Differential and Integral Equations

On superposition operators between higher-order Sobolev spaces and a multivariate Faà di Bruno formula: the subcritical case

George Dinca and Florin Isaia

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Abstract

In this paper, superposition operators, $ (N_gu ) (x )= g (u (x ) )$, between two arbitrary Sobolev spaces are studied. Sufficient conditions which ensure the well-definedness, the continuity, and the validity of the higher-order chain rule for such operators are given in the subcritical case (see Remark 1.1). As a consequence of these properties, it is proved that $N_g (W^{m,p} (\Omega )\cap W_0^{k,p} (\Omega ) )\subset W_0^{l,q} (\Omega )$.

Article information

Source
Differential Integral Equations, Volume 26, Number 1/2 (2013), 11-58.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1355867505

Mathematical Reviews number (MathSciNet)
MR3058696

Zentralblatt MATH identifier
1278.47058

Subjects
Primary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 26B05: Continuity and differentiation questions 26A46: Absolutely continuous functions

Citation

Dinca, George; Isaia, Florin. On superposition operators between higher-order Sobolev spaces and a multivariate Faà di Bruno formula: the subcritical case. Differential Integral Equations 26 (2013), no. 1/2, 11--58. https://projecteuclid.org/euclid.die/1355867505


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