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

Annales Polonici Mathematici (1993)

- Volume: 58, Issue: 2, page 139-146
- ISSN: 0066-2216

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topStanisław Brzychczy. "Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type." Annales Polonici Mathematici 58.2 (1993): 139-146. <http://eudml.org/doc/262341>.

@article{StanisławBrzychczy1993,

abstract = {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].},

author = {Stanisław Brzychczy},

journal = {Annales Polonici Mathematici},

keywords = {nonlinear differential-functional equations of elliptic type; monotone iterative technique; Chaplygin's method; Dirichlet problem; upper and lower solutions; Agmon-Douglis-Nirenberg theorem; strongly uniformly elliptic operator; nonlinear differential-functional equation; regular solution},

language = {eng},

number = {2},

pages = {139-146},

title = {Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type},

url = {http://eudml.org/doc/262341},

volume = {58},

year = {1993},

}

TY - JOUR

AU - Stanisław Brzychczy

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

JO - Annales Polonici Mathematici

PY - 1993

VL - 58

IS - 2

SP - 139

EP - 146

AB - 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].

LA - eng

KW - nonlinear differential-functional equations of elliptic type; monotone iterative technique; Chaplygin's method; Dirichlet problem; upper and lower solutions; Agmon-Douglis-Nirenberg theorem; strongly uniformly elliptic operator; nonlinear differential-functional equation; regular solution

UR - http://eudml.org/doc/262341

ER -

## References

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- [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. Zbl0093.10401
- [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). Zbl0613.35041
- [6] O. Diekmann and N. M. Temme, Nonlinear Diffusion Problems, MC Syllabus 28, Mathematisch Centrum, Amsterdam 1982.
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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. Zbl0549.35002
- [13] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000. Zbl0223.35038
- [14] J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), 257-282. Zbl0008.25502
- [15] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York 1983. Zbl0508.35002
- [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. Zbl0192.45802
- [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.

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