Minimal functions with unbounded integral

We prove theorems which imply the following results. (1) “Most” almost periodic functionsb(t) with unbounded integral oscillate in a strong sense. (2) IfB is a continuous function on a minimal flow (Ω,R), then either the time averages all converge, or they diverge on a residual set.     

An integral representation of functions

We obtain an integral representation of functions of real variables which can be considered (at least formally) as some analog of the Fourier formula.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 67–72, 1974.     

An integral representation for decomposable measures of measurable functions

Summary We start with a measurem on a measurable space (,A), decomposable with respect to an Archimedeant-conorm on a real interval [0,M], which generalizes an additive measure. Using the integral introduced by the second author, a Radon-Nikodym type theorem, needed in what follows, is given.The integral naturally leads to a -decomposable measurem on the space of all measurable functions from to [0, 1]. The main result of the present paper is the converse of this, namely that, under natural conditions, any -decomposable measurem on can be represented as an integral of a certain Markov-kernelK. We extend this representation to measures on which have values in a set of distribution functions.These results generalize the work done by the first author in the case of additive measures.     

Pontryagin functions for multiple integral control problems

An optimal control problem is considered whose performance index is represented by anm-fold multiple integral, the state equations being given by a system of first-order partial differential equations. The concept of a field of optimal control variables with respect to an independent integral is introduced, the significance of these fields being due to the fact that a control pair is optimal whenever it is imbedded in such a field. Since there arem distinctm-fold independent integrals, it is possible to constructm distinct fields of this kind. For each of these, a Pontryagin function is defined, and it is shown that, if an optimal pair is embedded in one of these fields, it satisfies a corresponding Pontryagin maximum principle.This research was supported in part by NSF Grant No. GP-32830     

On the integral representation of superbiharmonic functions

We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary. Research supported in part by IPM under the grant number 83310011.     

A local integral representation of strictly plurisubharmonic functions

Translated from Matematicheskie Zametki, Vol. 53, No. 6, pp. 108–115, June, 1993.     

An integral representation of functions in gas-kinetic models

Motivated by the theory of kinetic models in gas dynamics, we obtain an integral representation of lower semicontinuous functions on \({{\mathbb{R}}^d,}\)\({d\geq1}\). We use the representation to study the problem of compactness of a family of the solutions of the discrete time BGK model for the compressible Euler equations. We determine sufficient conditions for strong compactness of moments of kinetic densities, in terms of the measures from their integral representations.     

On variational problems with obstacles and integral constraints for vector-valued functions

The authors prove existence and regularity for vectorvalued solutions of n-dimensional variational problems with boundary conditions, integral constraints, and obstacles as side conditions. Main emphasis is given to the regularity proof in the case n=2 generalizing a well known technique due to C. B. Morrey. In addition, a regularity result is stated for the general n-dimensional case.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

A new integral representation for quasi-periodic scattering problems in two dimensions

Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

A coverage-based Box-Algorithm to compute a representation for optimization problems with three objective functions

In this paper, a method for computing a substitute of the nondominated set, a so-called representation system, is proposed for tricriteria optimization problems. Our algorithm is easy to implement, it is flexibly applicable to a wide range of problems, and it takes several quality measures which have been proposed in the literature into account. The resulting representation system satisfies provably certain quality levels and, thus, it may be utilized to make proper and confident decisions.