On the quasilinear Poisson equations in the complex plane

1Gutlyanskii, VYa.
1Nesmelova, OV
1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2020, 1:3-10
https://doi.org/10.15407/dopovidi2020.01.003
Section: Mathematics
Language: English
Abstract: 
First, we study the existence and regularity of solutions for the linear Poisson equations ∆U(z) = g(z) in bounded domains D of the complex plane £ with charges g in the classes \[L^{1}\left(D\right)\cap L_loc^p \left(D\right)\], p > 1. Then, applying the Leray— Schauder approach, we prove the existence of Höldercontinuous solutions U in the class \[W_loc^2\cdot^{p}\left(D\right)\] for the quasilinear Poisson equations of the form ∆U(z) = h(z)⋅ f (U(z)) with h in the same classes as g and continuous functions f : R → R such that f (t) / t → 0 as t → ∞. These results can be applied to various problems of mathematical physics.
Keywords: anisotropic and inhomogeneous media, potential theory, quasiconformal mappings, quasilinear Poisson equations, semilinear equations
References: 

1. Bers, L. & Nirenberg, L. (1954, August). On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications. Convegno Internazionale sulle equazioni lineari alle derivate parziali, Trieste (pp. 111140). Rome: Edizioni Cremonese.
2. Bojarski, B. V. (1955). Homeomorphic solutions of Beltrami systems. Dokl. Akad. Nauk SSSR (N.S.), 102, pp. 661664 (in Russian).
3. Bojarski, B. & Iwaniec, T. (1983). Analytical foundations of the theory of quasiconformal mappings in ¡n. Ann. Acad. Sci. Fenn. Ser. A. I. Math., 8, No. 2, pp. 257324. Doi: https://doi.org/10.5186/aasfm.1983.0806
4. Lavrentiev, M. (1938). Sur une crit re différentiel des transformations homéomorphes des domains à trois dimensions. Dokl. Acad. Nauk. SSSR, 20, pp. 241242.
5. Vekua, I. N. (1962). Generalized analytic functions. Oxford, New York: Pergamon Press.
6. Astala, K., Iwaniec, T. & Martin, G. (2009). Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series (Vol. 48). Princeton, NJ: Princeton University Press. Doi: https://doi.org/10.1515/9781400830114
7. Gutlyanskii, V. & Nesmelova, O. & Ryazanov, V. (2018). On quasiconformal maps and semilinear equations in the plane. J. Math. Sci., 229, No. 1, pp. 729. Doi: https://doi.org/10.1007/s10958-018-3659-6
8. Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2018). On the regularity of solutions of quasili near Pois son equations. Dopov. Nac. akad. nauk Ukr., No. 10, pp. 917. Doi: https://doi.org/10.15407/dopovidi2018.10.009
9. H rmander, L. (1983). The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften. (Vol. 256). Berlin: Springer.
10. Sobolev, S. L. (1991). Some applications of functional analysis in mathematical physics. Mathematical Monographs. (Vol. 90). Providence, RI: AMS.
11. Giaquinta, M. & Martinazzi, L. (2012). An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. 2nd ed. Lecture Notes (Scuola Normale Superiore) (Vol. 11). Pisa: Edizioni della Normale. Doi: https://doi.org/10.1007/978-88-7642-443-4
12. Ransford, T. (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. (Vol. 28). Cambridge: Cambridge University Press. Doi: https://doi.org/10.1017/CBO9780511623776
13. Leray, J. & Schauder, Ju. (1934). Topologie et quations fonctionnelles. Ann. Sci. Ecole Norm. Sup., 51, No. 3, pp. 4578. Doi: https://doi.org/10.24033/asens.836
14. Dunford, N. & Schwartz, J.T. (1958). Linear operators. Pt. I. General theory. Pure and Applied Mathematics. (Vol. 7). New York: Interscience.
15. Mihlin, S. G. (1977). Linear partial differential equations. Moscow: Vysshaya Shkola (in Russian).