On superposition operators between higher-order Sobolev spaces and a multivariate Faà di Bruno formula: the subcritical case. (English) Zbl 1278.47058

Dinca, George; Isaia, Florin

Differ. Integral Equ. 26, No. 1-2, 11-58 (2013).

The present article deals with the superposition operators \((N_gu)(x) = g(u(x)): W^{m,p}(\Omega) \to W^{l,q}(\Omega)\) between two Sobolev spaces \(W^{p,m}(\Omega)\) and \(W^{l,q}(\Omega)\) of functions on an open bounded domain \(\Omega \subset \mathbb {R}^n\) with the cone property. It is assumed that the function \(g: {\mathbb R} \to \mathbb {R}\) is of class \(C^{l-1}\) and the derivative \(g^{(l-1)}: \mathbb {R} \to \mathbb {R}\) is locally Lipschitz. In the case \(1 \leq p <\frac {n}{m}\) and \(1 \leq q \leq \frac {np}{l[n - (m - 1)p]}\), the operator \(N_g\) acts between \(W^{p,m}(\Omega)\) and \(W^{l,q}(\Omega)\), provided that the following growth condition \[ | g^{(l)}(t)| \leq a| t| ^{\frac {n(p-lq)+(m-1)lpq}{q(n-mp)}} + b \] holds a.e.in \(\mathbb {R}\); moreover, in this case, the following inequality \[ \| N_gu\|_{W^{l,q}(\Omega)} \leq C\bigg [\| u\|_{W^{m,p}(\Omega)}^{\frac {p(n-lq)}{q(n-mp)}} + 1\bigg ], \quad u \in W^{m,p}(\Omega), \] is true; \(C\) is a constant depending only on \(\Omega ,n,m,l,p,q,a,b,g(0),\dots ,g^{(l-1)}(0)\). In the case \(p =\frac {n}{m}\), \(1 \leq q <\frac {n}{l}\), the operator \(N_g\) acts between \(W^{p,m}(\Omega)\) and \(W^{l,q}(\Omega)\), provided that the following growth condition \[ | g^{(l)}(t)| \leq a| t| ^d + b \] holds; in this case, the following inequality \[ \| N_gu\|_{W^{l,q}(\Omega)} \leq C\bigg [\| u\|_{W^{m,p}(\Omega)}^{d+l} + 1\bigg ], \quad u \in W^{m,p}(\Omega), \] is true. Similar statements are formulated in some important partial cases, in particular when \(g\) is a polynomial of degree \(h \leq l\). The sufficient conditions presented in the article guarantee the continuity of the operator \(N_g\), the validity of the higher-order chain rule (the Faá de Bruno formula, cf. [C. F. Faà di Bruno, “Note sur une nouvelle formule du calcul différentiel”, Quart. J. Math. 1, 359–360 (1855)]) and also the important embedding \(N_g(W^{m,p}(\Omega) \cap W_0^{k,p}(\Omega)) \subset W_0^{l,q}(\Omega)\). A comparison of these results with classical theorems by M. Marcus and V. J. Mizel [Trans. Am. Math. Soc. 251, 187–218 (1979; Zbl 0417.46035)] and G. Bourdaud [Invent. Math. 104, No. 2, 435–446 (1991; Zbl 0699.46019)] is given.

Reviewer: Peter Zabreiko (Minsk)

MSC:

47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B05	Continuity and differentiation questions
26A46	Absolutely continuous real functions in one variable

Keywords:

superposition operator; Sobolev space; well-definedness; growth condition

Citations:

Zbl 0417.46035; Zbl 0699.46019