Density of smooth functions in weighted Sobolev spaces with variable exponent

SynopsisIn this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the spaceRnor a half-space ofRn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.

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A MEAN CONTINUITY TYPE RESULT FOR CERTAIN SOBOLEV SPACES WITH VARIABLE EXPONENT

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000762 ◽

2002 ◽

Vol 04 (03) ◽

pp. 587-605 ◽

Cited By ~ 11

Author(s):

ALBERTO FIORENZA

Keyword(s):

Sobolev Spaces ◽

Variable Exponent ◽

Lower Semicontinuous ◽

Type Result ◽

Smooth Functions ◽

Open Set

For functions f in Sobolev spaces W1,p(x)(Ω) with exponent lower semicontinuous, bounded away from 1 and ∞ and with the property of the density of smooth functions, it is shown that for each open set ω ⊂⊂ Ω, for each h ∈ RN such that ω+th ⊂ Ω ∀ t∈ [0,1], the following inequality holds [Formula: see text] where min p(x, x + h) denotes the minimum of p along the segment whose endpoints are x, x + h. As a consequence, if p(x) is also continuous, for mollifiers (ρn){n ∈ N the liminf and the limsup of [Formula: see text] are respectively minorized and majorized by expressions equivalent to ‖|∇f|‖Lp(x)(Ω).

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Nonlinear smoothing and unconditional uniqueness for the Benjamin–Ono equation in weighted Sobolev spaces

Nonlinear Analysis ◽

10.1016/j.na.2020.112227 ◽

2021 ◽

Vol 205 ◽

pp. 112227

Author(s):

Simão Correia

Keyword(s):

Sobolev Spaces ◽

Weighted Sobolev Spaces ◽

Nonlinear Smoothing ◽

Unconditional Uniqueness

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Pseudo-monotonicity and degenerated or singular elliptic operators

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032184 ◽

1998 ◽

Vol 58 (2) ◽

pp. 213-221 ◽

Cited By ~ 9

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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