Maximal Monotone Relations Research Articles

Fitzpatrick transform of monotone relations in Hadamard spaces

Abstract In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

Stochastic homogenization of maximal monotone relations and applications

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity.

Nonlinear maximal monotone extensions of symmetric operators

Given a linear semi-bounded symmetric operator $S\ge -\omega$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_\Theta$ of type $\lambda>\omega$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $\lambda$) which coincide with the Friedrichs extension of $S$ on a convex set containing ${\mathscr D}(S)$. The extension parameter $\Theta\subset{\mathfrak h}\times{\mathfrak h}$ ranges over the set of nonlinear maximal monotone relations on an auxiliary Hilbert space $\mathfrak h$ isomorphic to the deficiency subspace of $S$. Moreover $A_\Theta+\lambda$ is a sub-potential operator (i.e. is the sub-differential of a lower semicontinuos convex function) whenever $\Theta$ is sub-potential. Examples describing Laplacians with nonlinear singular perturbations supported on null sets and Laplacians with nonlinear boundary conditions on a bounded set are given.

Random variables, monotone relations, and convex analysis

Random variables can be described by their cumulative distribution functions, a class of nondecreasing functions on the real line. Those functions can in turn be identified, after the possible vertical gaps in their graphs are filled in, with maximal monotone relations. Such relations are known to be the subdifferentials of convex functions. Analysis of these connections yields new insights. The generalized inversion operation between distribution functions and quantile functions corresponds to graphical inversion of monotone relations. In subdifferential terms, it corresponds to passing to conjugate convex functions under the Legendre---Fenchel transform. Among other things, this shows that convergence in distribution for sequences of random variables is equivalent to graphical convergence of the monotone relations and epigraphical convergence of the associated convex functions. Measures of risk that employ quantiles (VaR) and superquantiles (CVaR), either individually or in mixtures, are illuminated in this way. Formulas for their calculation are seen from a perspective that reveals how they were discovered. The approach leads further to developments in which the superquantiles for a given distribution are interpreted as the quantiles for an overlying "superdistribution." In this way a generalization of Koenker---Basset error is derived which lays a foundation for superquantile regression as a higher-order extension of quantile regression.

A note on elliptic type boundary value problems with maximal monotone relations

In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as “coefficients”. A reformulation of the respective problems is constructed such that they turn out to be unitarily equivalent to inverting a maximal monotone relation in a Hilbert space. The method is based on the idea of “tailor‐made” distributions as provided by the construction of extrapolation spaces, see e.g. [Picard, McGhee: Partial Differential Equations: A unified Hilbert Space Approach (De Gruyter, 2011)]. The abstract framework is illustrated by various examples.

Well-posedness of non-autonomous evolutionary inclusions

A class of non-autonomous differential inclusions in a Hilbert space setting is considered. The well-posedness for this class is shown by establishing the mappings involved as maximal monotone relations. Moreover, the causality of the so established solution operator is addressed. The results are exemplified by the equations of thermoplasticity with time dependent coefficients and by a non-autonomous version of the equations of viscoplasticity with internal variables.

Scale-transformations and homogenization of maximal monotone relations with applications

In homogenization, two-scale models arise, e.g., by applying Nguetseng's notion of two-scale convergence to non- linear PDEs. A homogenized single-scale problem may then be derived via scale-transformations. A variational formulation due to Fitzpatrick is here used for the scale-integration of two-scale maximal monotone relations, and for the converse opera- tion of scale-disintegration. These results are applied to the periodic homogenization of a quasilinear model of Ohmic electric conduction with Hall effect: � E ∈ � α( � J, x/e) + h(x/e) � J × � B(x/e) + � Ea(x/e), ∇× � E = � g (x/e), ∇· � J = 0i n Ω, with � α (·, x/e) maximal monotone, � B, � Ea, h, � g prescribed fields. (This corresponds to a quasilinear second-order elliptic equa- tion in curl form: ∇× �( ∇× u, x/e) = � g (x/e).) This result is also retrieved via De Giorgi's Γ -convergence.

Structural stability of rate-independent nonpotential flows

Several phenomena may be represented by doubly-nonlinear equations of the form $$ \alpha(D_tu) - \nabla\cdot \gamma(\nabla u)\ni h, $$ with $\alpha$ and $\gamma$ (possibly multivalued) maximal monotone mappings. Hysteresis effects are characterized by rate-independence, which corresponds to $\alpha$ positively homogeneous of zero degree. &nbsp Fitzpatrick showed that any maximal monotone relation may be represented variationally. On this basis, an initial- and boundary-value problem associated to the equation above is here formulated as a null-minimization problem, without assuming $\gamma$ to be cyclically monotone. Existence of a solution $u\in H^1(0,T; H^1(\Omega))$ is proved, as well as its stability with respect to variations of the data, of the mapping $\gamma$, and of the domain $\Omega$.

Scale-Transformations in the Homogenization of Nonlinear Magnetic Processes

A class of nonlinear first-order processes is formulated as a minimization principle. In the presence of oscillating data, a two-scale model is then derived, via Nguetseng’s notion of two-scale convergence. The dependence on the fine-scale variable is eliminated by averaging with respect to the fine-scale (scale-integration or upscaling); conversely, any two-scale solution is retrieved from a coarse-scale one (scale-disintegration or downscaling). These results are first developed in a general functional framework, and are then applied to the homogenization of a relaxation dynamics in magnetic composites: A(x/e) ∂ Be ∂t + α(Be,x/e) � He ∇· Be = 0, ∇× He = J (x) ∀ e> 0. Here J is a prescribed current density.A(y)is a positive-definite symmetric tensor, α(·,y) is a maximal monotone operator; both are assumed to depend periodically on the fine-scale variable y. The homogenized problem consists in coupling the magnetostatic equations with a maximal monotone relation B ∈ γ( H ) that is local in space but not in time. This scale-transformation procedure is also interpreted in terms of (single-scale) Γ -convergence.

Maximal monotone relations and the second derivatives of nonsmooth functions

Maximal monotone relations serve as a prototype from which properties can be derived for the subdifferential relations associated with convex functions, saddle functions, and other important classes of functions in nonsmooth analysis. It is shown that the Clarke tangent cone at any point of the graph of a maximal monotone relation is actually a linear subspace. This fact clarifies a number of issues concerning the generalized second derivatives of nonsmooth functions.

Characterization of the subdifferentials of convex functions

Each lower semi-continuous proper convex function / on a Banach space E defines a certain multivalued mapping df from E to E* called the subdifferential of /. It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal cyclically relations on E X E*, and that each of these is also a maximal monotone relation. Furthermore, it is proved that df determines / uniquely up to an additive constant. These facts generally fail to hold when E is not a Banach space. The proofs depend on establishing a new result which relates the directional derivatives of / to the existence of approximate subgradients.