scholarly journals On the discreteness of the spectra of the Dirichlet and Neumannp-biharmonic problems

2004 ◽  
Vol 2004 (9) ◽  
pp. 777-792 ◽  
Author(s):  
Jiří Benedikt
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Neumann Problem ◽  
Nonlinear Boundary ◽  
Neumann Boundary ◽  
Unbounded Sequence ◽  
The Neumann Problem ◽  
Biharmonic Problems ◽  
Zero Points

We are interested in a nonlinear boundary value problem for(|u″|p−2u″)′​′=λ|u|p−2uin[0,1],p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to thenth eigenvalue, has preciselyn−1zero points in(0,1). Eigenvalues of the Neumann problem are nonnegative and isolated,0is an eigenvalue which is not simple, and the positive eigenvalues are simple and they form an increasing unbounded sequence. An eigenfunction, corresponding to thenth positive eigenvalue, has preciselyn+1zero points in(0,1).

scholarly journals Eigenstructure of the equilateral triangle, Part II: The Neumann problem

2002 ◽  
Vol 8 (6) ◽  
pp. 517-539 ◽  
Author(s):  
Brian J. McCartin
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Equilateral Triangle ◽  
Orthonormal System ◽  
Neumann Boundary ◽  
Complete Orthonormal System ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Lines ◽  
Mathematical Techniques ◽  
The Neumann Problem

Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

A Factorization Method for an Inverse Neumann Problem for Harmonic Vector Fields

2003 ◽  
Vol 10 (3) ◽  
pp. 549-560
Author(s):  
R. Kress
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Neumann Problem ◽  
Vector Fields ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Factorization Method ◽  
Harmonic Vector ◽  
Neumann Boundary Value Problem ◽  
Harmonic Vector Fields

Abstract Extending the previous work on the corresponding inverse Dirichlet problem, we present a factorization method for the solution of an inverse Neumann boundary value problem for harmonic vector fields.

scholarly journals TWO-PARAMETER NONLINEAR OSCILLATIONS: THE NEUMANN PROBLEM

2011 ◽  
Vol 16 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Armands Gritsans ◽  
Felix Sadyrbaev
Keyword(s):  
Dirichlet Problem ◽  
Neumann Problem ◽  
Nonlinear Oscillations ◽  
Solution Set ◽  
Nontrivial Solutions ◽  
Number Of Zeros ◽  
Solution Surface ◽  
The Neumann Problem ◽  
Two Parameter ◽  
Zero Points

Boundary value problems of the form are considered, where In our considerations functions f and g are generally nonlinear. We give a description of a solution set of the problem (i), (ii). It consist of all triples () such that (λ,μ,x(t)) nontrivially ′solves the problem(i),(ii) and |x (z)| = α at zero points z of the function x(t) (iii). We show that this solution set is a union of solution surfaces which are centro-affine equivalent. Each solution surface is associated with nontrivial solutions with definite nodal type. Properties of solution surfaces are studied. It is shown, in particular, that solution surface associated with solutions with exactly i zeroes in the interval (a,b) is centro-affne equivalent to a solution surface of the Dirichlet problem (i), x(a) = 0 = x(b), (iii) corresponding to solutions with odd number of zeros 2j − 1 (i ≠ 2j)in the interval (a,b).

The Heat Equation for the -Neumann Problem, II

1988 ◽  
Vol 40 (2) ◽  
pp. 502-512 ◽  
Author(s):  
Richard Beals ◽  
Nancy K. Stanton
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Neumann Problem ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Hermitian Manifold ◽  
Neumann Boundary Conditions ◽  
Compact Manifolds ◽  
Elliptic Boundary Value ◽  
Image Position

Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

scholarly journals Nonresonance conditions on the potential for a nonlinear nonautonomous Neumann problem

2019 ◽  
Vol 38 (3) ◽  
pp. 79-96 ◽  
Author(s):  
Ahmed Sanhaji ◽  
A. Dakkak
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Elliptic Problems ◽  
Neumann Boundary ◽  
Existence Results ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
The First Eigenvalue

The aim of this paper is to establish the existence of the principal eigencurve of the p-Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p-Laplacian operator.

scholarly journals On a nonlinear boundary value problem involving a small parameter

1962 ◽  
Vol 2 (4) ◽  
pp. 425-439 ◽  
Author(s):  
A. Erdéyi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Boundary Value Problem ◽  
Image Position

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

scholarly journals Systems of differential equations with fully nonlinear boundary conditions

1997 ◽  
Vol 56 (2) ◽  
pp. 197-208 ◽  
Author(s):  
H.B. Thompson
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Nonlinear Boundary Conditions ◽  
Fully Nonlinear ◽  
Systems Of Differential Equations ◽  
Special Cases

We give sufficient conditions involving f, g and ω in order that systems of differential equations of the form y″ = f(x, y, y′), x in [0, 1] with fully nonlinear boundary conditions of the form g((y(0), y(1)), (y′(0), y′(1))) = 0 have solutions y with (x, y) in . We use Schauder degree theory in a novel space. Well known existence results for the Picard, the periodic and the Neumann boundary conditions follow as special cases of our results.

scholarly journals INTERVAL EXTENSION STRUCTURE BASIC TYPES OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

2018 ◽  
pp. 10-14
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Interval Function ◽  
The Third ◽  
Interval Extension ◽  
The Neumann Problem

. In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the second order of making the basic operations that compose interval arithmetic is developed. For the differential equation (1) of the type, when constructing the interval expansion of the structure of the formula, structural formulas were used to construct with the Rfunction method and 4 problems were studied — the Dirichlet problem, the Neumann problem, the third type problem, the mixed boundary conditions problem. For the Dirichlet problem, the solution is an interval expansion of the structure in the form (5), where 𝑃 = {𝜔𝛷 , 𝜔𝛷̅, 𝜔̅𝛷, 𝜔̅𝛷̅} и [ 𝛷, 𝛷̅]is an indefinite interval function. For the Neumann problem, a solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form. For the problem of the third type, the solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [𝛷2, 𝛷2̅̅̅̅] -indefinite, interval function, 𝐷1 - differential operator of the form (9). For the problem, mixed boundary conditions are treated. The solution In the interval extension of the structure,[ 𝛷1, 𝛷1], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form.

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