Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type

• Autorzy: Stanisław Brzychczy
• Języki publikacji: EN

Abstrakty

EN

Consider a nonlinear differential-functional equation
(1) Au + f(x,u(x),u) = 0
where
$Au := ∑_{i,j=1}^m a_{ij}(x) (∂²u)/(∂x_i ∂x_j)$,
$x=(x_1,...,x_m) ∈ G ⊂ ℝ^m$, G is a bounded domain with $C^{2+α}$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real $L^p(G̅)$ function.
For the equation (1) we consider the Dirichlet problem with the boundary condition
(2) u(x) = h(x) for x∈ ∂G.
We use Chaplygin's method [5] to prove that problem (1), (2) has at least one regular solution in a suitable class of functions.
Using the method of upper and lower functions, coupled with the monotone iterative technique, H. Amman [3], D. H. Sattinger [13] (see also O. Diekmann and N. M. Temme [6], G. S. Ladde, V. Lakshmikantham, A. S. Vatsala [8], J. Smoller [15]) and I. P. Mysovskikh [11] obtained similar results for nonlinear differential equations of elliptic type.
A special case of (1) is the integro-differential equation
$Au + f(x,u(x), ∫_G u(x)dx) = 0$.
Interesting results about existence and uniqueness of solutions for this equation were obtained by H. Ugowski [17].

Słowa kluczowe

EN

nonlinear differential-functional equations of elliptic type; monotone iterative technique; Chaplygin's method; Dirichlet problem

Kategorie tematyczne

• 35J65: Nonlinear boundary value problems for linear elliptic equations
• 35R10: Partial functional-differential equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1993
• Tom: 58
• Numer: 2
• Strony: 139-146

Daty

wydano

1993

otrzymano

1992-05-12

poprawiono

1992-09-08

poprawiono

1993-02-20

Twórcy

autor

Stanisław Brzychczy

• Institute of Mathematics, University of Mining And Metallurgy, Al. Mickiewicza 30, 30-059 Kraków, Poland

Bibliografia

• [1] R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.
• [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623-727.
• [3] H. Amman, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146.
• [4] J. Appell and P. Zabreĭko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge 1990.
• [5] S. Brzychczy, Chaplygin's method for a system of nonlinear parabolic differential-functional equations, Differentsial'nye Uravneniya 22 (1986), 705-708 (in Russian).
• [6] O. Diekmann and N. M. Temme, Nonlinear Diffusion Problems, MC Syllabus 28, Mathematisch Centrum, Amsterdam 1982.
• [7] M. A. Krasnosel'skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford 1963.
• [8] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston 1985.
• [9] O. A. Ladyzhenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York 1968.
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• [11] I. P. Mysovskikh, Application of Chaplygin's method to the Dirichlet problem for elliptic equations of a special type, Dokl. Akad. Nauk SSSR 99 (1) (1954), 13-15 (in Russian).
• [12] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York 1984.
• [13] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000.
• [14] J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), 257-282.
• [15] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York 1983.
• [16] N. M. Temme (ed.), Nonlinear Analysis, Vol. II, MC Syllabus 26.2, Mathematisch Centrum, Amsterdam 1976.
• [17] H. Ugowski, On integro-differential equations of parabolic and elliptic type, Ann. Polon. Math. 22 (1970), 255-275.
• [18] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco 1964.
• [19] J. Wloka, Funktionalanalysis und Anwendungen, de Gruyter, Berlin 1971.
• [20] J. Wloka, Grundräume und verallgemeinerte Funktionen, Lecture Notes in Math. 82, Springer, Berlin 1969.