Hilbert and Poincaré problems for semi-linear equations in rectifiable domains
DOI:
https://doi.org/10.12775/TMNA.2022.044Keywords
Dirichlet, Hilbert, Neumann and Poincaré boundary value problems, generalized analytic and generalized harmonic functions with sources, semi-linear Poisson equations, nonlinear Vekua type equationsAbstract
The study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in \cite{R7}, we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.References
L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966.
R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vol. I–II, Clarendon Press, Oxford, 1975.
G.I. Barenblatt, Ja.B. Zel’dovic, V.B. Librovich and G.M. Mahviladze, Matematicheskaya Teoriya Goreniya i Vzryva, “Nauka”, Moscow, 1985 (in Russian); English transl.: The Mathematical Theory of Combustion and Explosion, Consult. Bureau, New York, 1980.
J.I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries I. Elliptic Equations, Research Notes in Mathematics, vol. 106, Pitman, Boston, 1985.
N. Dunford and J.T. Schwartz, Linear Operators I. General Theory, Pure and Applied Mathematics, vol. 7, Interscience Publishers, New York, London, 1958.
P.L. Duren, Theory of Hp Spaces, Pure and Applied Mathematics, vol. 38, Academic Press, New York London, 1970.
F.W. Gehring, On the Dirichlet problem, Michigan Math. J. 3 (1955–1956), 201.
B.R. Gelbaum and J.M.H. Olmsted, Counterexamples in Analysis, San Francisco etc., Holden-Day, 1964.
G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society, Providence, R.I., 1969.
V. Gutlyanskiı̆, O. Nesmelova and V. Ryazanov, On quasiconformal maps and semilinear equations in the plane, Ukr. Mat. Visn. 14 (2017), no. 2, 161–191; English transl.: J. Math. Sci. 229 (2018), no. 1, 7–29.
P. Koosis, Introduction to H p Spaces, Cambridge Tracts in Mathematics, vol. 115, Cambridge Univ. Press, Cambridge, 1998.
M.A. Lavrentiev, On some boundary problems in the theory of univalent functions, Mat. Sb. N.S. 1 (43) (1936), no. 6, 815–846 (in Russian).
J. Leray and Ju. Schauder, Topologie et equations fonctionnelles, Ann. Sci. Ecole Norm. Sup. 51 (1934), no. 3, 45–78 (in French); Topology and functional equations, Uspehi Matem. Nauk (N.S.) 1 (1846), no. 3–4 (13–14), 71–95.
N.N. Luzin, On the main theorem of integral calculus, Mat. Sb. 28 (1912), 266–294 (in Russian).
N.N. Luzin, Integral and trigonometric series, Dissertation, Moskwa, 1915 (in Russian).
N.N. Luzin, Integral and Trigonometric Series, (N.K. Bari and D.E. Men’shov, eds), Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, Leningrad, 1951 (in Russian).
N.N. Luzin, Sur la notion de l’integrale, Ann. Mat. Pura Appl. 26 (1917), no. 3, 77–129.
J. Mawhin, Leray–Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal. 9 (1997), no. 1, 179–200.
S.I. Pokhozhaev, On an equation of combustion theory, Mat. Zametki 88 (2010), no. 1, 53–62; Math. Notes 88 (2010), no. 1–2, 48–56.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 299, Springer–Verlag, Berlin, 1992.
I.I. Priwalow, Randeigenschaften Analytischer Funktionen, Hochschulbücher für Mathematik, vol. 25, Deutscher Verlag der Wissenschaften, Berlin, 1956.
V. Ryazanov, On the Riemann–Hilbert problem without index, Ann. Univ. Buchar. Math. Ser. 5 (LXIII) (2014), no. 1, 169–178, https://arxiv.org/pdf/1308.2486.pdf.
V. Ryazanov, Infinite dimension of solutions of the Dirichlet problem, Open Math. (the former Central European J. Math.) 13 (2015), no. 1, 348–350.
V. Ryazanov, On Neumann and Poincaré problems for Laplace equation, Anal. Math. Phys. 7 (2017), no. 3 (2017), 285–289.
V. Ryazanov, On the theory of the boundary behavior of conjugate harmonic functions, Complex Analysis and Operator Theory 13 (2019), no. 6, 2899–2915.
V. Ryazanov, On Hilbert and Riemann problems for generalized analytic functions and applications, Anal. Math. Phys. 11 (2021), 5.
S. Saks, Theory of the Integral, Warsaw, 1937; English transl.: Dover Publications Inc., New York, 1964.
S.L. Sobolev, Applications of functional analysis in mathematical physics, Transl. Math. Monogr., vol. 7, Amer. Math. Soc., Providence, R.I., 1963.
I.N. Vekua, Generalized Analytic Functions, Pergamon Press, London, Paris, Frankfurt,; Addison-Wesley Publishing Co., Inc., Reading, Mass, 1962.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Vladimir Ryazanov
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
