Усреднение уравнения Лаврентьева – Бицадзе в области, периодически перфорированной вдоль линии смены типа уравнения
Статья подготовлена при финансовой поддержке Московского центра фундаментальной и прикладной математики (грант Минобрнауки России № 075-15-2025-345).
DOI:
https://doi.org/10.52575/2687-0959-2026-58-2-137-146Ключевые слова:
уравнение Лаврентьева – Бицадзе, перфорированная область, уравнение смешанного типа, усреднениеАннотация
Рассмотрена задача для уравнения Лаврентьева – Бицадзе в модельной области, перфорированной вдоль линии смены типа уравнения и имеющей характерный размер микронеоднородностей ε, с однородным краевым условием Дирихле на границе полостей и однородным условием Дирихле на внешней части границы. Для этой задачи построена усреднённая задача и доказана сходимость решений исходной задачи к решению усреднённой. При этом обнаружено, что предельная (усреднённая) задача вырождается в «гиперболической» части области.
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Amirat Y., Bodart O., Chechkin GA., Piatnitski AL. Asymptotics of a spectral-sieve problem. Journal of Mathematical Analysis and Applications. 2016; 435(2): 1652–1671.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV. "Strange term"in homogenization of attractors of reaction–diffusion equation in perforated domain. Chaos, Solitons & Fractals. 2020;140:110208.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV. Application of Fatou’s lemma for strong homogenization of attractors to reaction–diffusion systems with rapidly oscillating coefficients in orthotropic media with periodic obstacles. Mathematics. 2023; 11(6): 1448.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV. Homogenization of attractors to reaction–diffusion system in a medium with random obstacles. Discrete and Continuous Dynamical Systems. 2024; 44(11): 3474–3490.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV., Tolemis AA. Homogenization of attractors to Ginzburg–Landau equations in media with locally periodic obstacles: critical case. Bulletin of the Karaganda University. Mathematics Series. 2023; 3:11–27.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV., Tolemis AA. Attractors of Ginzburg–Landau equations with oscillating terms in porous media. Homogenization procedure. Applicable Analysis. 2024; 103(1): 29–44.
Bekmaganbetov KA., Chechkin GA., Chepyzhov VV., Tolemis AA. Homogenization of attractors to Ginzburg–Landau equations in media with locally periodic obstacles: sub- and supercritical cases. Bulletin of the Karaganda University. Mathematics Series. 2024; 2:40–56.
Bekmaganbetov KA., Chechkin GA., Toleubay AM. Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium. Bulletin of the Karaganda University. Mathematics Series. 2022; 3:35–50.
Chechkin GA. The Meyers Estimates for Domains Perforated Along the Boundary. Mathematics. 2021; 9(23): 3015.
Chechkin GA., D’Apice C., De Maio U., Gadyl’shin RR. On Singularly Perturbed Steklov Problem in Domain Perforated Along the Boundary. Comptes Rendus Me´canique. 2016; 344(1): 12–18.
Chechkin GA., Friedman A., Piatnitski AL. The Boundary Value Problem in Domains with Very Rapidly Oscillating Boundary. Journal of Mathematical Analysis and Applications. 1999;231(1):213–234.
Chechkin GA., Koroleva YO., Persson LE., Wall P. On Spectrum of the Laplacian in a Circle Perforated Along the Boundary: Application to a Friedrichs–Type Inequality. International Journal of Differential Equations. 2011;2011:619623.
Chechkin GA., Koroleva YO., Persson LE., Wall P. A New Weighted Friedrichs–Type Inequality for a Perforated Domain with a Sharp Constant. Eurasian Mathematical Journal. 2011;2(1):81–103.
Chechkin GA., Piatnitski AL. Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain. Applicable Analysis. 1999; 71(1-4): 215–235.
Cioranescu D., Donato P. On a Robin Problem in Perforated Domains. In: Homogenization and Applications to Material Sciences. Tokyo: Gakko¯tosho; 1997. p. 123–136.
Cioranescu D., Saint Jean Paulin J. Truss Structures: Fourier Conditions and Eigenvalue Problems. In: Boundary Variation. Berlin–New York: Springer; 1992. p. 125–141.
Ene HI., Sanchez-Palencia E. Equations et phe´nome`nes de surface pour l’e´coulement dans un mode`le de milieu poreux. Journal de Me´canique. 1975; 14: 73–108.
Jäger W., Mikelic´ A. On the Flow Conditions at the Boundary between a Porous Medium and an Impermeable Solid. Progress in partial differential equations. London: Longman Scientific & Technical; 1994. p. 145–161.
Jäger W., Mikelic´ A. On the Boundary Conditions at the Contact Interface between a Porous Medium and a Free Fluid. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 1996;23(3):403–465.
Jäger W., Mikelic´ A. Homogenization of the Laplace Equation in a Partially Perforated Domain. Homogenization: In Memory of Serguei Kozlov. River Edge: World Scientific Publishing; 1999. p. 259–284.
Larson RE., Higdon JJL. Microscopic Flow Near the Surface of Two-Dimensional Porous Media. Part I — Axial Flow. Journal of Fluid Mechanics. 1986;166:449–472.
Larson RE., Higdon JJL. Microscopic Flow near the surface of Two-Dimensional Porous Media. Part II — Traverse Flow. Journal of Fluid Mechanics. 1986;178:119–136.
Lions JL., Magenes E. Proble`mes aux limites non homoge`nes et applications. Vol. I. Paris: Dunod; 1968. 372 p.
Lobo M., Oleinik OA., Pérez ME., Shaposhnikova TA. On homogenization of solutions of boundary value problems in domains perforated along manifolds. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 1997; 25(3–4): 611–629.
Osher S. Boundary Value Problems for Equations of Mixed Type I. The Lavrent’ev–Bitsadze Model. Communications in Partial Differential Equations. 1977; 2(5): 499–547.
Saffman PG. On the Boundary Condition at the Surface of a Porous Medium. Studies in Applied Mathematics. 1971;1: 93–101.
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