scholarly journals Existence, comparison, and compactness results for quasilinear variational-hemivariational inequalities

2005 ◽  
Vol 2005 (3) ◽  
pp. 401-417 ◽  
Author(s):  
S. Carl ◽  
Vy K. Le ◽  
D. Motreanu
Keyword(s):  
Convex Set ◽  
Indicator Function ◽  
Hemivariational Inequalities ◽  
Fundamental Notion ◽  
Locally Lipschitz Functional ◽  
Locally Lipschitz ◽  
Closed Convex Set ◽  
Quasilinear Elliptic

We consider quasilinear elliptic variational-hemivariational inequalities involving the indicator function of some closed convex set and a locally Lipschitz functional. We provide a generalization of the fundamental notion of sub- and supersolutions, on the basis of which we then develop the sub-supersolution method for variational-hemivariational inequalities, including existence, comparison, compactness, and extremality results.

scholarly journals Infinitely many solutions for a class of hemivariational inequalities involving p(x)-Laplacian

2017 ◽  
Vol 25 (2) ◽  
pp. 65-83
Author(s):  
Fariba Fattahi ◽  
Mohsen Alimohammady
Keyword(s):  
Main Tool ◽  
Neumann Boundary ◽  
Hemivariational Inequalities ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Smooth Bounded Domain ◽  
Laplacian Equation ◽  
Locally Lipschitz Functional ◽  
Locally Lipschitz ◽  
Symmetric Mountain Pass Lemma

AbstractIn this paper hemivariational inequality with nonhomogeneous Neumann boundary condition is investigated. The existence of infinitely many small solutions involving a class of p(x) - Laplacian equation in a smooth bounded domain is established. Our main tool is based on a version of the symmetric mountain pass lemma due to Kajikiya and the principle of symmetric criticality for a locally Lipschitz functional.

scholarly journals ON EIGENVALUE PROBLEMS FOR ELLIPTIC HEMIVARIATIONAL INEQUALITIES

2008 ◽  
Vol 51 (2) ◽  
pp. 407-419 ◽  
Author(s):  
Zhenhai Liu ◽  
Guifang Liu
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problems ◽  
Hemivariational Inequalities ◽  
Generalized Gradient ◽  
At Resonance ◽  
Quasilinear Elliptic ◽  
First Eigenfunction

AbstractThis paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

scholarly journals Auerbach Bases and Minimal Volume Sufficient Enlargements

2011 ◽  
Vol 54 (4) ◽  
pp. 726-738
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Linear Space ◽  
Isometric Embedding ◽  
Normed Linear Space ◽  
Convex Set ◽  
Linear Projection ◽  
Minimal Volume ◽  
Finite Dimensional ◽  
Dimensional Normed Space ◽  
Closed Convex Set

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

Points of egress in problems of Hamiltonian dynamics

1991 ◽  
Vol 109 (2) ◽  
pp. 405-417 ◽  
Author(s):  
C. J. Amick ◽  
J. F. Toland
Keyword(s):  
Convex Set ◽  
Hamiltonian Dynamics ◽  
Empty Interior ◽  
Natural Assumption ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Convexity Assumptions ◽  
Image Position ◽  
Closed Convex Set ◽  
Well Posed

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equationHere the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

On two pursuit differential game problems with state and geometric constraints in a Hilbert space

2021 ◽  
Vol 65 (3) ◽  
pp. 5-16
Author(s):  
Abbas Ja’afaru Badakaya ◽  
Keyword(s):  
Differential Game ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Nonempty Closed Convex Subset ◽  
Geometric Constraints ◽  
First Order ◽  
Control Functions ◽  
Closed Convex Set ◽  
The Given ◽  
First Order Differential Equations

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

On Convex Fundamental Regions for a Lattice

1961 ◽  
Vol 13 ◽  
pp. 177-178 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Lebesgue Measure ◽  
Convex Set ◽  
Fundamental Region ◽  
Linearly Independent ◽  
Whole Space ◽  
Closed Convex Set ◽  
Set Of Points

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1+ … + εnanwherea1… ,anare linearly independent vectors and the ε run over all integers. Letμdenote the Lebesgue measure. A closed convex setFis called afundamental regionfor ∧ if the setsF+x (x∈∧) cover the whole space without overlapping; that is, ifF0is the interior ofF,and 0 ≠x∈ ∧, thenF0∩(F0+x) =ϕ.

Extreme Points in H1(R)

1967 ◽  
Vol 19 ◽  
pp. 312-320 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Set ◽  
Complete Solution ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Riemann Surface ◽  
Harmonic Majorant ◽  
Closed Convex Set

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).

scholarly journals Some hemivariational inequalities in the Euclidean space

2019 ◽  
Vol 9 (1) ◽  
pp. 958-977 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Hemivariational Inequalities ◽  
Existence Of Weak Solutions ◽  
Lipschitz Continuous ◽  
Variational Structure ◽  
Locally Lipschitz ◽  
Sign Changing Solutions ◽  
Continuous Functionals

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

scholarly journals Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1824
Author(s):  
Stanisław Migórski ◽  
Long Fengzhen
Keyword(s):  
Directional Derivative ◽  
Upper Semicontinuity ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Strong Monotonicity ◽  
Monotonicity Condition ◽  
Elliptic Type ◽  
Existence Of A Solution ◽  
Locally Lipschitz ◽  
Clarke Subgradient

In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

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