On the Boundedness of Solutions of Degenerate Anisotropic Elliptic Variational Inequalities

In this article, for a class of degenerate anisotropic elliptic second-order variational inequalities we give conditions on the right-hand side and the set of constraints under which solutions of the variational inequalities are bounded. Our conditions on the set of constraints admit the consideration of a sufficiently large class of problems with pointwise constraints, and in particular, unilateral and bilateral problems. They also admit the consideration of the Dirichlet problem for the corresponding equations. We provide a series of examples which demonstrate the essentiality of the imposed conditions. In particular, we show that the condition assumed for the right-hand side of the variational inequalities in general is unimprovable in the scale of Lebesgue spaces.     

On variational problems with obstacles and integral constraints for vector-valued functions

The authors prove existence and regularity for vectorvalued solutions of n-dimensional variational problems with boundary conditions, integral constraints, and obstacles as side conditions. Main emphasis is given to the regularity proof in the case n=2 generalizing a well known technique due to C. B. Morrey. In addition, a regularity result is stated for the general n-dimensional case.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

ON THE COUPLING OF BOUNDARY INTEGRAL AND FINITE ELEMENT METHODS FOR SIGNORINI PROBLEMS

1.IntroductionPartialdifferentialequationssubjecttounilateralboundaryconditionsareusuallycalledSignoriniproblemsintheliterature.TheseproblemshavebeenstudiedbymanyauthodssincetheappearenceofthehistoricalpaperbyA.Signoriniin1933[25].Signoriniproblemsaroseinmanyareasofapplicationse.g.,theelasticitywithunilateralconditions[lo],thefluidmechnicsproblemsinmediawithsemipermeableboundaries[8,12],theelectropaintprocess[1]etc.Fortheexistence,uniquenessandregularityresultsforSignorinitypeproblemswerefer…     

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

Fractional variational problems depending on indefinite integrals

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of the previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered.     

入水冲击问题变分原理及其它

首先建立入水前后两个衔接阶段的较为严密的场方程.再得到与之对应的各类变分原理,界限定理,第二阶段问题的边界积分方程.证明了解的存在性并提供了求解实施方案.最后以船舶兴波阻力问题的算例,论证了第二阶段问题的一种特殊应用及其正确性.从而为求取较为精确的入水冲击问题基本方程的变分有限元及边界元方法奠定了严密的理论基础.     

A variational inequality arising from European option pricing with transaction costs

In this paper we present a method which can transform a variational inequality with gradient constraints into a usual two obstacles problem in one dimensional case.The prototype of the problem is a parabolic variational inequality with the constraints of two first order differential inequalities arising from a two-dimensional model of European call option pricing with transaction costs.We obtain the monotonicity and smoothness of two free boundaries.     

Analysis of stationary filtration problems with a multivalued law in the presence of several point sources

We suggest a generalized statement of stationary filtration problems for an incompressible fluid obeying a multivalued filtration law with limit gradient in an arbitrary bounded nonone-dimensional domain in the presence of several point sources modeled by delta functions. The function determining the filtration law is assumed to grow linearly at infinity. The problems are stated in the form of an integral variational inequality of the second kind. We prove existence theorems and study the properties of solutions. To solve the problem, we suggest an iteration method whose each step essentially amounts to solving the Dirichlet problem for the Poisson equation.     

Minimization of Energy Functionals Applied to Some Inverse Problems

We consider a general class of problems of the minimization of convex integral functionals subject to linear constraints. Using Fenchel duality, we prove the equality of the values of the minimization problem and its associated dual problem. This equality is a variational criterion for the existence of a solution to a large class of inverse problems entering the class of generalized Fredholm integral equations. In particular, our abstract results are applied to marginal problems for stochastic processes. Such problems naturally arise from the probabilistic approaches to quantum mechanics. Accepted 26 March 2001. Online publication 19 July 2001.     

The Bang-Bang,Purification and Convexity Principles in Infinite Dimensions

The authors apply their recent work on the Lyapunov theorem in locally convex Hausdorff spaces to the bang-bang principle for control systems in infinite dimensions. They show that the bang-bang principle holds for every integrably bounded, measurable, weakly compact convex-valued multifunction if and only if the underlying measure space is saturated. They also demonstrate the equivalence of the bang-bang principle to what is termed the purification and convexity principles. Applications to variational problems with integral constraints are indicated.     

A Green's Function Approach to Wave Propagation and Potential Flow around Obstacles in Infinite Cylindrical Channels

We study (a) acoustic waves generated by a time-harmonic force distribution and (b) the potential flow with prescribed velocity at infinity in an infinite cylinder Ω₀ = Ω′×ℝ with bounded cross-section Ω′⊂ℝ² in the presence of m embedded obstacles B₁,…,B_m. By using Green's function G_κ(x,y) of the Neumann problem for the reduced wave equation ΔU+κ²U = 0 in the unperturbed domain Ω₀, both problems can be reduced to integral equations over the boundaries of the obstacles. The main properties of G_κ(x,y), which are required for this approach, are derived in the first part of this paper.     

Constraints on singularities under Ricci curvature bounds

We announce results giving constraints on the singularities of spaces which are Gromov-Hausdorf'f limits of sequences of Riemannian manifolds whose Ricci curvature and volume are bounded from below and whose curvature tensor is bounded in an integral sense.     

A dual-dual mixed formulation for nonlinear exterior transmission problems

We combine a dual-mixed finite element method with a Dirichlet-to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

     

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

Abstract

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.     

On the coupling of BEM and FEM for exterior problems for the Helmholtz equation

This paper deals with the coupled procedure of the boundary element method (BEM) and the finite element method (FEM) for the exterior boundary value problems for the Helmholtz equation. A circle is selected as the common boundary on which the integral equation is set up with Fourier expansion. As a result, the exterior problems are transformed into nonlocal boundary value problems in a bounded domain which is treated with FEM, and the normal derivative of the unknown function at the common boundary does not appear. The solvability of the variational equation and the error estimate are also discussed.

     

Asymptotics of the solution in a problem of optimal boundary control of a flow through a part of the boundary

We consider a problem of optimal control through a part of the boundary of solutions to an elliptic equation in a bounded domain with smooth boundary with a small parameter at the Laplace operator and integral constraints on the control. A complete asymptotic expansion of the solution to this problems in powers of the small parameter is constructed.     

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.     

Symmetric coupling of finite and boundary elements for exterior magnetic field problems

We consider a coupled finite element (fe)–boundary element (be) approach for three‐dimensional magnetic field problems. The formulation is based on a vector potential in a bounded domain (fe) and a scalar potential in an unbounded domain (be). We describe a coupled variational problem yielding a unique solution where the constraints in the trial spaces are replaced by appropriate side conditions. Then we discuss a Galerkin discretization of the coupled problem and prove a quasi‐optimal error estimate. Finally we discuss an efficient preconditioned iterative solution strategy for the resulting linear system. Copyright © 2002 John Wiley & Sons, Ltd.     

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.