Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity

V. N. Pavlenko, D. K. Potapov

Research output

Abstract

An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter λ multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum σ of the nonlinear problem under consideration is closed (σ consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, λ. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that σ = [λ, +∞) and that for each λ ∈ σ the problem has a nontrivial semiregular solution. If there exists a positive constant M such that the sum of the nonlinearity and Mu is a function which is nondecreasing in the phase variable u, then for any λ ∈ σ the boundary value problem has a minimal nontrivial solution uλ(x). The required solution is semiregular, and uλ(x) is a decreasing mapping with respect to λ on [λ, +∞). Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered.

Original languageEnglish
Pages (from-to)1043-1066
Number of pages24
JournalSbornik Mathematics
Volume210
Issue number7
DOIs
Publication statusPublished - 1 Jan 2019

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Discontinuous Nonlinearities
Elliptic Boundary Value Problems
Nonlinearity
Semiregular
Non-negative
Boundary Value Problem
Boundary conditions
Dirichlet Boundary Value Problem
Zero
Nontrivial Solution
Absolute value
Incompressible Fluid
Nonlinear Problem
Infinity
Mathematical Model
Lower bound
Upper bound
Closed

Scopus subject areas

  • Algebra and Number Theory

Cite this

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title = "Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity",
abstract = "An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter λ multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum σ of the nonlinear problem under consideration is closed (σ consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, λ∗. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that σ = [λ∗, +∞) and that for each λ ∈ σ the problem has a nontrivial semiregular solution. If there exists a positive constant M such that the sum of the nonlinearity and Mu is a function which is nondecreasing in the phase variable u, then for any λ ∈ σ the boundary value problem has a minimal nontrivial solution uλ(x). The required solution is semiregular, and uλ(x) is a decreasing mapping with respect to λ on [λ∗, +∞). Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered.",
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N2 - An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter λ multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum σ of the nonlinear problem under consideration is closed (σ consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, λ∗. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that σ = [λ∗, +∞) and that for each λ ∈ σ the problem has a nontrivial semiregular solution. If there exists a positive constant M such that the sum of the nonlinearity and Mu is a function which is nondecreasing in the phase variable u, then for any λ ∈ σ the boundary value problem has a minimal nontrivial solution uλ(x). The required solution is semiregular, and uλ(x) is a decreasing mapping with respect to λ on [λ∗, +∞). Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered.

AB - An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter λ multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum σ of the nonlinear problem under consideration is closed (σ consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, λ∗. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that σ = [λ∗, +∞) and that for each λ ∈ σ the problem has a nontrivial semiregular solution. If there exists a positive constant M such that the sum of the nonlinearity and Mu is a function which is nondecreasing in the phase variable u, then for any λ ∈ σ the boundary value problem has a minimal nontrivial solution uλ(x). The required solution is semiregular, and uλ(x) is a decreasing mapping with respect to λ on [λ∗, +∞). Applications of the results to the Gol’dshtik mathematical model for separated flows in an incompressible fluid are considered.

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