Solutions Of Nonlinear Variational Inequalities Research Articles

Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control

The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon. Supposing that some sufficiently smooth conditions hold, by the dynamic programming principle, the author builds the Hamilton-Jacobi-Bellman (HJB for short) variational inequality for the value function. The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variational inequality.

An adaptive wavelet Galerkin scheme for solving contact problems based on elliptic variational inequalities of the first kind

PurposeThe real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.Design/methodology/approachTo generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.FindingsAdaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.Practical implicationsThe proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.Originality/valueAlthough many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

Variational Inequalities over Perturbed Polyhedral Convex Sets

This paper provides conditions for existence of a locally unique, Lipschitzian solution of a linear variational inequality posed over a polyhedral convex set in ℝn under perturbation of either or both of the constant term in the variational inequality and the right-hand side of the system of linear constraints defining its feasible set. Conditions for perturbation of just the constant term are well known. Here we show that a suitable extension of those conditions suffices for the more general case in which the right-hand side of the constraints varies also. As a consequence, we obtain existence, uniqueness, and Lipschitz continuity properties of solutions of nonlinear variational inequalities posed over perturbed polyhedral convex sets.

Solutions for nonlinear variational inequalities with a nonsmooth potential

First we examine a resonant variational inequality driven by the p‐Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving the p‐Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the form φ = φ1 + φ2 with φ1 locally Lipschitz and φ2 proper, convex, lower semicontinuous.

General algorithm of solutions for nonlinear variational inequalities in Banach space

In this paper, we shall study a class of general nonlinear variational inequalities in reflexive Banach spaces. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the general nonlinear variational inequalities are proved. Next, by applying the auxiliary problem technique, we suggest an innovative iterative algorithm to compute the approximate solutions of the general nonlinear variational inequality. Finally, the convergence criteria is also discussed.

Estimates for solutions of nonlinear variational inequalities

We prove a pointwise estimate for the solution u of a nonlinear variational inequality in terms of a function which is solution of a suitable variational inequality with spherically symmetric data. Using this result a lower bound for the measure of the set {x : u(x) = 0} and a priori estimates for the LP-norm and the W01,p-norm of u are obtained. An existence result is also given.

Multiple Existence of Solutions for Nonlinear Variational Inequalities

Multiple Existence of Solutions for Nonlinear Variational Inequalities

Regularity of solutions of nonlinear variational inequalities with a gradient bound as constraint

Regularity of solutions of nonlinear variational inequalities with a gradient bound as constraint

Regularity of solutions of nonlinear variational inequalities

Regularity of solutions of nonlinear variational inequalities