scholarly journals Asymptotics and Uniqueness of Solutions of the Elasticity System with the Mixed Dirichlet–Robin Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2241
Author(s):  
Hovik A. Matevossian
Keyword(s):  
Generalized Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Robin Boundary Conditions ◽  
Uniqueness Theorems ◽  
Robin Problem ◽  
Uniqueness Of Solutions ◽  
Main Method ◽  
Elasticity System ◽  
Robin Boundary ◽  
Admissible Functions

We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the weight |x|a is finite. Depending on the value of the parameter a, we have proved uniqueness (or non-uniqueness) theorems for the mixed Dirichlet–Robin problem, and also given exact formulas for the dimension of the space of solutions. The main method for studying the problem under consideration is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions.

scholarly journals On the Mixed Dirichlet–Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

2019 ◽  
Vol 24 (1) ◽  
pp. 25 ◽  
Author(s):  
Hovik Matevossian
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Boundary ◽  
Unbounded Domains ◽  
Generalized Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Compact Set ◽  
Uniqueness Theorems ◽  
Dirichlet Integral ◽  
Elliptic Boundary Value ◽  
Biharmonic Problems

We studied the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we studied the unique solvability of the mixed Dirichlet–Steklov-type and Steklov-type biharmonic problems in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight | x | a . Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

scholarly journals Mixed Boundary Value Problems for the Elasticity System in Exterior Domains

2019 ◽  
Vol 24 (2) ◽  
pp. 58
Author(s):  
Hovik A. Matevossian
Keyword(s):  
Asymptotic Behavior ◽  
Variational Principle ◽  
Mixed Boundary ◽  
Asymptotic Behavior Of Solutions ◽  
Compact Set ◽  
Uniqueness Theorems ◽  
Energy Integral ◽  
Exterior Domains ◽  
Mixed Problems ◽  
Elasticity System

We study the properties of solutions of the mixed Dirichlet–Robin and Neumann–Robin problems for the linear system of elasticity theory in the exterior of a compact set and the asymptotic behavior of solutions of these problems at infinity under the assumption that the energy integral with weight | x | a is finite for such solutions. We use the variational principle and depending on the value of the parameter a, obtain uniqueness (non-uniqueness) theorems of the mixed problems or present exact formulas for the dimension of the space of solutions.

scholarly journals ON AN EIGENVALUE PROBLEM INVOLVING THE HARDY POTENTIAL

2010 ◽  
Vol 12 (06) ◽  
pp. 953-975 ◽  
Author(s):  
J. CHABROWSKI ◽  
I. PERAL ◽  
B. RUF
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Principal Eigenvalue ◽  
Robin Boundary Conditions ◽  
Robin Problem ◽  
Hardy Potential ◽  
High Dimensions ◽  
The Neumann Problem ◽  
Robin Boundary ◽  
Second Eigenvalue

In this note we consider the eigenvalue problem for the Laplacian with the Neumann and Robin boundary conditions involving the Hardy potential. We prove the existence of eigenfunctions of the second eigenvalue for the Neumann problem and of the principal eigenvalue for the Robin problem in "high" dimensions.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

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