Title | On the quasilinear Poisson equations in the complex plane |
Publication Type | Journal Article |
Year of Publication | 2020 |
Authors | Gutlyanskii, VYa., Nesmelova, OV, Ryazanov, VI |
Abbreviated Key Title | Dopov. Nac. akad. nauk Ukr. |
DOI | 10.15407/dopovidi2020.01.003 |
Issue | 1 |
Section | Mathematics |
Pagination | 3-10 |
Date Published | 1/2020 |
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 |
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).