Local Variational Principle Research Articles

A local variational principle for fracture

The seminal paper of Francfort and Marigo (1998) introduced a variational formulation for Griffith fracture (Griffith, 1920) that has resulted in substantial theoretical and practical progress in modeling and simulating fracture. In particular, it led to the phase-field approximation proposed in Bourdin et al. (2000), which has been widely implemented. However, the formulation in Francfort and Marigo (1998) is known to have limitations, including its inability to treat applied loads and its reliance on global minimization. In addition, the phase-field model (Bourdin et al., 2000) and its extensions, as implemented, are not generally approximations of the global minimizers in Francfort and Marigo (1998). In this paper, we show that there is a local variational principle satisfied by global and local minimizers of the energy introduced in Francfort and Marigo (1998), which is compatible with loads, and which is a generalization of the stress intensity factor. We use this principle to reformulate variational fracture, including formulations that, for the first time, can include all forms of applied loads. We conclude by showing the connection between phase-field models, as implemented, and our formulations.

A local conditional variational principle of pressures and local conditional equilibrium states

In this article, we introduce the notions of topological conditional pressures and two measure-theoretical conditional pressures of a finite open cover conditioned by a fixed finite measurable partition. We establish a local conditional variational principle and show that local conditional pressures determine local conditional measure-theoretical entropies. As applications, we study properties of both local and global conditional equilibrium states for continuous, real-valued functions.

The Local Variational Principle of Weighted Entropy and Its Applications

The Local Variational Principle of Weighted Entropy and Its Applications

Metric Versus Topological Receptive Entropy of Semigroup Actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.

To continue the optical-mechanical analogy

On the basis of the V-function method, an optical-mechanical analogy is carried out; and further it receives a new continuation. In particular, the equality of the phase velocity of the wave and the velocity of the particle is shown. It is noted that according to its physical meaning the wave characterizes the properties of the action that manifests itself in the motion of the particle. The V-function method is based on the concept of a process-state and it consists of a local variational principle, a new formulation of the direct and inverse problem of dynamics. According to the V-function method, the trajectory motion of an object is associated with its wave motion. In this case it is assumed that there is an explicit connection between the trajectory and wave equations of the object. The application of the V-function method for the case of rectilinear motion of an object is shown.

Local unstable entropies of partially hyperbolic diffeomorphisms

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

Local conditional entropy in measure for covers with respect to a fixed partition

In this paper we introduce two measure theoretical notions of conditional entropy for finite measurable covers conditioned to a finite measurable partition and prove that they are equal. Using this we state a local variational principle with respect to the notion of conditional entropy defined by Misiurewicz (1976 Stud. Math. 55 176–200) for the case of open covers. This in particular extends the work done in Romagnoli (2003 Ergod. Theor. Dynam. Syst. 23 1601–10), Glasner and Weiss (2006 Handbook of Dynamical Systems vol 1B (Amsterdam: Elsevier)) and Huang et al (2006 Ergod. Theor. Dynam. Syst. 26 219–45).

Lie symmetry analysis, conservation laws and analytical solutions for the constant astigmatism equation

In this paper, the constant astigmatism equation is investigated, which finds numerous applications in geometry and physics describing surfaces of constant astigmatism. Utilizing a set of non-singular local multipliers, we present a set of local conservation laws for the equation, based on which, we further find a tree of nonlocally related PDE systems. Moreover, by considering these nonlocally related PDE systems, we systematically present Lie symmetries, optimal systems and some interesting analytical solutions of the constant astigmatism equation. Here it is the first time to investigate the nonlocally related PDE systems for this model, which can be used to expand the solution space of the given PDE system. Finally, by using its local conservation laws and variational principle, we obtain four kinds of Lagrangians.

Topological obstructions in Lagrangian field theories, with an application to 3D Chern–Simons gauge theory

We relate the existence of Noether global conserved currents associated with locally variational field equations to the existence of global solutions for a local variational problem generating global equations. Both can be characterized as the vanishing of certain cohomology classes. In the case of a 3-dimensional Chern–Simons gauge theory, the variationally featured cohomological obstruction to the existence of global solutions is sharp and equivalent to the usual obstruction in terms of the Chern characteristic class for the flatness of a principal connection. We suggest a parallelism between the geometric interpretation of characteristic classes as obstruction to the existence of flat principal connections and the interpretation of certain de Rham cohomology classes to be the obstruction to the existence of global extremals for a local variational principle.

A LOCAL VARIATIONAL PRINCIPLE FOR RANDOM BUNDLE TRANSFORMATIONS

We introduce local topological entropy [Formula: see text] and two kinds of local measure-theoretic entropy [Formula: see text] and [Formula: see text] for random bundle transformations. We derive a variational inequality of random local entropy for [Formula: see text]. As an application of such relation we prove a local variational principle in random dynamical system.

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.

Topological pressure for sub-additive potentials of amenable group actions

We introduce the topological pressure for any sub-additive potentials of a countable discrete amenable group action and any given open cover, and establish a local variational principle for it.

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.

The local variational principle of topological pressure for sub-additive potentials

Let ( X , T ) be a topological dynamical system and F = { f n } n = 1 ∞ be a sub-additive potential on C ( X , R ) . Let U be an open cover of X . Then for any T -invariant measure μ , let F ∗ ( μ ) = lim n → ∞ 1 n ∫ f n d μ . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle: P ( T , F , U ) = sup μ { h μ ( T , U ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } where h μ ( T , U ) denotes the measure-theoretic entropy of μ relative to U and the supremum can be attained by a T -invariant ergodic measure. The main purpose of this paper is to generalize a result of Huang and Yi (2007) [17]. In the paper [17], they proved the local variational principle of pressure for additive potentials. Furthermore, we prove the result P ( T , F ) = lim d i a m ( U ) → 0 P ( T , F ; U ) . Moreover, we obtained P ( T , F ) = sup μ { h μ ( T ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } , which gives another proof of the topological pressure variational principle for sub-additive potentials from Cao et al. (2008) [14].

A relative local variational principle for topological pressure

A relative local variational principle for topological pressure

A relative local variational principle for topological pressure

We define the relative local topological pressure for any given factor map and open cover, and prove the relative local variational principle of this pressure. More precisely, for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover U of X, a continuous, real-valued function f on X and an S-invariant measure ν on Y, we show that the corresponding relative local pressure P(T, f, U, y) satisfies $$ \mathop {\sup }\limits_{\mu \in \mathcal{M}(X,T)} \left\{ {h_\mu (T,\mathcal{U}|Y) + \int_X {f(x)d\mu (x):\pi \mu = \nu } } \right\} = \int_Y {P(T,f,\mathcal{U},y)d\nu (y)} , $$ where \( \mathcal{M} \)(X, T) denotes the family of all T-invariant measures on X. Moreover, the supremum can be attained by a T-invariant measure.

Variational principles of pressure

Given a topological dynamical system $(X, T)$, a Borel cover$\mathcal{U}$ of $X$ and a sub-additive sequence $\mathcal{F}$ ofreal-valued continuous functions on $X$, two notions ofmeasure-theoretical pressure $P_\mu^- (T, \mathcal{U}, \mathcal{F})$and $P_\mu^+ (T, \mathcal{U}, \mathcal{F})$ for an invariant Borelprobability measure $\mu$ are introduced. When $\mathcal{U}$ is anopen cover, a local variational principle between topological andmeasure-theoretical pressure is proved; it is also established theupper semi-continuity of P•+$(T, \mathcal{U}, \mathcal{F})$and P•+$(T, \mathcal{U}, \mathcal{F})$ on the space of allinvariant Borel probability measures. The notions ofmeasure-theoretical pressure $P_\mu^- (T, X, \mathcal{F})$ and$P_\mu^+ (T, X, \mathcal{F})$ for an invariant Borel probabilitymeasure $\mu$ are also introduced. A global variational principlebetween topological and measure-theoretical pressure is alsoobtained.

A local variational principle of pressure and its applications to equilibrium states

We prove a local variational principle of pressure for any given open cover. More precisely, for a given dynamical system (X, T), an open cover \(\mathcal{U}\) of X, and a continuous, real-valued function f on X, we show that the corresponding local pressure P(T, f; \(\mathcal{U}\)) satisfies $$P(T,f;\mathcal{U}) = \sup \left\{ {h_\mu (T,\mathcal{U}) + \int_X {f(x)\mu (x):\mu } is a T - invariant measure} \right\}$$ , moreover, the supremum can be attained by a T-invariant ergodic measure. By establishing the upper semi-continuity and affinity of the entropy map relative to an open cover, we further show that $$P(T,f;\mathcal{U}) = \sup \left\{ {h_\mu (T,\mathcal{U}) + \int_X {f(x)\mu (x):\mu } is a T - invariant measure} \right\}$$ for any T-invariant measure μ of (X, T), i.e., local pressures determine local measure-theoretic entropies. As applications, properties of both local and global equilibrium states for a continuous, real-valued function are studied.

Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

A local variational principle for conditional entropy

For a given factor map is an open cover. The relative variational principle is a consequence of the results.