Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation

Non-convex variational/boundary-value problems are studied usinga modified version of the Ericksen bar model in nonlinear elasticity.The strain-energy function is a general fourth-order polynomialin a suitable measure of strain that provides a convenient modelfor the study of, for example, phase transitions. On the basisof a canonical duality theory, the nonlinear differential equationfor the non-convex, non-homogeneous variational problem, herewith either mixed or Dirichlet boundary conditions, is convertedinto an algebraic equation, which can, in principle, be solvedto obtain a complete set of solutions. It should be emphasizedthat one important outcome of the theory is the identificationand characterization of the local energy extrema and the globalenergy minimizer. For the soft loading device criteria for theexistence, uniqueness, smoothness and multiplicity of solutionsare presented and discussed. The iterative finite-differencemethod (FDM) is used to illustrate the difficulty of capturingnon-smooth solutions with traditional FDMs. The results illustratethe important fact that smooth analytic or numerical solutionsof a nonlinear mixed boundary-value problem might not be minimizersof the associated potential variational problem. From a ‘dual’perspective, the convergence (or non-convergence) of the FDMis explained and numerical examples are provided.     

The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation

The asymptotic behaviors of solutions of an initial-boundary value problem for the generalized BBM equation with non-convex flux are discussed in this paper. It is proved that under the conditions of constant boundary data and small perturbation for the initial data, the global solutions exist and converge time-asymptotically to a stationary wave or the superposition of a stationary wave and a rarefaction wave. The proof is given by a technical L ²-weighted energy method.     

An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces

Abstract

An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg–de Vries equation is considered and results on global well-posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a “boundary potential” type for this linearized KdV equation.     

Uniform and optimal estimates for solutions to singularly perturbed parabolic equations

The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L²-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations.     

On the solvability of equations of a nonlinear viscous-ideal fluid

A model of the one-dimensional motion of a viscous compressible fluid is considered. A class of nonlinear stressed-state equations for which the initial boundary-value problem has global (in time) solutions in the class of generalized solutions satisfying the energy identity is described. In particular, media exhibiting viscous properties only for large strain rates are studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 13–23, July, 2000.     

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.     

Resonance phenomena in the nonlinear equations of a proper semiconductor h2Δu=shu

One considers a Steklov-type boundary-value problem for the nonlinear equation of a semiconductor. Under the assumption of the existence on the surface of the semiconductor of a closed geodesic, stable in a linear approximation, one constructs asymptotic solutions which are concentrated in the neighborhood of this geodesic. The obtained solutions are expressed in terms of the known asymptotic eigenfunctions of the Laplace operator on a Riemann manifold and in terms of the multisoliton solutions of the Sine-Gordon equation. Similar solutions are obtained for the mixed boundary-value problem.     

On the decay rate of solutions of the wave equation in domains with star-shaped boundaries

One studies the large-time decay rate of the weighted energy of solutions of the first mixed problem for the wave equation in domains with smooth boundaries which are star-shaped with respect to the origin. Estimates are established for solutions of the first boundary-value problem for the Helmholtz equation in the upper half-plane. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 390–406, 2007.     

A penalty approximation for a unilateral contact problem in non-linear elasticity

Using Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear-elastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials. Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.     

具材料阻尼的非线性双曲型方程整体解的不存在性

考虑了一类具材料阻尼的非线性双曲型方程初边值问题整体解的不存在性．分别采用能量方法、Jensen不等式和凹性方法证明了该问题整体解的不存在性定理．作为主要结果的应用,给出了3个例子．     

Third-order right-focal multi-point problems on time scales

We are concerned with the existence and form of positive solutions to a third-order multi-point boundary-value problem on time scales with mixed derivatives. We find and utilize the Green function for the corresponding homogeneous right-focal problem as the kernel of an integral equation of Hammerstein-type. Two examples are included to illustrate the results.     

Decay estimates for a thermoelastic system in waveguides

We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity.     

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior.     

Transformation of a singular boundary-value problem on a half-line to an integral equation with a discontinuous operator

An investigation of a boundary-value problem on a half-line for a nonlinear ordinary second order differential equation whose free term has a discontinuity in a strip. A method is proposed for the transformation of the boundary-value problem into an integral equation with a discontinuous operator. Some results have recently been obtained concerning the existence, the comparison, and integral representations of solutions of this integral equation.Translated from Matematicheskie Zematki, Vol. 9, No. 1, pp. 77–82, January, 1971.     

Integrated data envelopment analysis: Global vs. local optimum

Chiou et al. (2010) (A joint measurement of efficiency and effectiveness for non-storable commodities: integrated data envelopment analysis approaches. European Journal of Operational Research 201, 477–489) propose an integrated data envelopment analysis model in measuring decision making units (DMUs) that have a two-stage internal network structure with multiple inputs, outputs, and consumptions. They claim that any optimal solutions determined by their DEA model are a global optimum, not a local optimum. We show that such a conclusion is a false statement due to their misuse of Hessian matrix in examining the concavity of the objective function, and their DEA model is actually a non-convex optimization problem. As a result, their DEA model is unusable in practice due to a lack of efficient algorithm for this particular non-convex DEA model. We further show that Chiou et al.’s (2010) model is a special case of a well-known two-stage network DEA model, and it can be transformed into a parametric linear program for which an approximate global optimal solution can be obtained by solving a sequence of linear programs in combination with a simple search algorithm.     

Spatial behaviour of solutions of the three-phase-lag heat equation

In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial-time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solutions.     

Symmetry analysis of initial-value problems

Symmetry analysis is a powerful tool that enables the user to construct exact solutions of a given differential equation in a fairly systematic way. For this reason, the Lie point symmetry groups of most well-known differential equations have been catalogued. It is widely believed that the set of symmetries of an initial-value problem (or boundary-value problem) is a subset of the set of symmetries of the differential equation. The current paper demonstrates that this is untrue; indeed, an initial-value problem may have no symmetries in common with the underlying differential equation. The paper also introduces a constructive method for obtaining symmetries of a particular class of initial-value problems.     

Estimates for the solutions of systems of linear algebraic equations and their application to the investigation of the convergence of the method of finite differences

One proves theorems regarding the estimates of the solutions of systems of linear algebraic equations and inequalities. On the basis of these theorems one suggests a method for estimating the norm of the inverse matrix of a system of difference equations which approximates a boundary-value problem for an integrodifferential equation. The method allows us to eliminate the restrictions which are usually imposed on the coefficients of the integrodifferential equation in order to ensure the diagonal dominance in the system of the difference equations. One considers an application to nonlinear problems.