Dirichlet Principle Research Articles

Non-reversible metastable diffusions with Gibbs invariant measure I: Eyring–Kramers formula

In this article, we prove the Eyring–Kramers formula for non-reversible metastable diffusion processes that have a Gibbs invariant measure. Our result indicates that non-reversible processes exhibit faster metastable transitions between neighborhoods of local minima, compared to the reversible process considered in Bovier et al. (J Eur Math Soc 6:399–424, 2004). Therefore, by adding non-reversibility to the model, we can indeed accelerate the metastable transition. Our proof is based on the potential theoretic approach to metastability through accurate estimation of the capacity between metastable valleys. We carry out this estimation by developing a novel method to compute the sharp asymptotics of the capacity without relying on variational principles such as the Dirichlet principle or the Thomson principle.

Energy Method for the Elliptic Boundary Value Problems with Asymmetric Operators in a Spherical Layer

Three-dimensional elliptic boundary value problems arising in the mathematical modeling of quasi-stationary electric fields and currents in conductors with gyrotropic conductivity tensor in domains homeomorphic to the spherical layer are considered. The same problems are mathematical models of thermal conductivity or diffusion in moving or gyrotropic media. The operators of the problems in the traditional formulation are non-symmetric. New statements of the problems with symmetric positive definite operators are proposed. For the four boundary value problems the quadratic energy functionals, to the minimization of which the solutions of these problems are reduced, are constructed. Estimates of the obtained quadratic forms are made in comparison with the form appearing in the Dirichlet principle for the Poisson equation

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. Analyzing singular Dirichlet-to-Neumann operators, we will establish Laurent expansion near singularities as well as Mittag–Leffler expansion for the related quadratic forms. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the framework of Lebesgue spaces. The obtained results are supported by some examples where we analyze properties of singular Dirichlet-to-Neumann operators related to Neumann and Robin Laplacian on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boundary may affect positivity and derive Mittag-Leffler expansion for the eigenvalues of singular Dirichlet-to-Neumann operators.

The Dirichlet principle for inner variations

We are concerned with the Dirichlet energy of mappings defined on domains in the complex plane. The Dirichlet Principle, the name coined by Riemann, tells us that the outer variation of a harmonic mapping increases its energy. Surprisingly, when one jumps into details about inner variations, which are just a change of independent variables, new equations and related questions start to matter. The inner variational equation, called the Hopf–Laplace equation, is no longer the Laplace equation. Its solutions are generally not harmonic; we refer to them as Hopf harmonics. The natural question that arises is how does a change of variables in the domain of a Hopf harmonic map affect its energy? We show, among other results, that in case of a simply connected domain the energy increases. This should be viewed as Riemann’s Dirichlet Principle for Hopf harmonics. The Dirichlet Principle for Hopf harmonics in domains of higher connectivity is not completely solved. What complicates the matter is the insufficient knowledge of global structure of trajectories of the associated Hopf quadratic differentials, mainly because of the presence of recurrent trajectories. Nevertheless, we have established the Dirichlet Principle whenever the Hopf differential admits closed trajectories and crosscuts. Regardless of these assumptions, we established the so-called Infinitesimal Dirichlet Principle for all domains and all Hopf harmonics. Precisely, the second order term of inner variation of a Hopf harmonic map is always nonnegative.

Statements of the boundary value problems in mathematical simulation of a quasistationary electric field in the atmosphere and ionosphere

Boundary-value problems which have to be solved in the frame of the mathematical models of quasi-stationary electric fields and currents in the global conductor, that consists of the Earth’s ionosphere and the atmosphere are formulated. The approach based on the domain decomposition is used. A quadratic energy functional is constructed. It allows us to reduce the solution of the boundary value problem for a three-dimensional elliptic differential equation with asymmetric coefficients to minimizing of the functional. Estimates of the quadratic form in comparison with the Dirichlet principle for the Poisson equation are obtained. These estimates allow us to assess the condition number of the matrix of the system of linear algebraic equations, arising in the numerical solution to the problem.

Modelling the flexoelectric effect in solids: A micromorphic approach

Flexoelectricity is characterised by the coupling of the second gradient of the motion and the electrical field in a dielectric material. The presence of the second gradient is a significant obstacle to obtaining the approximate solution using conventional numerical methods, such as the finite element method, that typically require a C1-continuous approximation of the motion. A novel micromorphic approach is presented to accommodate the resulting higher-order gradient contributions arising in this highly-nonlinear and coupled problem within a classical finite element setting. Our formulation accounts for all material and geometric nonlinearities, as well as the coupling between the mechanical, electrical and micromorphic fields. The highly-nonlinear system of governing equations is derived using the Dirichlet principle and approximately solved using the finite element method. A series of numerical examples serve to elucidate the theory and to provide insight into this intriguing effect that underpins or influences many important scientific and technical applications.

Do wine buyers behave differently in brick and mortar vs online stores?

PurposeTo measure the extent to which wine buyers behave differently when purchasing wine online vs in two brick and mortar stores. The article aims to extend the use of the Double Jeopardy principle and empirical-based methodology to the wine category in a European retailing context.Design/methodology/approachCustomer loyalty data of two brick and mortar stores and the website orders of a Belgian retailer have been gathered for a one-year period. Data have been analysed based on three specific wine attributes: country of origin, grape variety and brand. Double Jeopardy measurements have been calculated for each of these attributes.FindingsThis study enlarges the scope of use of the Dirichlet principles. All three hypotheses derived from the Double Jeopardy patterns across all attributes are confirmed. From the perspective of these principles, we demonstrated that wine buyers do not behave differently in brick and mortar vs online stores.Originality/valueVery few studies have analysed and understood wine buyers' behaviour using actual purchasing data from retail stores, and none have been released comparing online and brick and mortar stores owned by the same retail brand. From that perspective, our study demystifies the way people really buy, and confirms what has been found in other product categories.

Alternative statements of the Rayleigh monotonicity law for linear time-invariant resistor networks driven by only voltages or only currents

Networks of linear time-invariant (LTI) resistors are a classical topic with many applications, used to model a variety of flow phenomena. In this short communication, we provide conceptually simple proofs of the following intuitively appealing bounds, which also imply the classical Rayleigh monotonicity law (RML): (i) For an LTI resistor network, driven by some node voltages, when a set of resistors is removed the total power dissipation is reduced by at least an amount that was consumed in the removed resistors. (ii) The complementary or dual result for such a network, but when current driven, is that shorting some resistors decreases the total power dissipation by at least as much as used to occur in these resistors before shorting. In fact, ideal diodes can be allowed as network branches and the same results hold. The short derivations assume that Dirichlet's and Thomson's minimum principles are known. With the expanding applications of a linear network theory, we hope that these variants of the monotonicity theorem will prove pedagogically useful.

Lebesgue’s criticism of Carl Neumann’s method in potential theory

In the 1870s, Carl Neumann proposed the so-called method of the arithmetic mean for solving the Dirichlet problem on convex domains. Neumann’s approach was considered at the time to be a reliable existence proof, following Weierstrass’s criticism of the Dirichlet principle. However, in 1937 H. Lebesgue pointed out a serious gap in Neumann’s proof. Curiously, the erroneous argument once again involved confusion between the notions of infimum and minimum. The objective of this paper is to show that Lebesgue’s sharp criticism of Neumann’s proof was only partially justified.

Lateral torsional buckling of compressed open thin walled beams: experimental confirmations

This paper provides an approximate solution for the differential equations that govern the buckling of beams with a gradually changing of the thin-walled C cross section. This is a coupled problem of flexural and torsional buckling, whose exact solution is hard to get. We have therefore chosen to use an energy approach through the Dirichlet's principle. It allows, using the Ritz-Rayleigh algorithm, the quick implementation of a solution close to the real value of the critical load. Because of the complexity of the problem, it was considered appropriate to provide an experimental validation of the theoretical results with proper laboratory tests.

A Dirichlet's principle for the k-Hessian

The k-Hessian operator σk is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σk(D2u)=f with Dirichlet boundary condition u=0 is variational; indeed, this problem can be studied by means of the k-Hessian energy −∫uσk(D2u). We construct a natural boundary functional which, when added to the k-Hessian energy, yields as its critical points solutions of k-Hessian equations with general non-vanishing boundary data. As a consequence, we establish a Dirichlet's principle for k-admissible functions with prescribed Dirichlet boundary data.

Boundary operators associated to the σk-curvature

We study conformal deformation problems on manifolds with boundary which include prescribing σk≡0 in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear operators as a key technical tool.

The quantum pigeonhole principle as a violation of the principle of bivalence

In the paper, it is argued that the phenomenon known as the quantum pigeonhole principle (namely, three quantum particles are put in two boxes, yet no two particles are in the same box) can be explained not as a violation of Dirichlet's box principle in the case of quantum particles but as a nonvalidness of a bivalent logic for describing not-yet verified propositions relating to quantum mechanical experiments.

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D2 ⊆ D1 to any quadratic differential on D1 with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (∂q)2 for a singular harmonic function q on D1. As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].

A Practical Index for Approximate Dictionary Matching with Few Mismatches

Approximate dictionary matching (checking if a pattern occurs in a collection of strings) is a classic problem with applications in e.g. spellchecking, online catalogs, and web searchers. We present a simple solution called split index, which is based on the Dirichlet principle, for matching a keyword with few mismatches, and experimentally show that it offers competitive space-time tradeoffs. Our implementation in the C++ language is focused mostly on data compaction, which is beneficial for the search speed. We compare our solution with other algorithms and we show that it is faster when the Hamming distance is used. Query times in the order of 1 microsecond were reported for one mismatch for a few-megabyte natural language dictionary on a medium-end PC. We also demonstrate that a basic compression technique consisting in q-gram substitution can significantly reduce the index size (up to 50 % of the input text size for the DNA sequences).

A contribution to approximate analytical evaluation of Fourier series via an Applied Analysis standpoint; an application in turbulence spectrum of eddies

In the present paper, we shall attempt to make a contribution to approximate analytical evaluation of the harmonic decomposition of an arbitrary continuous function. The basic assumption is that the class of functions that we investigate here, except the verification of Dirichlet's principles, is concurrently able to be expanded in Taylor's representation, over a particular interval of their domain of definition. Thus, we shall take into account the simultaneous validity of these two properties over this interval, in order to obtain an alternative equivalent representation of the corresponding harmonic decomposition for this category of functions. In the sequel, we shall also implement this resultant formula in the investigation of turbulence spectrum of eddies according to known from literature Von Karman's formulation, making the additional assumption that during the evolution of such stochastic dynamic effects with respect to time, the occasional time-returning period can be actually supposed to tend to infinity.

Statistical model of stress corrosion cracking based on extended form of Dirichlet energy: Part 2

The mechanism of stress corrosion cracking (SCC) has been discussed for decades. Here I propose a model of SCC reflecting the feature of fracture in brittle manner based on the variational principle under approximately supposed thermal equilibrium. In that model the functionals are expressed with extended forms of Dirichlet energy, and Dirichlet principle is applied to them to solve the variational problem that represents SCC and normal extension on pipe surface. Based on the model and the maximum entropy principle, the statistical nature of SCC colony is discussed and it is indicated that the crack has discrete energy and length under ideal isotropy of materials and thermal equilibrium.

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in [26], we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. Three approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped with complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work [1]. The second approach, which complements the first one, is variational and based on the Dirichlet principle. The last approach, which complements the second one, is also variational and uses the multiplier technique. Using these approaches, we are able to obtain new results on the well-posedness of these equations for which the conditions on the coefficients are imposed “partially” or “not strictly” on the interfaces of sign changing coefficients. This allows us to rediscover and extend known results obtained by the integral method, the pseudo differential operator theory, and the T-coercivity approach. The unique solution, obtained by the limiting absorption principle, is not in Hloc1(Rd) as usual and possibly not even in Lloc2(Rd). The optimality of our results is also discussed.

Variational Approach for Resolving the Flow of Generalized Newtonian Fluids in Circular Pipes and Plane Slits

We use a generic and general variational method to obtain solutions to the flow of generalized Newtonian fluids through circular pipes and plane slits. The new method is not based on the use of the Euler-Lagrange variational principle and hence it is totally independent of our previous approach which is based on this principle. Instead, the method applies a very generic and general optimization approach which can be justified by the Dirichlet principle although this is not the only possible theoretical justification. The results that were obtained from the new method using nine types of fluid are in total agreement, within certain restrictions, with the results obtained from the traditional methods of fluid mechanics as well as the results obtained from the previous variational approach. In addition to being a useful method in its own for resolving the flow field in circular pipes and plane slits, the new variational method lends more support to the old variational method as well as for the use of variational principles in general to resolve the flow of generalized Newtonian fluids and obtain all the quantities of the flow field which include shear stress, local viscosity, rate of strain, speed profile, and volumetric flow rate.

COMPACTNESS AND DIRICHLET'S PRINCIPLE

In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet‘s principle. We emphasize on the intuition in Riemann‘s statement on the principle criticized byWeierstrass‘ requirement of rigor followed by Hilbert‘s restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion. Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly afterWeierstrass’s famous criticism of Riemann’s use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar´e and Hilbert defended Riemann’s use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.