю.В., Пенкин О.М., Прядиев В.Л., Боровских А.В., Лазарев К.П., Шабров С.А. Дифференциальные уравнения на геометрических графах. М.: Физматлит; 2005. 272 с.
Даирбеков Н.С., Пенкин О.М., Сарыбекова Л.О. Аналог неравенства Соболева на стратифицированном множестве. Алгебра и анализ. 2018;30(5):149-158. DOI: https://doi.org/10.1090/spmj/1573
Даирбеков Н.С., Пенкин О.М., Сарыбекова Л.О. Неравенство Пуанкаре и p-связность стратифицированного множества. Сибирский математический журнал. 2018;59(6):1291-1302. DOI: https://doi.org/10.17377/smzh.2018.59.606
Фридман А. Вариационные принципы и задачи со свободными границами. М.: Наука; 1990. 536 с.
Gavrilov A., Nicaise S., Penkin O. Poincare’s inequality on stratified sets and applications. Progress in Nonlinear Differential Equations and Their Applications. 2003;55:195-213.
Ali Mehmeti F. Nonlinear waves in networks. Mathematical Research, 80, Academie Verlag, Berlin, 1994. 171 p.
Below J. von. A characteristic equation associated to an eigenvalue problem on c2-networks. Linear algebra and appl. 1985, 71, 309-325.
Nicaise S. Polygonal interface problems. Peter Lang Verlag, 1993. 250 p.
References
Pham F. Introduction a l’´etude topologique des singularit´es de Landau. Paris: Gauthier-Villars ´Editeur; 1967. 142 p.
Pokornyi YV, Penkin OM, Pryadiev VL, Borovskikh AV, Lazarev KP, Shabrov SA. Differentsial’nye uravneniya na geometricheskikh grafakh [Differential equations on geometric graphes]. Moscow: Fizmatlit; 2005. 272 p. (In Russ.)
Dairbekov NS, Penkin OM, Sarybekova LO. An analog of the Sobolev inequality on a stratified set. St. Petersburg Mathematical Journal, 2019;30:869–875. DOI: https://doi.org/10.1090/spmj/1573
Dairbekov NS, Penkin OM, Sarybekova LO. The Poincar´e inequality and p -connectedness of a stratified set Siberian Mathematical Journal. 2018;59(6):1024-1033. DOI: https://doi.org/10.1134/S003744661806006X
Friedman A. Variational principles and free-boundary problems. Wiley; 1982. 710 p. (Friedman A. Variational principles and free-boundary problems. Moscow: Nauka; 1990. 536 с.)
Gavrilov A, Nicaise S, Penkin O. Poincare’s inequality on stratified sets and applications. Progress in Nonlinear Differential Equations and Their Applications. 2003;55:195-213.
Ali Mehmeti F. Nonlinear waves in networks. Mathematical Research, 80, Academie Verlag, Berlin, 1994. 171 p.
Below J. von. A characteristic equation associated to an eigenvalue problem on c2-networks. Linear algebra and appl. 1985, 71, 309-325.
Nicaise S. Polygonal interface problems. Peter Lang Verlag, 1993. 250 p.
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