scholarly journals A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 850 ◽  
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Type Inequality ◽  
Coupled System ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Necessary Condition ◽  
Laplacian Operator ◽  
Lyapunov Type Inequality

We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals A Regularization Method for the Elliptic Equation with Inhomogeneous Source

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tuan H. Nguyen ◽  
Binh Thanh Tran
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Regularization Method ◽  
Dirichlet Boundary ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Ill Posed ◽  
Sharp Error

We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x=0 and x=π. The problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.

scholarly journals Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 647
Author(s):  
Michał Bełdziński ◽  
Marek Galewski ◽  
Igor Kossowski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Non Linear ◽  
The Stability

We study the stability and the solvability of a family of problems −(ϕ(x′))′=g(t,x,x′,u)+f* with Dirichlet boundary conditions, where ϕ, u, f* are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted.

scholarly journals Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator

2017 ◽  
Vol 15 (1) ◽  
pp. 1075-1089 ◽  
Author(s):  
Mohsen Khaleghi Moghadam ◽  
Johnny Henderson
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Local Minimum ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Discrete Problem ◽  
Dirichlet Boundary Value Problem ◽  
One Dimensional ◽  
Minimum Theorem

Abstract Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

Dual fat boundary method: the fat boundary method in elasticity with an extension of the application scope

2020 ◽  
pp. 108128652096449
Author(s):  
Kui Liu ◽  
Ang Zhao ◽  
Zhendong Hu
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Dirichlet Boundary ◽  
Mathematical Proof ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Stress Concentrations ◽  
Conditional Convergence ◽  
Boundary Method ◽  
Wide Range

The fat boundary method (FBM) is a fictitious domain method, proposed to solve Poisson problems in a domain with small perforations. It can achieve higher accuracy around holes, which makes it very suitable to solve elasticity problems because stress concentrations often appear around holes. However, there are some strict restrictions of the FBM limiting the wide range of applications. For example, the original FBM deals with perforated rectangular domain with only Dirichlet boundary conditions. Furthermore, because the global domain is extended to the holes, analytical solutions in holes corresponding to the Dirichlet boundary conditions around holes are required. This limits both the boundary conditions around holes and the shape of holes, because for arbitrary holes it is difficult to get the analytical solutions. This article makes an attempt to break these limitations and apply the FBM to elasticity. Firstly, we review the FBM and introduce Neumann boundary conditions to the rectangular domain. A mathematical proof of the conditional convergence of the algorithm is presented. Furthermore, the FBM is compared with the Lagrange multiplier method to clarify that the FBM is one kind of weak imposition method. Then we apply the FBM to linear elasticity and the dual fat boundary method is proposed to solve problems without analytical solutions in holes. Some numerical examples are presented to verify the method proposed here.

Sign in / Sign up

Export Citation Format

Share Document