Abstract
In the previous chapter, when studying the model problem − Δu = f(u), we have already met operators of the form u↦f(u) where f is a mapping from \({\mathbb R}\) to \({\mathbb R}\), and u belongs to some function space defined over an open subset of \({\mathbb R}^d\). This type of operator is called a superposition operator or Nemytsky operator. Let us now study these operators in more detail in various functional contexts.
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Notes
- 1.
As of now, we stop distinguishing between function f and superposition operator \(\tilde f\) for a lighter notation.
- 2.
Take two sequences with two different Young measures and mix them.
- 3.
As well as taking any other value almost everywhere.
- 4.
Of course, T′(u) may well be undefined elsewhere, but that would be on a negligible set.
- 5.
We need the “loc” here, because we have no regularity hypothesis on Ω.
- 6.
The space \(W^{1,\infty }_0\) can be defined as the space of W 1, ∞ functions that continuously extend by 0 on ∂ Ω.
- 7.
Same proof as the Carathéodory theorem, or using the Lipschitz character of T.
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Le Dret, H. (2018). Superposition Operators. In: Nonlinear Elliptic Partial Differential Equations. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-78390-1_3
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DOI: https://doi.org/10.1007/978-3-319-78390-1_3
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