Weak solutions for the exterior Stokes problem in weighted Sobolev spaces

2000 ◽  
Vol 23 (6) ◽  
pp. 575-600 ◽  
Author(s):  
F. Alliot ◽  
C. Amrouche
Keyword(s):  
Sobolev Spaces ◽  
Weak Solutions ◽  
Stokes Problem ◽  
Weighted Sobolev Spaces

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.

Multiplicity results for a class of asymptotically p-linear equations on ℝN

2016 ◽  
Vol 18 (01) ◽  
pp. 1550031 ◽  
Author(s):  
Rossella Bartolo ◽  
Anna Maria Candela ◽  
Addolorata Salvatore
Keyword(s):  
Elliptic Equation ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Quasilinear Elliptic Equation ◽  
Linear Equations ◽  
Weighted Sobolev Spaces ◽  
Embedding Theorem ◽  
Multiplicity Results ◽  
Quasilinear Elliptic

The aim of this paper is investigating the multiplicity of weak solutions of the quasilinear elliptic equation [Formula: see text] in [Formula: see text], where [Formula: see text], the nonlinearity [Formula: see text] behaves as [Formula: see text] at infinity and [Formula: see text] is a potential satisfying suitable assumptions so that an embedding theorem for weighted Sobolev spaces holds. Both the non-resonant and resonant cases are analyzed.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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