The Euler Method in the Field of Operators

Approximate solutions of a class of nonlinear differential equations are constructed in the field of Mikusiński operators by using a method similar to Euler's. Their character is analyzed and the error of approximation is estimated. The results are applied to a class of partial integro–differential equations.     

Solutions of stochastic partial differential equations as extremals of convex functionals

Summary Stochastic evolution equations with monotone operators in Banach spaces are considered. The solutions are characterized as minimizers of certain convex functionals. The method of monotonicity is interpreted as a method of constructing minimizers to these functionals, and in this way solutions are constructed via Euler-Galerkin approximations.     

Minimization of entropy functionals

Entropy (i.e. convex integral) functionals and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Primal attainment, dual equalities, dual attainment and characterizations of the minimizers are obtained with weak constraint qualifications. These results improve several aspects of the literature on the subject.     

On matrices with operator entries

In this paper we consider certain matrix equations in the field of Mikusiński operators, and construct a method for obtaining an approximate solution which allows working with numerical constants instead of operators. The theory of diagonally dominant matrices is applied for the analysis, existence and character of the obtained solutions. We introduce a method for determining approximate solutions of a discrete analogue for operational differential equations and give conditions for their existence. The error of the approximation is estimated.     

γ-Convergence of integral functionals and the variational dirichlet problem in variable domains

By using special local characteristics of domains Ω _s ⊂Ω,s=12,..., we establish necessary and sufficient conditions for the γ-convergence of sequences of integral functionalsI _λs :W ^k,m (Ω _s )→ℝ, λ⊂Ω to interal functionals defined on W ^k,m (Ω). Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1236–1254, September, 1996.     

Averaging of integral functionals related to domains of frame-type periodic structure with thin channels

We establish the Γ-convergence of a sequence of integral functionals related to domains of frame-type periodic structure with thin channels. We obtain a representation for the integrand of a Γ-limit functional.     

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces.We prove that the operators are bounded on L^p provided that the kernels satisfy weak conditions.     

An Operatorial approach to singular integral equations of a modified type

In this article,the authors discuss a kind of modified singular integral equations on a disjoint union of closed contours or a disjoint union of open arcs.The authors introduce some singular integral operators associated with this kind of singular integral equations,and obtain some useful properties for them.An operatorial approach is also given together with some illustrated examples.     

Second-order epi-derivatives of integral functionals

Epi-derivatives have many applications in optimization as approached through nonsmooth analysis. In particular, second-order epi-derivatives can be used to obtain optimality conditions and carry out sensitivity analysis. Therefore the existence of second-order epi-derivatives for various classes of functions is a topic of considerable interest. A broad class of composite functions on ⁿ called fully amenable functions (which include general penalty functions composed withC ² mappings, possibly under a constraint qualification) are now known to be twice epi-differentiable. Integral functionals appear widely in problems in infinite-dimensional optimization, yet to date, only integral functionals defined by convex integrands have been shown to be twice epi-differentiable, provided that the integrands are twice epi-differentiable. Here it is shown that integral functionals are twice epi-differentiable even without convexity, provided only that their defining integrands are twice epi-differentiable and satisfy a uniform lower boundedness condition. In particular, integral functionals defined by fully amenable integrands are twice epi-differentiable under mild conditions on the behavior of the integrands.This work was supported in part by the National Science Foundation under grant DMS-9200303.     

Numerical solution of boundary integral equations by means of attenuation factors

We consider first-kind boundary integral equations with logarithmickernel such as those arising from solving Dirichlet problemsfor the Laplace equation by means of single-layer potentials.The first-kind equations are transformed into equivalent equationsof the second kind which contain the conjugation operator andwhich are then solved with a degenerate-kernel method basedon Fourier analysis and attenuation factors. The approximationswe consider, among them spline interpolants, are linear andtranslation invariant. In view of the particularly small kernel,the linear systems resulting from the discretization can besolved directly by fixed-point iteration.     

Convergence of spline collocation for Volterra integral equations

Estimates for step-by-step interpolation projections are established. Depending on the spectrum of the transfer matrix these estimates allow to obtain the pointwise convergence of the projectors to the identity operator or, in some limit cases, to prove stable convergence of the corresponding approximate operators of integral equations. This, via general convergence theorems for operator equations, permits to get the convergence of collocation method for Volterra integral equations of the second kind in spaces of continuous or certain times continuously differentiable functions. Applications in the case of the most practical types of splines are analyzed.     

Boundary value problems for difference equations with causal operators

By using the monotone iterative method, some new results are established for nonlinear boundary conditions of difference problems with causal operators. We formulate sufficient conditions under which such problems have extremal solutions. Difference inequalities with causal operators are also discussed. Two examples are added to illustrate the results.     

Two-scale convergence of some integral functionals

Nguetseng’s notion of two-scale convergence is reviewed, and some related properties of integral functionals are derived. The coupling of two-scale convergence with convexity and monotonicity is then investigated, and a two-scale version is provided for compactness by strict convexity. The div-curl lemma of Murat and Tartar is also extended to two-scale convergence, and applications are outlined.     

Weak lower semicontinuity of integral functionals

A lower semicontinuity theorem for integral functionals is proved underL ₁-strong convergence of the trajectories andL ₁-weak convergence of the control functions. An alternative statement is also proved under pointwise convergence of the trajectories.     

Convex integral functionals of regular processes

This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be identified with the space of optional random measures with essentially bounded variation. Combined with classical Banach space techniques, our results allow for a systematic treatment of stochastic optimization problems over BV processes and, in particular, yields a maximum principle for a general class of singular stochastic control problems.     

Second-order subgradients of convex integral functionals

The purpose of this work is twofold: on the one hand, we study the second-order behaviour of a nonsmooth convex function defined over a reflexive Banach space . We establish several equivalent characterizations of the set , known as the second-order subdifferential of at relative to . On the other hand, we examine the case in which is the functional integral associated to a normal convex integrand . We extend a result of Chi Ngoc Do from the space to a possible nonreflexive Banach space . We also establish a formula for computing the second-order subdifferential .

     

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.