Differential and Integral Equations

Closure of smooth maps in $W^{1,p}(B^3;S^2)$

Augusto C. Ponce and Jean Van Schaftingen

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For every $2 < p < 3$, we show that $u \in W^{1,p}(B^3;S^2)$ can be strongly approximated by maps in $C^\infty(\overline B \,\!^3;S^2)$ if, and only if, the distributional Jacobian of $u$ vanishes identically. This result was originally proved by Bethuel-Coron-Demengel-H\'elein, but we present a different strategy which is motivated by the $W^{2,p}$-case.

Article information

Differential Integral Equations, Volume 22, Number 9/10 (2009), 881-900.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58D15: Manifolds of mappings [See also 46T10, 54C35]
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 46T20: Continuous and differentiable maps [See also 46G05]


Ponce, Augusto C.; Van Schaftingen, Jean. Closure of smooth maps in $W^{1,p}(B^3;S^2)$. Differential Integral Equations 22 (2009), no. 9/10, 881--900. https://projecteuclid.org/euclid.die/1356019513

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