A variational theory for monotone vector fields
نویسندگان
چکیده
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such a vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian LT on phase space X ×X ∗ that can be seen as a “potential” for T , in the sense that the problem of inverting T reduces to minimizing the convex energy LT . This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods –computational or not– that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish many old and new results concerned with the identification, superposition, and resolution of such vector fields.
منابع مشابه
Existence solutions for new p-Laplacian fractional boundary value problem with impulsive effects
Fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, engineering, etc. Differential equations with impulsive effects arising from the real world describe the dyn...
متن کاملMaximal monotone operators are selfdual vector fields and vice-versa
If L is a selfdual Lagrangian L on a reflexive phase space X ×X∗, then the vector field x → ∂̄L(x) := {p ∈ X∗; (p, x) ∈ ∂L(x, p)} is maximal monotone. Conversely, any maximal monotone operator T on X is derived from such a potential on phase space, that is there exists a selfdual Lagrangian L on X ×X∗ (i.e, L∗(p, x) = L(x, p)) such that T = ∂̄L. This solution to problems raised by Fitzpatrick can...
متن کاملDecomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators
The majority of First Order methods for large-scale convex-concave saddle point problems and variational inequalities with monotone operators are proximal algorithms which at every iteration need to minimize over problem’s domain X the sum of a linear form and a strongly convex function. To make such an algorithm practical, X should be proximal-friendly – admit a strongly convex function with e...
متن کاملA Relaxed Extra Gradient Approximation Method of Two Inverse-Strongly Monotone Mappings for a General System of Variational Inequalities, Fixed Point and Equilibrium Problems
متن کامل
Variational representations for N -cyclically monotone vector fields
Given a convex bounded domain Ω in R and an integer N ≥ 2, we associate to any jointly Nmonotone (N − 1)-tuplet (u1, u2, ..., uN−1) of vector fields from Ω into R, a Hamiltonian H on R × R...×R, that is concave in the first variable, jointly convex in the last (N − 1) variables such that for almost all x ∈ Ω, (u1(x), u2(x), ..., uN−1(x)) = ∇2,...,NH(x, x, ..., x). Moreover, H is N -sub-antisymm...
متن کامل