Monotone Vector Fields Research Articles

Capacity solution to a coupled system of parabolic\u2013elliptic equations in Orlicz\u2013Sobolev spaces

The existence of a capacity solution to a coupled nonlinear parabolic–elliptic system is analyzed, the elliptic part in the parabolic equation being of the form -,mathrm{div}, a(x,t,u,nabla u). The growth and the coercivity conditions on the monotone vector field a are prescribed by an N-function, M, which does not have to satisfy a Delta _2 condition. Therefore we work with Orlicz–Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.

Existence results for a non-linear obstacle parabolic semicoercive problems with lower order term

ABSTRACTIn this paper, we shall be concerned with the existence result of a non-linear parabolic equations associated to the form, , where the right-hand side and the initial data belong to and the lower order term is a non-coercive Carathéodory function. The growth and coercivity conditions on the monotone vector field a are prescribed by an N-function M, which does not have to satisfy the condition. Therefore, we work with an Orlicz–Sobolev spaces which are not necessarily reflexive.

Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider−divA(x,∇u)=f∈L1(Ω), on a Lipschitz bounded domain in RN. The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate M⁎ (neither Δ2, nor ∇2). The condition we impose is log-Hölder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

In this paper, we shall be concerned with the existence result of the following problem, 0.1 $$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\left( a(x,u,\nabla u)\right) -\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ with the second term f belongs to $$L^1(\Omega )$$ . The growth and the coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M. We assume any restriction on M, therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term $$\Phi $$ is a Caratheodory function satisfying only a growth condition.

Existence of renormalized solution of nonlinear elliptic problems with lower order term in Orlicz spaces

In this paper, we shall be concerned with the existence of renormalized solution of the following problem, $$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\Big (a(x,u,\nabla u)\Big )-\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ with the second term f belongs to \(L^1(\Omega )\). The growth and the coercivity conditions on the monotone vector field a are prescribed by a N-function M. We assume any restriction on M, therefore we work with Orlicz-Sobolev spaces which are not necessarily reflexive. The lower order term \(\Phi \) is a Caratheodory function which is not coercive.

Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds

In this paper an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced and studied its convergence properties. The main tool used for presenting the method is the concept of enlargement of monotone vector fields, which generalizes the concept of enlargement of monotone operators from the linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained.

Metric selfduality and monotone vector fields on manifolds

We develop a “metrically selfdual” variational calculus for c-monotone vector fields between general manifolds X and Y, where c is a coupling on X×Y. Remarkably, many of the key properties of classical monotone operators known to hold in a linear context extend to this non-linear setting. This includes an integral representation of c-monotone vector fields in terms of c-convex selfdual Lagrangians, their characterization as a partial c-gradients of antisymmetric Hamiltonians, as well as the property that these vector fields are generically single-valued. We also use a symmetric Monge–Kantorovich transport to associate to any measurable map its closest possible c-monotone “rearrangement”. We also explore how this metrically selfdual representation can lead to a global variational approach to the problem of inverting c-monotone maps, an approach that has proved efficient for resolving non-linear equations and evolutions driven by monotone vector fields in a Hilbertian setting.

Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces

We will present the proof of existence of renormalized solutions to a nonlinear parabolic problem ∂tu−diva(⋅,Du)=f with right-hand side f and initial data u0 in L1. The growth and coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M which is anisotropic and inhomogeneous with respect to the space variable. In particular, M does not have to satisfy an upper growth bound described by a Δ2-condition. Therefore we work with generalized Musielak–Orlicz spaces which are not necessarily reflexive. Moreover we provide a weak sequential stability result for a more general problem: ∂tβ(⋅,u)−div(a(⋅,Du)+F(u))=f, where β is a monotone function with respect to the second variable and F is locally Lipschitz continuous. Within the proof we use truncation methods, Young measure techniques, the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive Musielak–Orlicz spaces.

Non-existence of strictly monotone vector fields on certain Riemannian manifolds

The main goal of this paper is to prove that Riemannian manifolds that admit monotone vector fields that satisfy a weaker condition than strictly monotone has infinite volume, in a sense, generalizing a result that Bishop and O’Neill proved for convex functions.

Variational representations forN-cyclically monotone vector fields

Given a convex bounded domain $\Omega $ in ${{\mathbb{R}}}^{d}$ and an integer $N\geq 2$, we associate to any jointly $N$-monotone $(N-1)$-tuplet $(u_1, u_2,..., u_{N-1})$ of vector fields from $% \Omega$ into $\mathbb{R}^{d}$, a Hamiltonian $H$ on ${\mathbb{R}}^{d} \times {\mathbb{R}}^{d} ... \times {\mathbb{R}}^{d}$, that is concave in the first variable, jointly convex in the last $(N-1)$ variables such that for almost all $% x\in \Omega$, \hbox{$(u_1(x), u_2(x),..., u_{N-1}(x))= \nabla_{2,...,N} H(x,x,...,x)$. Moreover, $H$ is $N$-sub-antisymmetric, meaning that $\sum% \limits_{i=0}^{N-1}H(\sigma ^{i}(\mathbf{x}))\leq 0$ for all $\mathbf{x}% =(x_{1},...,x_{N})\in \Omega ^{N}$, $\sigma $ being the cyclic permutation on ${\mathbb{R}}^{d}$ defined by $\sigma (x_{1},x_2,...,x_{N})=(x_{2},x_{3},...,x_{N},x_{1})$. Furthermore, $H$ is $N$% -antisymmetric in a sense to be defined below. This can be seen as an extension of a theorem of E. Krauss, which associates to any monotone operator, a concave-convex antisymmetric saddle function. We also give various variational characterizations of vector fields that are almost everywhere $N$-monotone, showing that they are dual to the class of measure preserving $N$-involutions on $\Omega$.

Rate of convergence for proximal point algorithms on Hadamard manifolds

In this paper, an estimate of convergence rate concerned with an inexact proximal point algorithm for the singularity of maximal monotone vector fields on Hadamard manifolds is discussed. We introduce a weaker growth condition, which is an extension of that of Luque from Euclidean spaces to Hadamard manifolds. Under the growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. The main results presented in this paper generalize and improve some corresponding known results.

Concepts and techniques of optimization on the sphere

In this paper some concepts and techniques of Mathematical Programming are extended in an intrinsic way from the Euclidean space to the sphere. In particular, the notion of convex functions, variational problem and monotone vector fields are extended to the sphere and several characterizations of these notions are shown. As an application of the convexity concept, necessary and sufficient optimality conditions for constrained convex optimization problems on the sphere are derived.

ON THE CONVERGENCE OF INEXACT PROXIMAL POINT ALGORITHM ON HADAMARD MANIFOLDS

In this paper we consider the proximal point algorithm to approximate a singularity of a multivalued monotone vector field on a Hadamard manifold. We study the convergence of the sequence generated by an inexact form of the algorithm. Our results extend the results of [3, 25] to Hadamard manifolds as well as the main result of [11] with more general assumptions on the control sequence. We also give some application to optimization.

An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds

In this paper, an inexact proximal point algorithm concerned with the singularity of maximal monotone vector fields is introduced and studied on Hadamard manifolds, in which a relative error tolerance with squared summable error factors is considered. It is proved that the sequence generated by the proposed method is convergent to a solution of the problem. Moreover, an application to the optimization problem on Hadamard manifolds is given. The main results presented in this paper generalize and improve some corresponding known results given in the literature.

Some questions of qualitative theory in dynamics of systems with the variable dissipation

In this work, we consider some problems of the qualitative theory of ordinary differential equations; the study of dissipative systems, as well as variable dissipation system considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and in the oscillation theory, depends on solutions of these problems. We consider such problems as existence and uniqueness problems for trajectories having infinitely remote points as limit sets for systems on the plane, elements of qualitative theory of monotone vector fields, and also existence problems for families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system and the comparison system to the many-dimensional case.

Modified proximal point algorithms on Hadamard manifolds

Given a maximal monotone vector field A on a Hadamard manifold, we devise a modified proximal point algorithm which, under reasonable assumptions, converges to a singular point of A. This algorithm is a natural extension to the Hadamard manifold of the one introduced by Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), pp. 240–256 in Hilbert spaces.

Homogenization of Maximal Monotone Vector Fields via Selfdual Variational Calculus

Abstract We 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.

Resolvents of Set-Valued Monotone Vector Fields in Hadamard Manifolds

Firmly nonexpansive mappings are introduced in Hadamard manifolds, a particular class of Riemannian manifolds with nonpositive sectional curvature. The resolvent of a set-valued vector field is defined in this setting and by means of this concept, a strong relationship between monotone vector fields and firmly nonexpansive mappings is established. This fact is then used to prove that the resolvent of a maximal monotone vector field has full domain. The Yosida approximation of a set-valued vector field is also introduced, analyzing its properties from which the asymptotic behavior of the resolvent is studied. Regarding the singularities of a set-valued monotone vector field, existence results are proved under certain boundary condition. As a consequence, the existence of fixed points for continuous pseudo-contractive mappings is obtained.

Monotone vector fields and the proximal point algorithm on Hadamard manifolds

The maximal monotonicity notion in Banach spaces is extended to Riemannian manifolds of nonpositive sectional curvature, Hadamard manifolds, and proved to be equivalent to the upper semicontinuity. We consider the problem of finding a singularity of a multivalued vector field in a Hadamard manifold and present a general proximal point method to solve that problem, which extends the known proximal point algorithm in Euclidean spaces. We prove that the sequence generated by our method is well defined and converges to a singularity of a maximal monotone vector field, whenever it exists. Applications in minimization problems with constraints, minimax problems and variational inequality problems, within the framework of Hadamard manifolds, are presented.

A variational theory for monotone vector fields

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 vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L T on the 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 a convex energy functional derived from L T . 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.