A unified approach for embedded boundary conditions for fourth-order elliptic problems

2014 ◽  
Vol 104 (7) ◽  
pp. 655-675 ◽  
Author(s):  
Isaac Harari ◽  
Eran Grosu
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Elliptic Problems ◽  
Unified Approach ◽  
Embedded Boundary

scholarly journals Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

scholarly journals A Paneitz–Branson type equation with Neumann boundary conditions

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Denis Bonheure ◽  
Hussein Cheikh Ali ◽  
Robson Nascimento
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Type Equation ◽  
Neumann Boundary ◽  
Rayleigh Quotient ◽  
Neumann Boundary Conditions ◽  
Asymptotic Estimates ◽  
Best Constant ◽  
Order Elliptic Problem ◽  
Nonconstant Solution

AbstractWe consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.

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