AN EXISTENCE AND UNIQUENESS OF THE SOLUTION OF SEMILINEAR MONOTONE ELLIPTIC EQUATION WITH THE DATA IN STUMMEL CLASSES

  • Nicky Kurnia Tumalun Mathematics Department, Faculty of Mathematics Natural Sciences and Earth, Universitas Negeri Manado, Indonesia
DOI: https://doi.org/10.30598/barekengvol17iss2pp1123-1130
Keywords: Semilinear elliptic equations, Stummel classes

Abstract

Let  be a bounded open subset of , ,  be a function in Stummel classes , where , and

be a semilinear monotone elliptic equation, where  is  symmetric matrix, elliptic, bounded, and  is non decreasing and Lipschitz. By proving a weighted estimation for a function in Stummel class with its weight in , which allows us to use Stampacchia’s lemma, we obtained the existence and uniqueness of the solution of this equation.

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References

N. K. Tumalun, "Fungsi elemen kelas Stummel modulus terbatas tetapi bukan elemen dari kelas Stummel," BAREKENG: J. Math. & App., vol. 15, no. 1, pp. 077-084, 2021.

N. K. Tumalun, D. I. Hakim and H. Gunawan, "Inclusion between generalized Stummel classes and other function spaces," Math. Inequal. Appl., vol. 23, no. 2, pp. 547-562, 2020.

N. K. Tumalun and H. Gunawan, "Morrey spaces are embedded between weak Morrey spaces and Stummel classes," J. Indones. Math. Soc., vol. 25, no. 3, pp. 203-209, 2019.

N. K. Tumalun, H. Gunawan and J. Lindiarni, "On the inclusion between weak Lebesgue spaces and Stummel classes," in 5th ICRIEMS Proceedings, Yogyakarta, 2018.

N. K. Tumalun, D. I. Hakim and H. Gunawan, "Some function spaces and their applications to elliptic partial differential equations," Matematički Vesnik, vol. 75, no. 2 (in press), pp. 1-16, 2023.

G. Di Fazio, "Regularity for elliptic equations under minimal assumptions," J. Indones. Math. Soc., vol. 26, no. 1, pp. 101-127, 2020.

M. D. Surnachev, "Estimates of solutions to the noncoercive Dirichlet problem for a second order elliptic equation in divergence form with drift from a Kato class," Zap. Nauchn. Sem. POMI, vol. 519, pp. 229-263, 2022.

G. Di Fazio, "Poisson Equation and Morrey Spaces," Journal of Mathematical Analysis and Applications, vol. 163, pp. 157-167, 1992.

S. Samko, "Morrey spaces are closely embedded between vanishing Stummel class," Math. Ineq. Appl., vol. 17, no. 2, pp. 627-639, 2014.

N. K. Tumalun, "An existence and uniqueness of the weak solution of the Dirichlet problem with the data in Morrey spaces," BAREKENG: J. Math. & App., vol. 16, no. 3, pp. 829-834, 2022.

N. K. Tumalun and P. E. A. Tuerah, "A regularity of the weak solution gradient of the Dirichlet problem for divergent form elliptic equations in Morrey spaces," Aust. J. Math. Anal. Appl, vol. 18, no. 2, pp. 14.1-14.7, 2021.

P. E. A. Tuerah and N. K. Tumalun, "Some notes on the inclusion between Morrey spaces," J. Math. Inequal., vol. 16, no. 1, pp. 355-362, 2022.

N. K. Tumalun, "Proper inclusion between vanishing Morrey spaces and Morrey spaces," Tensor: Pure and Applied Mathematics Journal, vol. 2, no. 1, pp. 1-4, 2021.

G. R. Cirmi and S. Leonardi, "Regularity results for the gradient of solutions of linear elliptic equations with L1,λ data," Annali di Mat. Pura e Appl., vol. 185, no. 4, pp. 537-553, 2006.

H. Brezis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations, New York: Springer, 2011.

G. Stampacchia, "Formes bilinéaires coercitives sur les ensembles convexes," C. R. Acad. Sci. Paris, vol. 258, pp. 4413-4416, 1964.

L. Boccardo and G. Croce, Elliptic Partial Differential Equations: Exsistence and Regularity of Distributional Solutions, Berlin: De Gruyter, 2014.

Published
2023-06-11
How to Cite
[1]
N. Tumalun, “AN EXISTENCE AND UNIQUENESS OF THE SOLUTION OF SEMILINEAR MONOTONE ELLIPTIC EQUATION WITH THE DATA IN STUMMEL CLASSES”, BAREKENG: J. Math. & App., vol. 17, no. 2, pp. 1123-1130, Jun. 2023.
Issue
Vol 17 No 2 (2023): BAREKENG: Journal of Mathematics and Its Applications
Section
Articles

Copyright (c) 2023 Nicky Kurnia Tumalun

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