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.

scholarly journals An extragradient-type algorithm for variational inequality on Hadamard manifolds

2020 ◽  
Vol 26 ◽  
pp. 63 ◽  
Author(s):  
E.E.A. Batista ◽  
G.C. Bento ◽  
O.P. Ferreira
Keyword(s):  
Variational Inequality ◽  
Vector Field ◽  
Vector Fields ◽  
Extragradient Method ◽  
Maximal Monotone ◽  
Hadamard Manifolds ◽  
Convergence Properties ◽  
Type Algorithm ◽  
Monotone Vector Fields ◽  
Maximal Monotone Vector Fields

This paper presents an extragradient method for variational inequality associated with a point-to-set vector field in Hadamard manifolds, and a study of its convergence properties. To present our method, the concept of ϵ-enlargement of maximal monotone vector fields is used, and its lower-semicontinuity is established to obtain the method convergence in this new context.

scholarly journals NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

2010 ◽  
Vol 83 (1) ◽  
pp. 22-29 ◽  
Author(s):  
FILOMENA CIANCIARUSO ◽  
GIUSEPPE MARINO ◽  
LUIGI MUGLIA ◽  
HONG-KUN XU
Keyword(s):  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Null Sequence ◽  
Graph Convergence ◽  
Common Domain

AbstractWe construct a sequence {An} of maximal monotone operators with a common domain and converging, uniformly on bounded subsets, to another maximal monotone operator A; however, the sequence {t−1nAn} fails to graph-converge for some null sequence {tn}.

scholarly journals Fundamental tone estimates for elliptic operators in divergence form and geometric applications

2006 ◽  
Vol 78 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Gregório P. Bessa ◽  
Luquésio P. Jorge ◽  
Barnabé P. Lima ◽  
José F. Montenegro
Keyword(s):  
Lower Bounds ◽  
Vector Fields ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Fundamental Tone ◽  
Space Forms ◽  
Closed Hypersurfaces ◽  
Laplace Eigenvalues ◽  
Cheeger’S Constant ◽  
Lr Operator

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

Invariant monotone vector fields on Riemannian manifolds

2009 ◽  
Vol 70 (5) ◽  
pp. 1850-1861 ◽  
Author(s):  
A. Barani ◽  
M.R. Pouryayevali
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Monotone Vector Fields
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