Cea's error estimate for strongly monotone variational inequalities

Céa's approximation lemma is extended to variational inequalities which are defined by strongly monotone operators in closed convex subsets of linear normed spaces. This abstract error estimate is applied to the finite element discretization of a nonlinear elliptic two-sided obstacle problem providing an asymptotic error estimate for a smooth enough solution.     

An unusual self-adjoint linear partial differential operator

In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

     

On left and right uninorms on a finite chain

The main concern of this paper is to introduce and characterize the class of operators on a finite chain L, having the same properties of pseudosmooth uninorms but without commutativity. Moreover, in this case it will only be required the existence of a one-side neutral element. These operators are characterized as combinations of AND and OR operators of directed algebras (smooth t-norms and smooth t-conorms) and the case of pseudosmooth uninorms is retrieved for the commutative case.     

Perturbations by lower order terms do not destroy the global hypoellipticity of certain systems of pseudodifferential operators defined on torus

We introduce a new class of smooth pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity with a finite loss of derivatives of certain systems of pseudodifferential operators is stable under perturbations by lower order systems of pseudodifferential operators whose order depends on the loss of derivatives. We also present some applications.     

Eigenvalue decay rates for positive integral operators

We present decay rates for the eigenvalues of positive integral operators with smooth kernels on special metric spaces endowed with a strictly positive measure. The smoothness is defined by either differentiability conditions or inequalities of Lipschitz type. We use the decay rates to place the operators in some Schatten p-classes.     

General Logic-Systems and Finite Consequence Operators

In this paper, the significance of using general logic-systems and finite consequence operators defined on non-organized languages is discussed. Results are established that show how properties of finite consequence operators are independent from language organization and that, in some cases, they depend only upon one simple language characteristic. For example, it is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language L that cannot be generated from any finite logic-system. On the other hand, it is shown that for any nonempty language L, a set map is a finite consequence operator if and only if it is defined by a general logic-system. Simple logic-system examples that determine specific consequence operator properties are given. Mathematics Subject Classification (2000): Primary 03B22, Secondary 03B65     

Operators with smooth functional calculi

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, which contains all operators of Helffer-Sjöstrand type and is closed under the action of smooth proper mappings. Moreover, the class is closed under tensor product of commuting operators. In general, and operator in this class has no resolvent in the usual sense, so the spectrum must be defined in terms of the functional calculus. We also consider invariant subspaces and spectral decompositions.     

<Emphasis Type="Italic">B</Emphasis>-essential and <Emphasis Type="Italic">B</Emphasis>-Weyl spectra of sum of two commuting bounded operators

In this paper, we devote our research to the B-essential spectra of the sum of two bounded linear operators defined on a Banach space by means of the B-essential spectra of each of the two operators where their products are finite rank operators.     

Properties of Toeplitz Operators on the Dirichlet Space Over the Ball

On the Dirichlet space of the unit ball, we study some algebraic properties of Toeplitz operators. We give a relation between Toeplitz operators on the Dirichlet space and their analogues defined on the Hardy space. Based on this, we characterize when finite sums of products of Toeplitz operators are of finite rank. Also, we give a necessary and sufficient condition for the commutator and semi-commutator of two Toeplitz operators being zero.     

Higher order limit q‐Bernstein operators

In this paper we give the estimates of the central moments for the limit q‐Bernstein operators. We introduce the higher order generalization of the limit q‐Bernstein operators and using the moment estimations study the approximation properties of these newly defined operators. It is shown that the higher order limit q‐Bernstein operators faster than the q‐Bernstein operators for the smooth functions defined on [0, 1]. Copyright © 2011 John Wiley & Sons, Ltd.     

The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles

Let L be a uniformly elliptic second order differential operator with nice coefficients, defined on a smooth, bounded domain in ℝ ^d , d ≥ 2, with either the Dirichlet or an oblique-derivative boundary condition. In this work we study the asymptotics for the principal eigenvalue of L under hard and soft obstacle perturbations. The hard obstacle perturbation of L is obtained by making a finite number of holes with the Dirichlet boundary condition on their boundaries. The main result gives the asymptotic shift of the principal eigenvalue as the holes shrink to points. The rates are expressed in terms of the Newtonian capacity of the holes and the principal eigenfunctions for the unperturbed operator and its formal adjoint. The soft obstacle corresponds to a finite number of compactly supported finite potential wells. Here we only consider the oblique-derivative Laplacian. The main difference from the hard obstacle problem is that phase transitions occur, due to the various scaling possibilities. Our results generalize known results on similar perturbations for selfadjoint operators. Our approach is probabilistic.     

Singular integral and maximal operators associated to hypersurfaces:L p theory

We consider singular integral and maximal operators associated to hypersurfaces given by the graph of a function whose level sets are defined by a convex function of finite type. We investigate the L^p theory for these operators which depend on geometric properties of the hypersurface.     

Minimization of a functional over the set of causal operators of causal Hilbert space

The minimization problem for a quadratic functional defined on the set of nonwarning (causal) operators acting in a causal Hilbert space can be regarded as an abstrat analog of the Wiener problem on constructing the optimal nonwarning filter. A similar problem also arises in the linear control problem with the quadratic performance criterion (in this case the transfer operators of a closed control system serve as causal ones). The introduction of causal operators in filtering theory and control theory is a mathematical expression of the causality principle, which must be taken into account for a number of problems. In the present paper we attempt to systematize the mathematical foundations of the abstract linear filtering theory, for which its basic results are expressed in terms of operators describing the filtering problem. We introduce and study a class of finite operators, a natural generalization of the class of causal operators, and give a solution of the minimization problem for a quadratic positive functional defined on the set of causal operators acting in a “discrete” causal space. Bibliography: 54 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995. pp. 143–187.     

Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility

Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

Characterizations of pseudodifferential operators on the circle

Globally defined operators are shown to be equivalent to the classical pseudodifferential operators on the circle. A characterization of the smooth operators for the regular representation of is also given.

     

Approximating spectral invariants of Harper operators on graphs II

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

     

First-order differential-operator equation with variable domains of piecewise smooth operators

We prove the existence, uniqueness, and smoothness of weak solutions of a first-order differential-operator equation with variable domains of nonself-adjoint piecewise smooth operators for which one has the corresponding majorant operators. We analyze the well-posedness and smoothness of weak solutions of three new mixed problems with piecewise smooth (in time) coefficients in the equations of finite and infinite order and in the boundary conditions.     

Reduction of variational inequalities with irregular operators on a ball to regular operator equations

We propose a method for reducing variational inequalities defined by general smooth irregular operators on a ball in a Hilbert space to equivalent regular operator equations. The mentioned equations involve the operator of metric projection on the boundary of the ball. We establish conditions which guarantee the local strong monotonicity of the obtained equations. We discuss applications to the problem of finding normed eigenvectors of nonlinear operators.     

Estimates for the classical parametrix for the Laplacian

We consider the integral operators which were used classically to give a parametrix and remainder for the Laplacian on a Riemannian manifold. Their kernels are defined in terms of the distance function. These operators are shown to be bounded operators on the L₂ Hilbert spaces of differential forms, under the hypotheses that the manifold be complete and of finite volume, and that it satisfy curvature bounds. Furthermore, the remainder is shown to be compact.