Degenerate elliptic boundary-value problems of second order in nonsmooth domains

Borsuk, Mikhail

doi:10.1007/s10958-007-0470-1

Cited by 9 publications

(5 citation statements)

References 86 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We derive a priori estimates in the Kondratiev spaces and an estimate of the type u(x) ¼ O(jxj ) with exact exponent . Similar estimates have been established in [6] for Dirichlet problem. We apply methods of [6,7].…”

Section: Introductionsupporting

confidence: 69%

See 1 more Smart Citation

On the degenerate oblique derivative problem for elliptic second-order equation in a domain with boundary conical point

Bodzioch

Borsuk

2012

Complex Variables and Elliptic Equations

Self Cite

View full text Add to dashboard Cite

We study the behaviour of strong solutions to the degenerate oblique derivative problem for linear second-order elliptic equation in a neighbourhood of the boundary conical point of a bounded domain. IntroductionIn our article we study the behaviour of strong solutions to the degenerate oblique derivative problem for linear second-order elliptic equation in a neighbourhood of the boundary conical point of an n-dimensional, !3, bounded domain. Solonnikov et al. [1,2] have studied a similar problem for the Laplace operator in a plane angle. They proved the unique existence of solutions and obtained a priori estimates for weak solutions in the Sobolev-Kondratiev weighted spaces.Lieberman [3][4][5] derived some pointwise estimates and regularity results for solution to the linear uniformly elliptic second-order oblique derivative problem.We study our problem for general second-order linear elliptic equation. The degeneration of the problem occurs in the boundary condition. We derive a priori estimates in the Kondratiev spaces and an estimate of the type u(x) ¼ O(jxj ) with exact exponent . Similar estimates have been established in [6] for Dirichlet problem. We apply methods of [6,7].Our article is an extension of [8,9]. In [8], such problem in a plain domain with angular point and in [9] for similar problem in a three-dimensional bounded domain with the boundary conical point were considered.

show abstract

Section: Introductionsupporting

confidence: 69%

“…Similar estimates have been established in [6] for Dirichlet problem. We apply methods of [6,7].Our article is an extension of [8,9]. In [8], such problem in a plain domain with angular point and in [9] for similar problem in a three-dimensional bounded domain with the boundary conical point were considered.…”

mentioning

confidence: 99%

On the degenerate oblique derivative problem for elliptic second-order equation in a domain with boundary conical point

Bodzioch

Borsuk

2012

Complex Variables and Elliptic Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Borsuk obtains a weak maximum principle, a Hopf lemma, and a strong maximum principle [13 1.7. Notation and conventions.…”

Section: Comparison With Previous Researchmentioning

confidence: 99%

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Feehan

2013

Communications in Partial Differential Equations

View full text Add to dashboard Cite

Abstract. We prove weak and strong maximum principles, including a Hopf lemma, for C 2 subsolutions to equations defined by second-order, linear elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C 2 subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C 2 solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W 1,2 solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions. Contents List of

show abstract

“…[13]- [15]). Отметим также работу [35], где получены точные коэр-цитивные оценки решения в весовых классах Гёльдера.…”

Section: § 1 введениеunclassified

О Разрешимости Первой Начально-Краевой Задачи Для Параболических И Вырождающихся Параболических Уравнений В Областях С Конической Точкой

Дегтярев¹,

Degtyarev²

2010

Матем. сб.

View full text Add to dashboard Cite

Общероссийский математический портал С. П. Дегтярев, О разрешимости первой начально-краевой задачи для параболических и вырождающих-ся параболических уравнений в областях с конической точкой, Матем. сб., 2010, том 201, номер 7, 67- О разрешимости первой начально-краевой задачи для параболических и вырождающихся параболических уравнений в областях с конической точкойНа основании одного известного подхода изучается первая начально-краевая задача для параболических и вырождающихся параболических уравнений второго порядка в области с конической или угловой точкой в весовых классах гладких или суммируемых функций. В терминах угло-вой меры телесного угла и показателя веса сформулированы достаточные условия существования единственного решения и наличия коэрцитивных оценок решения. Показано также, что если указанные условия не выпол-няются, то параболическая задача обладает свойствами эллиптической, т.е. может иметь ненулевое ядро или быть неразрешимой, причем в по-следнем случае она не является даже фредгольмовой. Рассмотрены па-раболическое уравнение и уравнение с вырождением или особенностью в конической точке.Библиография: 49 названий.Ключевые слова: параболическое уравнение, нерегулярная область, коэрцитивные оценки, спектральные свойства.

show abstract

Degenerate elliptic boundary-value problems of second order in nonsmooth domains

Cited by 9 publications

References 86 publications

On the degenerate oblique derivative problem for elliptic second-order equation in a domain with boundary conical point

On the degenerate oblique derivative problem for elliptic second-order equation in a domain with boundary conical point

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

О Разрешимости Первой Начально-Краевой Задачи Для Параболических И Вырождающихся Параболических Уравнений В Областях С Конической Точкой

Contact Info

Product

Resources

About