scholarly journals Applications of abstract parabolic quasi-variational inequalities to obstacle problems

2009 ◽  
Author(s):  
Risei Kano
Keyword(s):  
Variational Inequalities ◽  
Obstacle Problems

Noncoercive variational inequalities with application to friction problems

1991 ◽  
Vol 117 (3-4) ◽  
pp. 275-293 ◽  
Author(s):  
P. Shi ◽  
M. Shillor
Keyword(s):  
Variational Inequalities ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Obstacle Problems ◽  
Elastic Contact ◽  
Compatibility Conditions ◽  
Sublinear Functionals ◽  
Noncoercive Variational Inequalities ◽  
Necessary And Sufficient

SynopsisNoncoercive variational inequalities with sublinear functionals are considered. Necessary and sufficient conditions are given for the solvability of such problems. These conditions are in the form of compatibility conditions-for the data, as well as the boundedness of the solutions to related problems. These results are used for the obstacle problems for the membrance and the elastic contact in the presence of friction.

Boundary regularity for linear and quasilinear variational inequalities

1989 ◽  
Vol 112 (3-4) ◽  
pp. 319-326 ◽  
Author(s):  
Gary M. Lieberman
Keyword(s):  
Boundary Conditions ◽  
Variational Inequalities ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Parabolic Obstacle Problems ◽  
Second Derivatives ◽  
Quasilinear Problems ◽  
Derivatives Of

SynopsisA method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.

scholarly journals Optimal control for obstacle problems involving time-dependent variational inequalities with Liouville–Caputo fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Parinya Sa Ngiamsunthorn ◽  
Apassara Suechoei ◽  
Poom Kumam
Keyword(s):  
Optimal Control ◽  
Variational Inequalities ◽  
Obstacle Problem ◽  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Existence Results ◽  
The State ◽  
Time Dependent ◽  
Obstacle Problems ◽  
Caputo Fractional Derivatives

AbstractWe consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives.

scholarly journals On the asymptotic behavior of the solutions to parabolic variational inequalities

2020 ◽  
Vol 2020 (768) ◽  
pp. 149-182
Author(s):  
Maria Colombo ◽  
Luca Spolaor ◽  
Bozhidar Velichkov
Keyword(s):  
Asymptotic Behavior ◽  
Variational Inequalities ◽  
Stationary Solution ◽  
Global Solutions ◽  
Singular Points ◽  
Obstacle Problems ◽  
Abstract Result ◽  
Łojasiewicz Inequality ◽  
Lojasiewicz Inequality ◽  
Thin Obstacle

AbstractWe consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

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