Relative Variational Principle Research Articles

Existence of Ordered Solutions to Quasilinear Schrödinger Equations with General Nonlinear Term

In this paper, the existence of a pair of ordered solutions for the following class of equations in (1) was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was used to prove the existence of the positive solution. The existence of the ordered solutions is finally found.

Quasi-static responses and variational principles in gradient plasticity

Gradient models have been much discussed in the literature for the study of time-dependent or time-independent processes such as visco-plasticity, plasticity and damage. This paper is devoted to the theory of Standard Gradient Plasticity at small strain. A general and consistent mathematical description available for common time-independent behaviours is presented. Our attention is focussed on the derivation of general results such as the description of the governing equations for the global response and the derivation of related variational principles in terms of the energy and the dissipation potentials.It is shown that the quasi-static response under a loading path is a solution of an evolution variational inequality as in classical plasticity. The rate problem and the rate minimum principle are revisited. A time-discretization by the implicit scheme of the evolution equation leads to the increment problem. An increment of the response associated with a load increment is a solution of a variational inequality and satisfies also a minimum principle if the energy potential is convex. The increment minimum principle deals with stables solutions of the variational inequality. Some numerical methods are discussed in view of the numerical simulation of the quasi-static response.

Energy extremum principle for charged black holes

For a set of $N$ asymptotically flat black holes with arbitrary charges and masses, all initially at rest and well separated, we prove the following extremum principle: the extremal charge configuration ($|q_i|=m_i$ for each black hole) can be derived by extremizing the total energy, for variations of the black hole apparent horizon areas, at fixed charges and fixed Euclidean separations. We prove this result through second order in an expansion in the inverse separations. If all charges have the same sign, this result is a variational principle that reinterprets the static equilibrium of the Majumdar-Papapetrou-Hartle-Hawking solution as an extremum of total energy, rather than as a balance of forces; this result augments a list of related variational principles for other static black holes, and is consistent with the independently known Bogomol'nyi-Prasad-Sommerfield (BPS) energy minimum.

First law and a variational principle for static asymptotically Randall-Sundrum black holes

We give a new, intrinsic, mass definition for spacetimes asymptotic to the Randall-Sundrum braneworld models, RS1 and RS2. For this mass, we prove a first law for static black holes, including variations of the bulk cosmological constant, brane tensions, and RS1 interbrane distance. Our first law defines a thermodynamic volume and a gravitational tension that are braneworld analogs of the corresponding quantities in asymptotically AdS black hole spacetimes and asymptotically flat compactifications, respectively. We also prove the following related variational principle for asymptotically RS black holes: instantaneously static initial data that extremizes the mass yields a static black hole, for variations at fixed apparent horizon area, AdS curvature length, cosmological constant, brane tensions, and RS brane warp factors. This variational principle is valid with either two branes (RS1) or one brane (RS2), and is applicable to variational trial solutions.

A formula of Friedland’s entropy for Z<sub>+</sub><sup><italic>k</italic></sup>-actions on tori

In this paper, Friedland’s entropy for Z₊^{<italic>k</italic>}-actions on tori is investigated. For a Z₊^{<italic>k</italic>}-action on torus generated by pairwise different, commuting non-singular integer matrices, a formula of Friedland’s entropy is obtained via the relative variational principle for the related skew product transformation and topological pressure.

Wavelet Finite Element Method Analysis of Bending Plate Based on Hermite Interpolation

In this paper, a new kind of finite element method (FEM) is proposed, which use the two-dimensional Hermite interpolation scaling function constructed by tensor product as the basis interpolation function of field function, and then combine with the energy functional with related mechanics and variational principle, the wavelet finite element equations for solving elastic thin plate unit that constructed in this paper are derived. Then the bending problem of thin plate is solved very quickly and availably through the matlab program. The numerical example in this paper indicates the correctness and validity of this method, and has high calculation precision and convergence speed. Moreover, it also provides a reliable method to solve the free vibration problem of thin plate and the pipe crack problems.

APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

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.

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.

Hamilton’s principle applied to piezomagnetism and related variational principles

In piezomagnetism, the fundamental equations have been developed in differential form [e.g., V. I. Alshits and A. N. Darinskii, Wave Motion 15, 265–283 (1992)]. Alternatively, they may be expressed in variational form with its well-known features; this is the topic of this paper. First, the magnetic vector, that is, the gradient of the magnetic potential, is introduced [cf. the authors, Int. J. Solids Struct. 40, 4699–4706 (2003)]. Second, the sufficient conditions based on the energy argument are enumerated for a unique solution in the fundamental equations. Third, Hamilton’s principle is stated and a three-field variational principle is obtained. The principle yields only the divergence equations and some natural boundary conditions, and it has the remaining fundamental equations as its constraint conditions. The conditions are generally undesirable in computation, and they are accordingly removed through an involutory transformation [e.g., the authors, Int. J. Eng. Sci. 40, 457−489 (2002)]. Thus, a unified variational principle operating on all the field variables is derived in piezomagnetism. The principle is shown, as special cases, to recover some of earlier ones. [Work supported by TUBA.]

Algebraic $\mathbb {Z}^d$-actions of entropy rank one

We investigate algebraic $\mathbb Z^d$-actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic $\mathbb Z^d$-action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.

On the relative variational principle for fibre expanding maps

Relative transfer operators are used in Denker and Gordin's 1999 paper to construct Gibbs families of conditional measures on fibres of a uniformly-expanding and exact-fibred system. Here we investigate the relationship between these operators and the relative variational principle (extending results for non-fibred expanding systems).It turns out that the maximal value for the free energy in the relative variational problem can be represented in terms of the transfer operator. However, for a general potential, the possibility to reduce the construction of an equilibrium measure to the search for an appropriate family of conditional measures on the fibres, critically depends on the invertibility of the base transformation.A certain class of potentials (called basic) which allow the last-mentioned reduction is introduced and the properties of the corresponding equilibrium measures are studied. Any measure of this kind gives rise to a regular factor; under a natural assumption the latter property is shown to be equivalent to the validity of the relative version of Rokhlin's formula for the entropy of a measure preserving transformation. Several examples are presented, among them families of polynomial skew products in \mathbb{C}^n restricted to their Julia sets.

Nonlocal elasticity and related variational principles

Abstract The Eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied. The model is refined by assuming an attenuation function depending on the `geodetical distance' between material particles, such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented. A suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model. The nonlocal elasticity boundary-value problem for infinitesimal displacements and quasi-static loads is addressed and the conditions for the solution uniqueness are established. Three variational principles, nonlocal counterparts of classical ones, i.e. the total potential energy, the complementary energy, and the mixed Hu–Washizu principles, are provided. The former is used to derive a nonlocal-type FEM (NL-FEM) in which the (symmetric) global stiffness matrix reflects the nonlocality features of the problem. An alternative standard-FEM-based solution method is also provided, which consists in an iterative procedure of the type local prediction/nonlocal correction, in which the nonlocality is simulated by an imposed-like correction strain. The potentialities of these analysis methods are pointed out, their numerical implementations being the object of an ongoing research work.

A note on the existence of a largest topological factor with zero entropy

Given a topological system T T on a σ \sigma -compact Hausdorff space and its factor S S we show the existence of a largest topological factor S ^ \hat {S} containing S S such that for each S ^ \hat {S} -invariant measure μ \mu , h μ ( S ^ | S ) = 0 h_\mu (\hat {S}|S)=0 . When a relative variational principle holds, h ( S ^ ) = h ( S ) h(\hat {S})=h(S) .

Variational second-order estimates for nonlinear composites

It is shown here that the approximate ‘second–order’ method of Ponte Castaneda for estimating the effective behaviour of nonlinear composite materials can be given a variational status. This is accomplished by means of a slight generalization of the variational structure of Talbot and Willis consisting of the replacement of the infimum over the polarization field by the more general corresponding stationary operation. Since the relevant functions are not convex in general, this replacement provides additional flexibility that can be exploited advantageously. The net result is a stationary principle, from which stationary variational estimates can be generated. Under certain hypotheses on the choice of the linear comparison composite, it is shown that the resulting estimates are identical to the ‘second–order’ estimates, thus showing that these second–order estimates are, indeed, variational in character, although only of the stationary variety. The connection of these second–order estimates to earlier bounds, obtained from related extremum variational principles, is also explored.

Calculation of predissociation resonances for the A 2 Sigma + state of HeH with the complex coordinate method in a diabatic Hamiltonian representation

The predissociation lifetimes of the lowest rovibrational levels of the A 2 Sigma + state of HeH are computed with the aid of a diabatic Hamiltonian representation obtained by means of a pointwise diagonalization of adiabatic CI quadrupole moment matrices. Complex vibrational energy eigenvalues are computed employing the standard complex coordinate method and related variational principle in conjunction with the diabatic potential and coupling element curves. This procedure is marked by good convergence behaviour for the complex energy eigenvalues and yields lifetime values which are in good agreement with known experimental results for a series of HeH isotopomers without necessitating the computation of non-adiabatic nuclear kinetic energy matrix elements.

Nonlinear Stability of Vortex Patches

To establish the nonlinear (Liapunov) stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two-dimensional Euler equations of incompressible, inviscid hydrodynamics. This is accomplished by using a relative variational principle in terms of energy function. A counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate.

Nonlinear stability of vortex patches

To establish the nonlinear (Liapunov) stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two-dimensional Euler equations of incompressible, inviscid hydrodynamics. This is accomplished by using a relative variational principle in terms of energy function. A counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate.

Nonlinear stability of circular vortex patches

This paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the Euler equations of incompressible fluids. This is achieved by establishing a relative variational principle in terms of either energy or angular momentum. Thus, we exploit and extend Arnold's idea in (1965, 1969) to a nonsmooth setting as well.