Nonlinear potential theory and weighted Sobolev spaces

The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems

RESEARCH ARTICLE An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

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Estimating effective boundaries of population growth in a variable environment

We study the impact of age-structure and temporal environmental variability on the persistence of populations. We use a linear age-structured model with time-dependent vital rates. It is the same as the one presented by Chipot in (Arch. Ration. Mech. Anal. 82(1):13-25, 1983), but the assumptions on the vital rates are slightly different. Our main interest is in describing the large-time behavior of a population provided that we know its initial distribution and transient vital rates. Using upper and lower solutions for the characteristic equation, we define time-dependent upper and lower boundaries for a solution in a constant environment. Moreover, we estimate solutions for the general time-dependent case and also for a special case when the environment is changing periodically

An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method

Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H-1/2)* spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well. (C) 2016 Elsevier Ltd. All rights reserved

Relationships among anxiety, perceived stress, and burnout in young athletes

KOPSAVILKUMS Pētījuma mērķis bija noskaidrot, vai pastāv sakarība starp sportistu izdegšanas trim dimensijām – fizisko nogurumu, sporta vērtības samazināšanos un pazeminātu sniegumu - ar stresa, trauksmes līmeni, tā saistību ar traumatisma līmeni. Pētījumā piedalījās dažādu sporta veidu 104 abu dzimumu sportisti, Latvijas izlases dalībnieki, vecumā no astoņpadsmit līdz divdesmit gadiem. Pētījumā tika izmantota DASS-21 aptauja bez depresijas skalas un Sportistu izdegšanas aptauja. Aptauja tika papildināta ar četriem traumu smaguma pakāpi precizējošiem jautājumiem. Tika pierādīta cieši nozīmīga sakarība starp stresu un trauksmi, vidēji nozīmīga starp sportistu izdegšanas trim dimensijām un stresu, trauksmi. Netika apstiprināta fizisko traumu sakarība ar sportistu pašsajūtas aptaujas elementiem, stresu vai trauksmi. Korelācijas atklāja arī vidēji ciešu emocionālo traumu sakarību ar dzimumu, fizisko nogurumu, stresu un trauksmi. Atslēgas vārdi : stress, trauksmes līmenis, jauno sportistu pašsajūta, aptaujas, sakarība, traumas.SUMMARY The aim of the study was to find out whether there is a relationship between the three dimensions of athletes' well-being - physical fatigue, reduced sports value and reduced performance with stress, anxiety, how it relates to the level of trauma. The study involved 104 athletes of different sports, participants of both sexes, from 18 to twenty years of age. The study used the DASS-21 survey, with the depression subdivision removed, and the Athlete Well-Being Survey. Four questions were added to clarify the severity of the injuries. There was a strong relationship between stress and anxiety, a moderately significant relationship between the three dimensions of athletes' well-being and stress and anxiety. There was no correlation between physical injuries and the elements of the athletes' well-being survey, stress, anxiety. Showed moderate emotional correlation with gender, physical fatigue, stress, and anxiety. Keywords: stress, anxiety level, well-being of young athletes, surveys, correlation, injurie

Numerical Solution of the Cauchy Problem for the Helmholtz Equation

The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem