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

Stanisław Brzychczy

Annales Polonici Mathematici (1993)

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

Abstract

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Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where A u : = ∑ i , j = 1 m a i j ( x ) ( ∂ ² u ) / ( ∂ 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 ( G ̅ ) 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 A u + f ( x , u ( x ) , ∫ G u ( x ) d x ) = 0 . Interesting results about existence and uniqueness of solutions for this equation were obtained by H. Ugowski [17].

How to cite

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Stanisław 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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