On the minimal solution for some variational inequalities

Applying an asymptotic method, the existence of the minimal solution to some variational elliptic inequalities defined on bounded or unbounded domains is established. The minimal solution is obtained as limit of solutions to some classical variational inequalities defined on domains becoming unbounded when some parameter tends to infinity. The considered quasilinear operators are only monotone (not strictly) and noncoercive. Some related comparison principles are also investigated.     

Su talune disequazioni variazionali ellittiche non lineari del secondo ordine

Summary In this note, making use of a result of J. L. Lions, we examine some non linear elliptic variational inequalities defined on domains which may be unbounded. Such variational inequalities are associated to a uniformely second order elliptic operator. We start with the derivation of an existence theorem (on bounded domains) under non coerciveness assumptions. Next we examine the convergence for the solutions of a collection of variational inequalities. To this purpose we study convergence theorems for variational inequalities associated to operators belonging to a class of abstract mapping of pseudomonotone type between Banach spaces. The solvability of some variational inequalities on unbounded domains then follows directly.

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Entrata in Redazione il 6 aprile 1977.

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Lavoro eseguito nell'ambito del C.N.R., Laboratorio per la Matematica Applicata via L. B. Alberti 4, Genova.     

Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities

A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.     

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.     

Existence Results for Evolution Noncoercive Hemivariational Inequalities

This paper is devoted to the existence of solutions for evolution hemivariational inequalities as generalizations of evolution variational inequalities to nonconvex functionals. The operators involved are taken to be multivalued and noncoercive. Using the notion of the generalized gradient of Clarke and the recession method, some existence results of solutions are proved.     

Elliptic variational hemivariational inequalities

This paper is devoted to the existence of solutions for elliptic variational hemivariational inequalities. The operators involved are taken to be multivalued and noncoercive. Using the notion of the generalized gradient of Clarke and recession method, some existence results of solutions have been proved.     

The Alexandrov-Bakelman-Pucci Weak Maximum Principle for Fully Nonlinear Equations in Unbounded Domains

ABSTRACT

We obtain Alexandrov-Bakelman-Pucci type estimates for semicontinuous viscosity solutions of fully nonlinear elliptic inequalities in unbounded domains. Under suitable assumptions relating the geometry of the domain with structural conditions on the differential operator, we establish the validity of the weak maximum principle for solutions which are bounded from above. Two variants are also given, namely one for unbounded solutions in narrow domains and one for operators with possibly changing sign zero order coefficients in domains of small measure.     

Some existence results for a class of degenerate semilinear elliptic systems

Using a variational approach, we investigate a class of degenerate semilinear elliptic systems with measurable, unbounded nonnegative weights, where the domain is bounded or unbounded. Some existence results are obtained.     

Existence results for viscous polytropic fluids with vacuum

We consider the full Navier-Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover our results hold for both bounded and unbounded domains.     

On Some Noncoercive Variational Inequalities

We study existence and regularity of solutions of noncoercive variational inequalities.     

A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature

We prove a pointwise gradient bound for bounded solutions of Δu+F^′(u)=0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature.We also obtain some rigidity results when equality in the bound holds at some point.     

Nonlinear Elliptic Variational Inequalities

We consider the problem of finding a solution to a class of nonlinear elliptic variational inequalities. These inequalities may be defined on bounded or unbounded domains Ω, and the nonlinearity can depend on gradient terms. Appropriate definitions of sub-and supersolutions relative to the constraint sets are given. By using a mixture of maximal monotone operator theory and compactness arguments we prove the existence of a H²(Ω) solution lying between a given subsolution φ₁ and a given supersolution φ₂≧φ₁, when Ω is bounded, and a H¹(Ω) solution when Ω is unbounded.     

一类非强制的变分不等式

讨论了一类非强制的变分不等式解之存在性的一个必要条件和一个充分条件,改进了文[1]中的相应结果。     

Equilibrium problems and noncoercive variational inequalities

In this paper, we are interested to establish existence results for equilibrium problems in a noncoercive framework by using techniques of recession analysis. The abstract result is then applied to find solution of noncoercive monotone and pseudomonotone variational inequalities     

COMPARISON, SYMMETRY AND MONOTONICITY RESULTS FOR SOME DEGENERATE ELLIPTIC OPERATORS IN CARNOT－CARATHEODORY SPACES

This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces.The authors show some com-parison results for solutions of the relevant differential inequalities and use them to get somesymmetry and monotonicity properties of solutions,in bounded or unbounded domains.     

On the solvability of noncoercive linear variational inequalities in separable Hilbert spaces

In this paper, we build an existence theory for linear variational inequalities associated with an operator which generalizes in Hilbert space the class of copositive plus matrices. We show how this theory can be used to study some important engineering problems governed by noncoercive variational inequalities.Thanks are due to Professor V. H. Nguyen for many valuable discussions. The author thanks the Associate Editor and the referees for their helpful suggestions     

Enclosure results for quasilinear systems of variational inequalities

In this paper we consider systems of quasilinear elliptic variational inequalities, and prove the existence of minimal and maximal (in the set theoretical sense) solutions within some ordered interval of an appropriately defined pair of sub- and supersolutions. We show that the notion of sub- and supersolutions of variational inequalities introduced here is consistent with the usual notion of sub-supersolutions for (variational) equations. For weakly coupled quasimonotone systems of variational inequalities the existence of smallest and greatest solutions is proved.     

Blow-up of solutions of some nonlinear inequalities in unbounded domains

Results concerning the blow-up of nontrivial nonnegative solutions are obtained for several classes of nonlinear partial differential inequalities and systems in unbounded domains with coefficients having singularities near the boundary of the domain.     

Harmonic solutions for noncoercive dynamic unilateral problems

This paper studies the solvability of a general class of variational inequalities. Existence of periodic solutions for noncoercive variational inequalities will be proved.     

Infinitely many solutions of a Sturm-Liouville system with impulses

Using variational method and lower and upper solutions, we present a new approach to obtain the existence of infinitely many solutions of a second-order Sturm-Liouville system with impulse effects. As applications, we get arbitrary small sequence and unbounded sequence of nontrivial nonnegative solutions, assuming that some oscillatory behaviours of F, associated to the system, at 0 and +∞.