THE STOKES PROBLEM IN ℝn: AN APPROACH IN WEIGHTED SOBOLEV SPACES

1999 ◽  
Vol 09 (05) ◽  
pp. 723-754 ◽  
Author(s):  
F. ALLIOT ◽  
C. AMROUCHE
Keyword(s):  
Asymptotic Expansion ◽  
Sobolev Spaces ◽  
Vector Fields ◽  
Stokes Problem ◽  
Weighted Sobolev Spaces ◽  
Explicit Representation ◽  
Solenoidal Vector ◽  
Regularity Results

We prove some existence, uniqueness and regularity results for the solutions to the Stokes problem in ℝn, n≥2 in weighted Sobolev spaces [Formula: see text]. This framework enables us to characterise for which data the problem has solutions with prescribed decay or growth at infinity. Moreover, we obtain an explicit representation as well as an asymptotic expansion of the solution for non-smooth decaying data. We also establish the density of smooth solenoidal vector fields in the subspace of [Formula: see text] such that div v=0.

scholarly journals Regularity results for p-Laplacians in pre-fractal domains

2018 ◽  
Vol 8 (1) ◽  
pp. 1043-1056 ◽  
Author(s):  
Raffaela Capitanelli ◽  
Salvatore Fragapane ◽  
Maria Agostina Vivaldi
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Obstacle Problems ◽  
Convex Polygons ◽  
Regularity Results ◽  
Fractal Domains ◽  
Laplace Type

Abstract We study obstacle problems involving p-Laplace-type operators in non-convex polygons. We establish regularity results in terms of weighted Sobolev spaces. As applications, we obtain estimates for the FEM approximation for obstacle problems in pre-fractal Koch Islands.

Piecewise Solenoidal Vector Fields and the Stokes Problem

1990 ◽  
Vol 27 (6) ◽  
pp. 1466-1485 ◽  
Author(s):  
Garth A. Baker ◽  
Wadi N. Jureidini ◽  
Ohannes A. Karakashian
Keyword(s):  
Vector Fields ◽  
Stokes Problem ◽  
Solenoidal Vector

scholarly journals Two-Scale Regularity for Homogenization Problems with Nonsmooth Fine Scale Geometry

2003 ◽  
Vol 13 (07) ◽  
pp. 1053-1080 ◽  
Author(s):  
A.-M. Matache ◽  
J. M. Melenk
Keyword(s):  
Detailed Analysis ◽  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Unbounded Domains ◽  
Wave Number ◽  
Critical Parameters ◽  
Fine Scale ◽  
Cell Problem ◽  
Regularity Results ◽  
Homogenization Problems

Elliptic problems on unbounded domains with periodic coefficients and geometries are analyzed and two-scale regularity results for the solution are given. These are based on a detailed analysis in weighted Sobolev spaces of the so-called unit-cell problem in which the critical parameters (the period ε, the wave number t, and the differentiation order) enter explicitly.

Sobolev spaces of solenoidal vector fields

1982 ◽  
Vol 22 (3) ◽  
pp. 399-420 ◽  
Author(s):  
V. N. Maslennikova ◽  
M. E. Bogovskii
Keyword(s):  
Sobolev Spaces ◽  
Vector Fields ◽  
Solenoidal Vector

scholarly journals Regularity results for nonlocal evolution Venttsel’ problems

2020 ◽  
Vol 23 (5) ◽  
pp. 1416-1430 ◽  
Author(s):  
Simone Creo ◽  
Maria Rosaria Lancia ◽  
Alexander I. Nazarov
Keyword(s):  
Sobolev Spaces ◽  
Fractional Laplacian ◽  
A Priori Estimates ◽  
A Priori ◽  
Extension Theorem ◽  
Weighted Sobolev Spaces ◽  
Nonlocal Term ◽  
Two Dimensional ◽  
Priori Estimates ◽  
Regularity Results

Abstract We consider parabolic nonlocal Venttsel’ problems in polygonal and piecewise smooth two-dimensional domains and study existence, uniqueness and regularity in (anisotropic) weighted Sobolev spaces of the solution. The nonlocal term can be regarded as a regional fractional Laplacian on the boundary. The regularity results deeply rely on a priori estimates, obtained via the so-called Munchhausen trick, and sophisticated extension theorem for anisotropic weighted Sobolev spaces.

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