A Unifying Approach to Variational Relation Problems

The purpose of this paper is to present a unified approach to study the existence of solutions for two types of variational relation problems, which encompass several generalized equilibrium problems, variational inequalities and variational inclusions investigated in the recent literature. By using two well-known fixed point theorems, we establish several existence criteria for the solutions of these problems.     

A Variational Approach to Upper and Lower Solutions

Traditional proofs of the upper and lower solution existenceresults for nonlinear elliptic equations use monotone iterationschemes based on the maximum principle. In this paper, a directvariational method is employed which characterizes the solutionas the global minimum of a functional on a closed convex set.     

A Variational Approach to Dynamic Recrystallization Using Probability Distributions

We investigate a model of dynamic recrystallization in polycrystalline materials. A probability distribution function is introduced to characterize the state of individual grains by grain size and dislocation density. Specifying free energy and dissipation within the polycrystalline aggregate we are able to derive an evolution equation for the probability density function via a thermodynamic extremum principle. Once the distribution function is known macroscopic quantities like average strain and stress can be calculated. For distribution functions which are constant in time, describing a state of dynamic equilibrium, we obtain a partial differential equation in parameter space which we solve using a marching algorithm. Numerical results are presented and their physical interpretation is given. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Variational Approach to Nonlinear Singularly Perturbed Boundary-Value Problems

Carrier and Pearson introduced a nonlinear singularly perturbed boundary value problem that has served as a paradigm for problems where the method of matched asymptotic expansions (MAE) apparently fails. The “failure” of MAE is its inability to select the location of possible internal layers, though their structure is determined. Thus, a straightforward application of MAE leaves the positions of any internal layers arbitrary, though the asymptotic expansion of the exact solution to the problem exhibits internal layers only at specific locations. For this reason the solutions produced by MAE have been referred to as spurious solutions. We resolve the question of finding the positions of the interior layers by employing the variational approach of Grasman and Matkowsky. In addition, we show that this method tells how solutions bifurcate as the boundary values are varied, and give an alternative motivation for the variational approach via Newton”s method.     

Schrodinger方程的变分迭代解法

该文以Ｓｃｈｒｏｄｉｎｇｅｒ方程为例,分析变分迭代法的一些基本特点．在该方法中引进了一广义拉氏乘子构造了一迭代格式,拉氏乘子可由变分理论最佳识别．由于在识别拉氏乘子是应用了限制变分的概念,所以只能通过迭代才能得到收敛解．为了加快收敛速度,可以在初始近似引入待定常数,而待定常数又可用各种方式最佳识别．文中初步分析了该方法的收敛性,对于Ｓｃｈｒｏｄｉｎｇｅｒ方程,其一阶近似即可得到Ｊｏｓｔ解．     

Variational Approach to Scattering by Inhomogeneous Layers Above Rough Surfaces

In this paper, we study, via variational methods, the problem of scattering of time harmonic acoustic waves by unbounded inhomogeneous layers above a sound soft rough surface. We first propose a variational formulation and exploit it as a theoretical tool to prove the well-posedness of this problem when the media is non-absorbing for arbitrary wave number and obtain an estimate about the solution, which exhibit explicitly dependence of bound on the wave number and on the geometry of the domain. Then, based on the non-absorbing results, we show that the variational problem remains uniquely solvable when the layer is absorbing by means of a priori estimate of the solution. Finally, we consider the finite element approximation of the problem and give an error estimate.     

A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − Δu = λh(x)H(u − a)u^q + u^2* − 1 in R^N, where λ, a > 0 are parameters, h(x) is both nonnegative and integrable on R^N, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0 ≤ q < 2* − 1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0 ≤ q ≤ 1 and 1 < q < 2* − 1.     

A Variational Approach to Dynamical Systems and Its Numerical Simulation

We propose and describe an alternative perspective to the study and numerical approximation of dynamical systems. It is based on a variational approach that seeks to minimize the quadratic error understood as a deviation of paths from being a solution of the corresponding system. Although this philosophy has been examined recently from the point of view of the direct method, we exploit optimality conditions and steepest descent strategies to establish precise and easy-to-implement numerical schemes for the approximation. We show the practical performance in a number of selected examples and indicate how this strategy, with minor changes, may also be used to deal with boundary value problems. Our emphasis is placed more so on relevant results that justify the numerical implementation and less on abstract theoretical results under optimal sets of assumptions.     

A Variational Approach to a Kirchhoff-type Problem Involving Two Parameters

In this paper, the authors establish the existence of at least three weak solutions for the Kirchhoff-type problem $$\left\{\begin{array}{ll}-K \left( \int_{\Omega}| \nabla u(x)|^{2}dx \right) \Delta u(x)= \lambda f(x,u)+\mu g(x,u),\quad {\rm in}\; \Omega,\\u=0, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad {\rm on}\; \partial \Omega, \end{array} \right.$$ under appropriate hypotheses. The proofs are based on variational methods.     

Variational Approach to the Inviscid Compressible Fluid

It is well-known that not every partial differential equation admits a variational formula. A rigorous proof of the existence of a variational principle is very difficult. In this paper, the semi-inverse method proposed by Ji-Huan He is used to construct a variational principle for a one-dimensional inviscid compressible fluid.      

Variational Approach to Scattering by Inhomogeneous Layers Ab ove Rough Surfaces

In this paper, we study, via variational methods, the problem of scattering of time harmonic acoustic waves by unbounded inhomogeneous layers above a sound soft rough surface. We first propose a variational formulation and exploit it as a theoretical tool to prove the well-posedness of this problem when the media is non-absorbing for arbitrary wave number and obtain an estimate about the solution, which exhibit explicitly dependence of bound on the wave number and on the geometry of the domain. Then, based on the non-absorbing results, we show that the variational problem remains uniquely solvable when the layer is absorbing by means of a priori estimate of the solution. Finally, we consider the finite element approximation of the problem and give an error estimate.     

A Variational Approach to the Maximization of Preferences Without Numerical Representation

Journal of Optimization Theory and Applications - We introduce a variational approach to study a maximization problem of preferences that cannot be represented by a utility function. In such...     

A Variational Approach for One-Dimensional Scalar Field Problems

In this paper, we are interested in the existence of infinitely many weak solutions for a onedimensional scalar field problem. By using variational methods, in an appropriate functional space which involves the potential V, we determine intervals of parameters such that our problem admits either a sequence of weak solutions strongly converging to zero provided that the nonlinearity has a suitable behavior at zero or an unbounded sequence of weak solutions if a similar behavior occurs at infinity.     

A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems

The existence of at least one nontrivial solution to a class of semilinear Tricomi problems is established via an application of the dual variational method which captures the solution as the preimage of a minimum of a suitable dual action functional. The boundary conditions are homogeneous Dirichlet conditions on a suitable part of the boundary, as dictated by uniqueness theorems for the linear problem. While there are good compactness properties for the inverse operator for the linear problem, there is a manifest asymmetry in the linear part due to the form of the boundary conditions. The linear part is symmetrized by introducing suitable re ection operators on symmetric domains, which then results in a nonlocal character of the nonlinearity. Received July 1997; Revised February 1998     

A Penalty Approach to Optimal Control of Allen-Cahn Variational Inequalities: MPEC-View

A scalar Allen-Cahn-MPEC problem is considered and a penalization technique is applied to show the existence of an optimal control. We show that the stationary points of the penalized problems converge to some stationary points of the limit problem, which however are weaker than C-stationarity conditions.