scholarly journals Γ-convergence for functionals depending on vector fields. I. Integral representation and compactness

2020 ◽  
Vol 139 ◽  
pp. 109-142 ◽  
Author(s):  
A. Maione ◽  
A. Pinamonti ◽  
F. Serra Cassano
Keyword(s):  
Integral Representation ◽  
Vector Fields ◽  
Γ Convergence

scholarly journals Homogenization of Maximal Monotone Vector Fields via Selfdual Variational Calculus

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Nassif Ghoussoub ◽  
Abbas Moameni ◽  
Ramón Zárate Sáiz
Keyword(s):  
Vector Fields ◽  
Maximal Monotone ◽  
Variational Calculus ◽  
Classical Case ◽  
Divergence Form ◽  
Graph Convergence ◽  
Monotone Vector Fields ◽  
Γ Convergence ◽  
Maximal Monotone Vector Fields ◽  
Periodic Family

AbstractWe use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

2020 ◽  
Vol 69 (1) ◽  
pp. 155-163
Author(s):  
B.Zh. Omarova ◽  
Keyword(s):  
Integral Representation ◽  
Vector Fields ◽  
Homogeneous System ◽  
Differentiation Operator ◽  
Second Order ◽  
Inhomogeneous System ◽  
Representation Of Solutions ◽  
Real Analytic ◽  
Time Variables ◽  
Second Order Systems

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

MODELING OF A MEMBRANE FOR NONLINEARLY ELASTIC INCOMPRESSIBLE MATERIALS VIA GAMMA-CONVERGENCE

2006 ◽  
Vol 04 (01) ◽  
pp. 31-60 ◽  
Author(s):  
KARIM TRABELSI
Keyword(s):  
Integral Representation ◽  
Internal Energy ◽  
Elastic Membrane ◽  
Gamma Convergence ◽  
Two Dimensional ◽  
Incompressible Materials ◽  
Plate Models ◽  
Γ Convergence ◽  
Nonlinearly Elastic

In this paper, we derive nonlinearly elastic membrane plate models for hyperelastic incompressible materials using Γ-convergence arguments. We obtain an integral representation of the limit two-dimensional internal energy owing to a result of singular functionals relaxation due to Ben Belgacem [6].

scholarly journals Linear operators on Köthe spaces of vector fields

2013 ◽  
Vol 21 (2) ◽  
pp. 53-80
Author(s):  
Ion Chiţescu ◽  
Liliana Sireţchi
Keyword(s):  
Integral Representation ◽  
Representation Theorem ◽  
Vector Fields ◽  
Linear Operators ◽  
Köthe Spaces

Abstract The study of Köthe spaces of vector fields was initiated by the present authors. In this paper linear operators on these spaces are studied. An integral representation theorem is given and special types of linear operators are introduced and studied.

scholarly journals Integral representation of local functionals depending on vector fields

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Fares Essebei ◽  
Andrea Pinamonti ◽  
Simone Verzellesi
Keyword(s):  
Integral Representation ◽  
Sobolev Spaces ◽  
Vector Fields ◽  
First Order ◽  
Bounded Set

Abstract Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.

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