MSC: 47J20, 49J40, 49J45
DOI: 10.21538/0134-4889-2025-31-4-132-148
This work was partially supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of the Program of development of Ural Federal University under the “Priority-2030” academic leadership program.
We consider variational inequalities with operators ${\mathcal A}_{s}\colon W^{1,p}(\Omega_{s})\to(W^{1,p}(\Omega_{s}))^{\ast}$ in divergence form and constraint sets $V_{s}=\{v\in W^{1,p}(\Omega_{s}): h(v)+\Phi_{s}(v) \leqslant\varphi_{s} \ \text{a.e. in} \ \Omega_{s}\}$, where $\Omega_s$ with $s\in\mathbb N$ is a domain in $\mathbb R^n$ contained in a bounded domain $\Omega\subset\mathbb R^n$ ($n\geqslant 2$), $p>1$, $h$ is a convex function on $\mathbb R$, $\varphi_{s}$ is a function on $\Omega_{s}$, and $\Phi_{s}$ is a continuous convex functional on $W^{1,p}(\Omega_{s})$. We describe conditions for a weak convergence of solutions of the considered variational inequalities to the solution of a variational inequality with an operator from $W^{1,p}(\Omega)$ to $(W^{1,p}(\Omega))^{\ast}$ and constraint set defined by the equality $V=\{v\in W^{1,p}(\Omega): h(v)+\Phi(v)\leqslant\varphi \ \text{a.e. in} \ \Omega\}$, where $\varphi$ is a limit function for $\varphi_{s}$ and $\Phi$ is a limit functional for $\Phi_{s}$. These conditions include some requirements on the involved domains, operators, and the mappings defining the constraint sets. In so doing, one of the main conditions is the $G$-convergence of the sequence $\{{\mathcal A}_{s}\}$ to an operator ${\mathcal A}\colon W^{1,p}(\Omega)\to(W^{1,p}(\Omega))^{\ast}$.
Keywords: nonlinear elliptic variational inequality, pointwise functional constraint, variable domains, $G$-convergence of operators, convergence of solutions
REFERENCES
1. Lions J.L. Quelques méthodes de résoltion des problèmes aux limites non linéaires. Paris: Dunod, Gauthier-Villars, 1969, 554 p.
2. Kinderlehrer D., Stampacchia G. An introduction to variational inequalities and their applications. NY: Acad. Press, 1980, 313 p. ISBN: 0080874045 .
3. Spagnolo S. Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Super. Pisa. Cl. Sci. (3), 1968, vol. 22, no. 4, pp. 571–597.
4. Zhikov V.V., Kozlov S.M., Oleinik O.A., Ha Tien Ngoan. Averaging and G-convergence of differential operators. Russian Math. Surveys, 1979, vol. 34, no. 5, pp. 69–147. https://doi.org/10.1070/RM1979v034n05ABEH003898
5. Zhikov V.V., Kozlov S.M., Oleinik O.A. G-convergence of parabolic operators. Russian Math. Surveys, 1981, vol. 36, no. 1, pp. 9–60. https://doi.org/10.1070/RM1981v036n01ABEH002540
6. Pankov A.A. Averaging and G-covergence of nonlinear elliptic operators of divergence type. Dokl. Akad. Nauk SSSR, 1984, vol. 278, no. 1, pp. 37–41.
7. Pankov A. G-convergence and homogenization of nonlinear partial differential operators. Dordrecht: Kluwer Academic Publishers, 1997, 249 p.
8. Kovalevskii A.A. G-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain. Russ. Acad. Sci. Izv. Math., 1995, vol. 44, no. 3, pp. 431–460.
https://doi.org/10.1070/IM1995v044n03ABEH001607
9. Murat F. Sur l’homogeneisation d’inequations elliptiques du 2ème ordre, relatives au convexe $K(\psi_{1},\psi_{2})=\{v\in H^{1}_{0}(\Omega)\,|\,\psi_{1}\leqslant v\leqslant\psi_{2}\text{ p.p. dans }\Omega\}$}. 1976. Publ. Laboratoire d’Analyse Numérique, no. 76013. Univ. Paris VI, 23 p.
10. Boccardo L., Murat F. Homogenization of nonlinear unilateral problems. In: Composite media and homogenization theory. Progr. Nonlinear Differential Equations Appl., vol. 5, Boston: Birkhäuser, 1991, pp. 81–105. https://doi.org/10.1007/978-1-4684-6787-1_6
11. Kovalevsky A.A. Nonlinear variational inequalities with variable regular bilateral constraints in variable domains. Nonlinear Differ. Equ. Appl., 2022, vol. 29, no. 6, Paper No. 70, 24 p.
https://doi.org/10.1007/s00030-022-00797-w
12. Dal Maso G., Paderni G. Variational inequalities for the biharmonic operator with variable obstacles. Ann. Mat. Pura Appl. (4), 1988, vol. 153, no. 1, pp. 203–227. https://doi.org/10.1007/BF01762393
13. Sandrakov G.V. Homogenization of variational inequalities for nonlinear diffusion problems in perforated domains. Izv. Math., 2005, vol. 69, no. 5, pp. 1035–1059. https://doi.org/10.1070/IM2005v069n05ABEH002287
14. Sandrakov G.V. Homogenization of variational inequalities and equations defined by pseudomonotone operators. Sb. Math., 2008, vol. 199, no. 1, pp. 67–98.
https://doi.org/10.1070/SM2008v199n01ABEH003911
15. Vorob’ev A.Yu., Shaposhnikova T.A. On the homogenization of variational inequalities in perforated domains with arbitrary perforation density. Mosc. Univ. Math. Bull., 2005, vol. 60, no. 1, pp. 7–15.
16. Gómez D., Lobo M., Pérez E., Podolskii A.V., Shaposhnikova T.A. Unilateral problems for the p-Laplace operator in perforated media involving large parameters. ESAIM, Control Optim. Calc. Var., 2018, vol. 24, no. 3,
pp. 921–964. https://doi.org/10.1051/cocv/2017026
17. Zubova M.N., Shaposhnikova T.A. Averaging of some variational inequalities with restrictions on subsets located ε-periodically along the domain boundary. Mosc. Univ. Math. Bull., 2007, vol. 62, no. 2, pp. 67–77.
18. Gómez D., Pérez E., Podolskii A.V., Shaposhnikova T.A. Homogenization of variational inequalities for the p-Laplace operator in perforated media along manifolds. Appl. Math. Optim., 2019, vol. 79, no. 3,
pp. 695–713.
https://doi.org/10.1007/s00245-017-9453-x
19. Kovalevsky A.A. On the convergence of solutions of variational problems with pointwise functional constraints in variable domains. J. Math. Sci. (N.Y.), 2021, vol. 254, no. 3, pp. 375–396.
https://doi.org/10.1007/s10958-021-05310-9
20. Kovalevsky A.A. On the convergence of minimizers and minimum values in variational problems with pointwise functional constraints in variable domains. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 246–257. https://doi.org/10.21538/0134-4889-2021-27-1-246-257
21. Kovalevsky A.A. Variational problems with variable regular bilateral constraints in variable domains. Rev. Mat. Complut., 2019, vol. 32, no. 2, pp. 327–351. https://doi.org/10.1007/s13163-018-0281-6
Received October 15, 2025
Revised October 28, 2025
Accepted November 3, 2025
Funding Agency: This work was partially supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of the Program of development of Ural Federal University under the “Priority-2030” academic leadership program.
Aleksandr Al’bertovich Kovalevsky, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: alexkvl71@mail.ru
Cite this article as: A.A. Kovalevsky. Nonlinear elliptic variational inequalities with unilateral pointwise functional constraints in variable domains. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 132–148.
Русский
А.А. Ковалевский. Нелинейные эллиптические вариационные неравенства с односторонними поточечно функциональными ограничениями в переменных областях
Рассмотрены вариационные неравенства с операторами ${\mathcal A}_{s}\colon W^{1,p}(\Omega_{s})\to(W^{1,p}(\Omega_{s}))^{\ast}$ дивергентного вида и множествами ограничений $V_{s}=\{v\in W^{1,p}(\Omega_{s}): h(v)+\Phi_{s}(v) \leqslant\varphi_{s} \ \text{п.в. в} \ \Omega_{s}\}$, где $\Omega_s$ с $s\in\mathbb N$ – область в $\mathbb R^n$, содержащаяся в ограниченной области $\Omega\subset\mathbb R^n$ ($n\geqslant 2$), $p>1$, $h$ – выпуклая функция на $\mathbb R$, $\varphi_{s}$ – функция на $\Omega_{s}$ и $\Phi_{s}$ – непрерывный выпуклый функционал на $W^{1,p}(\Omega_{s})$. Описаны условия некоторой слабой сходимости решений рассматриваемых вариационных неравенств к решению вариационного неравенства с оператором из $W^{1,p}(\Omega)$ в $(W^{1,p}(\Omega))^{\ast}$ и множеством ограничений, определенным равенством $V=\{v\in W^{1,p}(\Omega): h(v)+\Phi(v)\leqslant\varphi \ \text{п.в. в} \ \Omega\}$, где $\varphi$ – предельная функция для $\varphi_{s}$ и $\Phi$ – предельный функционал для $\Phi_{s}$. Эти условия включают некоторые требования на участвующие области, операторы и отображения, определяющие множества ограничений. При этом одним из основных условий является $G$-сходимость последовательности $\{{\mathcal A}_{s}\}$ к некоторому оператору ${\mathcal A}\colon W^{1,p}(\Omega)\to(W^{1,p}(\Omega))^{\ast}$.
Ключевые слова: нелинейное эллиптическое вариационное неравенство, поточечно функциональное ограничение, переменные области, $G$-сходимость операторов, сходимость решений
