The Dirichlet problem for the Poisson type equations in the plane

TitleThe Dirichlet problem for the Poisson type equations in the plane
Publication TypeJournal Article
Year of Publication2020
AuthorsGutlyanskii, VYa., Nesmelova, OV, Ryazanov, VI
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2020.05.010
Issue5
SectionMathematics
Pagination10-16
Date Published5/2020
LanguageEnglish
Abstract

We present a new approach to the study of semilinear equations of the form div [A(z)∇u] = f (u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f (u) is a continuous non-linear function. We establish a theorem on the existence of weak C(D)∩W1.2loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components and give applications to equations of mathematical physics in anisotropic media.

Keywordsanisotropic and inho mogeneous media, Dirichlet problem, quasiconformal maps, quasilinear Poisson equations, semilinear elliptic equations
References: 

1. Gutlyanskiĭ, V., Nesmelova, O. & Ryazanov, V. (2018). On quasiconformal maps and semilinear equations in the plane. J. Math. Sci., 229, No. 1, pp. 7-29. Doi: https://doi.org/10.1007/s10958-018-3659-6
2. Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2020). On the quasilinear Poisson equations in the complex plane. Dopov. Nac. akad. nauk Ukr., No. 1, pp. 3-10. Doi: https://doi.org/10.15407/dopovidi2020.01.003
3. Hurewicz, W. & Wallman, H. (1948). Dimension Theory. Princeton: Princeton Univ. Press.
4. Ignat’ev, A. A. & Ryazanov, V. I. (2005). Finite mean oscillation in the mapping theory. Ukr. Math. Bull., 2, No. 3, pp. 403-424.
5. Bojarski, B. V. (2009). Generalized solutions of a system of differential equations of the first order and ellip tic type with discontinuous coefficients. Report Dept. Math. Stat. (Vol. 118). Jyväskylä: University Printing House Jyväskylä.
6. Ahlfors, L. V. (1966). Lectures on quasiconformal mappings. Van Nostrand Mathematical Studies (Vol. 10). Toronto, New York; London: Van Nostrand Co., Inc.
7. Ransford, T. (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. (Vol. 28). Cambridge: Cambridge Univ. Press. Doi: https://doi.org/10.1017/CBO9780511623776
8. Koosis, P. (1998). Introduction to H p spaces. Cambridge Tracts in Mathematics. (Vol. 115). Cambridge: Cambridge Univ. Press.
9. Gutlyanskii, V. Ya. & Nesmelova, O. V. & Ryazanov, V. I. (2018). On the regularity of solutions of quasilinear Poisson equations. Dopov. Nac. akad. nauk Ukr., No. 10, pp. 9-17. Doi: https://doi.org/10.15407/dopovidi2018.10.009
10. Leray, J. & Schauder, J. (1934). Topologie ét equations fonctionnelles. Ann. Sci. Ecole Norm. Sup., Ser. 3, 51, pp. 45-78 (in French). Doi: https://doi.org/10.24033/asens.836
11. Dunford, N. & Schwartz, J. T. (1958). Linear operators. Pt. I. General theory. Pure and Applied Mathematics (Vol. 7). New York, London: Interscience Publ.
12. Diaz, J. I. (1985). Nonlinear partial differential equations and free boundaries. Vol. 1: Elliptic equations. Research Notes in Mathematics. (Vol. 106). Boston: Pitman.
13. Aris, R. (1975). The mathematical theory of diffusion and reaction in permeable catalysts. Vol. 1, 2. Oxford: Clarendon Press.
14. Barenblatt, G. I., Zel’dovic, Ja. B., Librovich, V. B. & Mahviladze, G. M. (1985). The mathematical theory of combustion and explosions. New York: Consult. Bureau.
15. Pokhozhaev, S. I. (2010). On an equation of combustion theory. Math. Notes, 88, No. 1-2, pp. 48-56. Doi: https://doi.org/10.1134/S0001434610070059