Variational Formula Research Articles

Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.

Gibbs Variational Formula for Thermal Equilibrium States in Terms of Quantum Relative Entropy Density

We prove the Gibbs variational formula in terms of quantum relative entropy density that characterizes translation invariant thermal equilibrium states in quantum lattice systems. It is a natural quantum extension of the similar statement established by Follmer for classical systems.

Iteration Variational Method for Solving Two-Dimensional Partial Integro-Differential Equations

The two-dimensional integro-differential partial equations is one of the so difficult problems to be solved analytically and/or approximately, and therefore, a method that is efficient for solving such type of problems seems to be necessary. Therefore, in this paper, the iteration methods, which is so called the variational iteration method have been used to provide a solution to such type of problems approximately, in which the obtained results are very accurate in comparison with the exact solution for certain well selected examples which are constructed so that the exact solution exist. Main results of this work is to derive first the variational iteration formula and then analyzing analytically the error term and prove its convergence to zero as the number of iteration increases.

Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps

We study the stability of exponentially subelliptic harmonic (e.s.h.) maps from a Carnot–Caratheodory complete strictly pseudoconvex pseudohermitian manifold $$(M, \theta )$$ into a Riemannian manifold (N, h). E.s.h. maps are $$C^\infty $$ solutions $$\phi : M \rightarrow N$$ to the nonlinear PDE system $$\tau _b (\phi ) + \phi _*\, \nabla ^H e_b (\phi ) = 0$$ [the Euler–Lagrange equations of the variational principle $$\delta \, E_b (\phi ) = 0$$ where $$E_b (\phi ) = \int _\Omega \exp \big [ e_b (\phi ) \big ] \; \Psi $$ and $$e_b (\phi ) = \frac{1}{2} \, \mathrm{trace}_{G_\theta } \left\{ \Pi _H \phi ^*h \right\} $$ and $$\Omega \subset M$$ is a Carnot–Caratheodory bounded domain]. We derive the second variation formula about an e.s.h. map, leading to a pseudohermitian analog to the Hessian (of an ordinary exponentially harmonic map between Riemannian manifolds) $$\begin{aligned} H(E_b )_\phi (V, W)= & {} \int _\Omega h^\phi \big ( J^\phi _{b, \, \exp } V, \, W \big ) \; \Psi \\&+\, \int _M \exp \big [ e_b (\phi ) \big ] \, (h^\phi )^*(D^\phi V, \; \Pi _H \phi _*) \, (h^\phi )^*(D^\phi W, \; \Pi _H \phi _*) \; \Psi J_{b, \, \exp }^\phi V\equiv & {} \big ( D^\phi \big )^*\big ( \exp \big [ e_b (\phi ) \big ] \; D^\phi V \big ) \\&-\, \exp \big [ e_b (\phi ) \big ] \; \mathrm{trace}_{G_\theta } \left\{ \Pi _H \, \big ( R^h \big )^\phi \big ( V, \; \phi _*\, \cdot \, \big ) \phi _*\cdot \right\} , \end{aligned}$$ [ $$\Psi = \theta \wedge (d \theta )^n$$ ]. Given a bounded domain $$\Omega \subset M$$ and an e.s.h. map $$\phi \in C^\infty \big ( \overline{\Omega }, \; N \big )$$ with values in a Riemannian manifold $$N = N^m (k)$$ of nonpositive constant sectional curvature $$k \le 0$$ , we solve the generalized Dirichlet eigenvalue problem $$J^\phi _{b, \, \exp } V = \lambda \, V$$ in $$\Omega $$ and $$V = 0$$ on $$\partial \Omega $$ for the degenerate elliptic operator $$J^\phi _{b, \, \exp }$$ , provided that $$\Omega $$ supports Poincare inequality $$\begin{aligned} \Vert V \Vert _{L^2} \le C \Vert D^\phi V \Vert _{L^2}, \;\; V \in C^\infty _0 \big ( \Omega , \, \phi ^{-1} T N \big ), \end{aligned}$$ and the embedding $$\mathring{W}^{1,2}_H (\Omega , \, \phi ^{-1} T N ) \hookrightarrow L^2 (\Omega , \, \phi ^{-1} T N)$$ is compact.

Martingales of stochastic Laplacian growth

A family of exponential martingales of a stochastic Laplacian growth problem is proposed. Stochastic Laplacian growth describes a regularized interface dynamics in a two-fluid system, where the viscous fluid is incompressible at a large scale, while compressible at a small scale in the vicinity of the interface. Hence, random fluctuations of pressure near the boundary are inevitable. By using Loewner–Kufarev equation, we study interface dynamics generated by nonlocal random Loewner measure, which produces the patterns with viscous fingers. We use a Schottky double construction to introduce a one-parametric family of functions of random processes on the double closely connected to the correlation functions of primary operators of the boundary conformal field theory in the Coulomb gas framework. For a specific value of the parameter, these functions are martingales with respect to stochastic Loewner flow on the Schottky double. A connection between the proposed algebraic construction and the physical problem of stochastic interface dynamics in the Hele-Shaw cell relies on the Hadamard’s variational formula. Namely, the variation of pressure in stochastic Laplacian growth in the vicinity of the interface is given by the variance of martingales on the double.

The Orlicz-Minkowski problem for torsional rigidity

In this paper, we prove a variational formula of torsional rigidity under the Orlicz sum, and then introduce the Orlicz mixed torsional rigidity which yields the Orlicz-Brunn-Minkowski inequality and Orlicz-Minkowski inequality for torsional rigidity. As an application, we prove the existence of solution to the Orlicz-Minkowski problem for discrete measure and general measure associated with torsional rigidity.

A note on equivariant biharmonic maps and stable biharmonic maps

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [20], [12]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a 4-dimensional space form into a 4-dimensional model space. We also give an improved second variation formula for biharmonic maps into a space form and use it to prove that there exists no stable proper biharmonic map with constant square norm of tension field from a compact Riemannian manifold without boundary into a space form of positive sectional curvature.

Nonlocal mixed variational formula for orthotropic nanoplates resting on elastic foundations

This paper deals with the examination of orthotropic nanoplates utilizing nonlocal mixed variational formula. For this model, the displacement and stress fields are considered to be arbitrary and it includes the impact of transverse normal stress which does not exist in numerous theories. The orthotropic nanoplates are thought to be refreshed on elastic foundations. The mixed theory is utilized to get the governing differential equations as per Eringen’s nonlocal elasticity model. The governing equations incorporate the small-scale effect as well as foundations and mechanical impacts. Analytical solutions of bending response for simply supported orthotropic nanoplates are derived. The present outcomes are compared well with those being in the literature. The effects played by elastic foundation parameters, small-scale parameter, aspect and thickness nanoplate ratios are examined. The present study can be used as benchmarks for future comparisons with other investigators applying models containing the transverse shear and normal strains.

Optimal Source Codes for Timely Updates

A transmitter observing a sequence of independent and identically distributed random variables seeks to keep a receiver updated about its latest observations. The receiver need not be apprised about each symbol seen by the transmitter, but needs to output a symbol at each time instant t. If at time t the receiver outputs the symbol seen by the transmitter at time U (t) ≤ t, the age of information at the receiver at time t is t - U(t). We study the design of lossless source codes that enable transmission with minimum average age at the receiver. We show that the asymptotic minimum average age can be attained up to a constant gap by the Shannon codes for a tilted version of the original pmf generating the symbols, which can be computed easily by solving an optimization problem. Furthermore, we exhibit an example with alphabet X where age of a factor O(√(log |X|)) more than that achieved by our Shannon codes for the original pmf incur an asymptotic average codes. Underlying our prescription for optimal codes is a new variational formula for integer moments of random variables, which may be of independent interest. Also, we discuss possible extensions of our formulation to randomized schemes and to the erasure channel, and include a treatment of the related problem of source coding for minimum average queuing delay.

Herglotz’s Variational Problem for Non-Conservative System with Delayed Arguments under Lagrangian Framework and Its Noether’s Theorem

Because Herglotz’s variational problem achieves the variational representation of non-conservative dynamic processes, its research has attracted wide attention. The aim of this paper is to explore Herglotz’s variational problem for a non-conservative system with delayed arguments under Lagrangian framework and its Noether’s theorem. Firstly, we derive the non-isochronous variation formulas of Hamilton–Herglotz action containing delayed arguments. Secondly, for the Hamilton–Herglotz action case, we define the Noether symmetry and give the criterion of symmetry. Thirdly, we prove Herglotz type Noether’s theorem for non-conservative system with delayed arguments. As a generalization, Birkhoff’s version and Hamilton’s version for Herglotz type Noether’s theorems are presented. To illustrate the application of our Noether’s theorems, we give two examples of damped oscillators.

Analysis of the element-free Galerkin method with penalty for general second-order elliptic problems

In this paper, an element-free Galerkin (EFG) method with penalty for solving general second-order elliptic problems with mixed boundary conditions is presented. A priori approximate estimations of the penalty method are obtained under some proper assumptions, which ensure the availability of the penalty method. By defining a special norm, the existence and uniqueness for the weak solution of the penalized continuous variational formula are proved, which guarantee the validity of the EFG discretization. Error estimates of the EFG method are provided in L2 and H1 norms for the Dirichlet and the mixed boundaries, respectively. Numerical examples are finally performed to illustrate the theoretical results.

Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning

Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for which we develop a calculus and provide representation formulas. Functions having a conservative field are called path differentiable: convex, concave, Clarke regular and any semialgebraic Lipschitz continuous functions are path differentiable. Using Whitney stratification techniques for semialgebraic and definable sets, our model provides variational formulas for nonsmooth automatic differentiation oracles, as for instance the famous backpropagation algorithm in deep learning. Our differential model is applied to establish the convergence in values of nonsmooth stochastic gradient methods as they are implemented in practice.

On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

Study and Analysis on the Tensile Test Elongation Variation Law for High-Strength Pipeline Steel

In view of the tensile test elongation conversion problems for high-strength pipeline steel, several kinds of high-strength pipeline steel were selected to study the corresponding tensile test elongation variation laws. Then, a statistical analysis was carried out on a large collection of test data. The test results showed that the Oliver formula, which was established based on old carbon steel and low-alloy steel, is still applicable to high-strength pipeline steel but with different material coefficients than those used for carbon steel and low-alloy steel. This study established tensile test elongation variation formulas for new-made X65 to X90 pipeline steels and a formula for high-strength pipeline steel with a fixed 50 mm gauge length. The test data and the requirements were analyzed and compared according to the new material coefficients and formulas, through which a revision was proposed for the standards.

The Lp Minkowski problem for torsion

In the present paper, we introduce the so-called Lp torsion measure, Lp mixed torsion. A Hadamard variational formula for torsion under the p-sum is established. As an application, the existence of solutions to a Lp Minkowski problem for torsion is proved for any fixed p>1 and p≠n+2, which extends a result of Colesanti and Fimiani [24]. A Lp Minkowski inequality for torsion is also obtained, which implies the uniqueness of the solutions for the Lp Minkowski problem mentioned above.

Variational formulae and estimates of O’Hara’s knot energies

O’Hara’s energies, introduced by Jun O’Hara, were proposed to answer the question what the canonical shape in a given knot type is, and were configured so that the less the energy value of a knot is, the “better” its shape is. The existence and regularity of minimizers has been well studied. In this paper, we calculate the first and second variational formulae of the [Formula: see text]-O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the [Formula: see text]-O’Hara energies, their techniques do not seem to be applicable to the case [Formula: see text]. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the [Formula: see text]-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.

Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula.

We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments with liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.

A Lp Brunn-Minkowski Theory for Logarithmic Capacity

In the present paper, we first introduce the concepts of the Lp n-dimensional Logarithmic-Capacity measure and Lp mixed n-dimensional Logarithmic-Capacity and then prove some geometric properties of Lp n-dimensional Logarithmic-Capacity measure and a Lp Minkowksi inequality for the n-dimensional Logarithmic-Capacity for any fixed n ≥ 2 and p ≥ 1. As an application, we establish a Hadamard variational formula for the n-dimensional Logarithmic-Capacity under p-sum for any fixed n ≥ 2 and p ≥ 1, which extends some results of Borell (Ann. Sci. Ecole Norm. Sup. 17(3), 451–467 1984), Jerison (Adv. Math. 122(2), 262–279 1996), Colesanti-Cuoghi (Potential Anal. 22(3), 289–304 2005), Akman-Lewis-Saari-Vogel (in press) and Xiao (2013, 2018). With the Hadamard variational formula and Minkowksi inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the Logarithmic-Capacity.

Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants

The $n$-dimensional hypergeometric integrals associated with a hypersphere arrangement $S$ are formulated by the pairing of $n$-dimensional twisted cohomology $H_\nabla^n (X, \Omega^\cdot (*S))$ and its dual. Under the condition of general position we present an explicit representation of the standard form by a special (NBC) basis of the twisted cohomology (contiguity relation in positive direction), the variational formula of the corresponding integral in terms of special invariant $1$-forms $\theta_J$ written by Calyley-Menger minor determinants, and a connection relation of the unique twisted $n$-cycle identified with the unbounded chamber to a special basis of twisted $n$-cycles identified with bounded chambers. Gauss-Manin connections are formulated and are explicitly presented in two simplest cases. In the appendix contiguity relation in negative direction is presented in terms of Cayley-Menger determinants.

Nonexponential Sanov and Schilder theorems on Wiener space: BSDEs, Schrödinger problems and control

We derive new limit theorems for Brownian motion, which can be seen as nonexponential analogues of the large deviation theorems of Sanov and Schilder in their Laplace principle forms. As a first application, we obtain novel scaling limits of backward stochastic differential equations and their related partial differential equations. As a second application, we extend prior results on the small-noise limit of the Schrödinger problem as an optimal transport cost, unifying the control-theoretic and probabilistic approaches initiated respectively by T. Mikami and C. Léonard. Lastly, our results suggest a new scheme for the computation of mean field optimal control problems, distinct from the conventional particle approximation. A key ingredient in our analysis is an extension of the classical variational formula (often attributed to Borell or Boué–Dupuis) for the Laplace transform of Wiener measure.