Elliptic Problems with Boundary Conditions of Higher Orders in Hörmander Spaces

Kasirenko, Tetiana; Murach, O. O.

doi:10.1007/s11253-018-1466-3

Cited by 4 publications

(1 citation statement)

References 46 publications

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“…These spaces form the extended Sobolev scale considered in [50,52] and [51,Section 2.4.2]. It has important applications to elliptic operators [54,55,73,74] and elliptic boundary-value problems [6,7,8,9,31]. Among them are applications to the investigation of various types of convergence of spectral expansions induced by elliptic operators.…”

Section: Introductionmentioning

confidence: 99%

An extended Hilbert scale and its applications

Mikhailets,

Murach,

Zinchenko

2021

Preprint

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We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of OR-varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, and spectral expansions induced by abstract and elliptic operators.

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Section: Introductionmentioning

confidence: 99%

An extended Hilbert scale and its applications

Mikhailets,

Murach,

Zinchenko

2021

Preprint

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Elliptic problems with nonclassical coundary conditions in an extended Sobolev scale

Murach¹,

Chepurukhina²

2020

Dopov. Nac. akad. nauk Ukr.

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Еліптичні задачі з некласичними крайовими умовами у розширеній соболєвській шкалі Представлено членом-кореспондентом НАН України А.Н. Кочубеєм Розглянуто еліптичні задачі з некласичними крайовими умовами, які містять додаткові невідомі функції на межі області задання еліптичного рівняння та крайові оператори порядків, вищих, ніж порядок цього рівняння. Досліджено розв'язність вказаних задач і властивості їх розв'язків у розширеній соболєвській шка лі. Вона складається з гільбертових узагальнених просторів Соболєва, для яких показником регулярності є довільна радіальна функція, RO-змінна за Авакумовичем на нескінченності. Встановлено теорему про нетеровість вказаних задач на відповідних парах цих просторів і теореми про регулярність та апріорну оцінку узагальнених розв'язків задач. Отримано точні достатні умови неперервної диференційовності компонент цих розв'язків.

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Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

Anop

Chepurukhina

Murach

2021

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In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

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Elliptic Problems with Boundary Conditions of Higher Orders in Hörmander Spaces

Cited by 4 publications

References 46 publications

An extended Hilbert scale and its applications

An extended Hilbert scale and its applications

Elliptic problems with nonclassical coundary conditions in an extended Sobolev scale

Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

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