scholarly journals Nonlocal characterizations of variable exponent Sobolev spaces

2021 ◽  
pp. 1-22
Author(s):  
Gianluca Ferrari ◽  
Marco Squassina
Keyword(s):  
Elasticity Theory ◽  
Nonlinear Elasticity ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Singular Limit ◽  
Electrorheological Fluids ◽  
Nonlinear Elasticity Theory ◽  
Limit Formula ◽  
Variable Exponent Sobolev Spaces ◽  
Anisotropic Case

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.

scholarly journals Nonlocal approximations to anisotropic Sobolev norms

2021 ◽  
pp. 1-20
Author(s):  
Ivan Cinelli ◽  
Gianluca Ferrari ◽  
Marco Squassina
Keyword(s):  
Image Processing ◽  
Nonlinear Elasticity ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Electrorheological Fluids ◽  
Variable Exponent Sobolev Spaces ◽  
Sobolev Norms

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent p ( x ) reaches the value 1.

Critical points approaches to a nonlocal elliptic problem driven by 𝑝(𝑥)-biharmonic operator

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shapour Heidarkhani ◽  
Shahin Moradi ◽  
Mustafa Avci
Keyword(s):  
Variational Methods ◽  
Elasticity Theory ◽  
Critical Points ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Variable Exponent ◽  
Electrorheological Fluids ◽  
Biharmonic Operator ◽  
Technical Approach ◽  
Nonlinear Elasticity Theory

Abstract Differential equations with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids. We study the existence of at least three weak solutions for the nonlocal elliptic problems driven by p ⁢ ( x ) p(x) -biharmonic operator. Our technical approach is based on variational methods. Some applications illustrate the obtained results. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

Influence of Distributed Dislocations on Stability of Hollow Nonlinearly Elastic Sphere

2020 ◽  
pp. 17-21
Author(s):  
Evgeniya V. Goloveshkina
Keyword(s):  
Boundary Value Problem ◽  
Elasticity Theory ◽  
Elastic Body ◽  
Nonlinear Elasticity ◽  
External Pressure ◽  
Elastic Sphere ◽  
Unperturbed State ◽  
Nonlinear Elasticity Theory ◽  
Edge Dislocations ◽  
Distributed Dislocations

The phenomenon of stability loss of a hollow elastic sphere containing distributed dislocations and loaded with external hydrostatic pressure is studied. The study was carried out in the framework of the nonlinear elasticity theory and the continuum theory of continuously distributed dislocations. An exact statement and solution of the stability problem for a three-dimensional elastic body with distributed dislocations are given. The static problem of nonlinear elasticity theory for a body with distributed dislocations is reduced to a system of equations consisting of equilibrium equations, incompatibility equations with a given dislocation density tensor, and constitutive equations of the material. The unperturbed state is caused by external pressure and a spherically symmet-ric distribution of dislocations. For distributed edge dislocations in the framework of a harmonic (semi-linear) mate-rial model, the unperturbed state is defined as an exact spherically symmetric solution to a nonlinear boundary value problem. This solution is valid for any function that characterizes the density of edge dislocations. The perturbed equilibrium state is described by a boundary value problem linearized in the neighborhood of the equilibrium. The analysis of the axisymmetric buckling of the sphere was performed using the bifurcation method. It consists in determining the equilibrium positions of an elastic body, which differ little from the unperturbed state. By solving the linearized problem, the value of the external pressure at which the sphere first loses stability is found. The effect of dislocations on the buckling of thin and thick spherical shells is analyzed.

scholarly journals Nonlocal Variational Principles with Variable Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yongqiang Fu ◽  
Miaomiao Yang
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Variable Exponent ◽  
Variational Principles ◽  
Young Measures ◽  
Bounded Set ◽  
Weak Lower Semicontinuity ◽  
Variable Exponent Sobolev Spaces ◽  
Regular Open

This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.

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