Constant Q-curvature metrics in arbitrary dimension

Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order non-linear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.     

On Conformal Metrics with Constant Q-Curvature

We review some recent results in the literature concerning existence of conformal metrics with constant Q-curvature.The problem is rather similar to the classical Yamabe problem:however辻 is characterized by a fourth-order operator that might lack in general a maximum principle.For several years existence of geometrically admissible solutions was known only in particular cases.Recently;there has been instead progress in this direction for some general classes of conformal metrics.     

Multiplicity Result for a Scalar Field Equation on Compact Surfaces

Abstract

We consider a scalar field equation on compact surfaces which has variational structure. When the surface is a torus and a physical parameter ρ belongs to (8π,4π²) we show under some extra assumptions that, besides a local minimum, the functional admits at least other two saddle points.     

On a General Projection Algorithm for Variational Inequalities

Let H be a real Hilbert space with norm and inner product denoted by and . Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let be a point-to-set mapping. We deal with the problem of finding uK such that g(u)K(u) and the following relation is satisfied: , where >0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.     

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.     

增广Lagrangian对偶分解方法求解变分不等式系统

本文考虑带约束的变分不等式系统.提出一个基于增广Lagrangian对偶的分解算法,本文给出了算法的收敛性分析.     

An Abstract Problem in Variational Analysis

The purpose of this paper is to present a unifying approach to study various models of equilibrium theory and variational inclusions. A simple condition is established for the existence of solutions of variational relations and is applied to a number of variational problems.     

The Level Set Method for Systems of PDEs

We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans (1996 Evans , L. C. ( 1996 ). A geometric interpretation of the heat equation with multivalued initial data . SIAM J. Math. Anal. 27 ( 4 ): 932 – 958 .[Crossref], [Web of Science ®] , [Google Scholar]) for the heat equation and by Giga and Sato (2001 Giga , Y. , Sato , M.-H. ( 2001 ). A level set approach to semicontinuous viscosity solution for Cauchy problems . Comm. Partial Differential Equations 26 ( 5–6 ): 813 – 839 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) for Hamilton–Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each nonzero sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method to a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction–diffusion equations.     

Variational inequalities with cone constraints

The present paper considers variational inequalities with cone and operator constraints. The characterization of solutions is given. An algorithm for numerical solution of the problem is proposed, and the convergence of the generated sequence of points is proved.This research was supported in part by the Science Foundation of the Republic of Serbia.The author thanks two anonymous referees for their helpful remarks.     

Variational minimizing solutions for the 2-fixed center problems

Using variational minimizing methods, we study the 2-fixed center problems.     

Tikhonov Regularization Methods for Variational Inequality Problems

Motivated by the work of Facchinei and Kanzow (Ref. 1) on regularization methods for the nonlinear complementarity problem and the work of Ravindran and Gowda (Ref. 2) for the box variational inequality problem, we study regularization methods for the general variational inequality problem. A sufficient condition is given which guarantees that the union of the solution sets of the regularized problems is nonempty and bounded. It is shown that solutions of the regularized problems form a minimizing sequence of the D-gap function under a mild condition.     

Extragradient Methods for Pseudomonotone Variational Inequalities

We consider and analyze some new extragradient-type methods for solving variational inequalities. The modified methods converge for a pseudomonotone operator, which is a much weaker condition than monotonicity. These new iterative methods include the projection, extragradient, and proximal methods as special cases. Our proof of convergence is very simple as compared with other methods.     

Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence

In this paper, we propose a modified extragradient method for solving variational inequalities (VI) which has the following nice features: (i) The generated sequence possesses an expansion property with respect to the starting point; (ii) the existence of the solution to a VI problem can be verified through the behavior of the generated sequence from the fact that the iterative sequence diverges to infinity if and only if the solution set is empty. Global convergence of the method is guaranteed under mild conditions. Our preliminary computational experience is also reported.     

Proximal Methods for Mixed Variational Inequalities

A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.     

Some Predictor-Corrector Algorithms for Multivalued Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued variational inequalities. The convergence of the proposed methods requires only the partially-relaxed strong monotonicity of the operator, which is weaker than cocoercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.     

Opérateurs différentiels magnétiques : stabilité des trous dans le spectre,invariance du spectre essentiel et applications

We study here the problem of geometry optimization for a crystal in the Thomas–Fermi–Von Weizsäcker (TFW) solid-state setting, i.e., the problem of minimizing the TFW energy with respect to the periodic lattice defining the positions of the nuclei. We show the existence of such a minimum, and use for that purpose the TFW models of polymers and thin films defined in a previous work (X. Blanc and C. Le Bris, Adv. Differential Equations, 5, 977–1032, 2000).     

Variational Inequalities and the Elastic-Plastic Torsion Problem

We show that, under suitable conditions, the variational inequality that expresses the elastic-plastic torsion problem is equivalent to a variational inequality on a convex set which depends on (x)=d(x, ). Such an equivalence allows us to find the related Lagrange multipliers and to exhibit a computational procedure based on the subgradient method.     

A Variational Analysis of the Toda System on Compact Surfaces

In this paper we consider the following Toda system of equations on a compact surface: We will give existence results by using variational methods in a noncoercive case. A key tool in our analysis is a new Moser‐Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u₁ and u₂. © 2012 Wiley Periodicals, Inc.     

Iterative Schemes for Nonconvex Variational Inequalities

In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets.     

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.