6.3. More advanced functional analysis

6.3.1. Nemitski operators

This section deals with a class of operations that is frequently encountered in non linear problems. We refer to [AP95] for more details.

Throughout this section, \(\Omega\) is a bounded set in \(\R^d\).

Definition 6.7

A function \(f : \Omega \times \R \to \R\) is called a Caratheodory function if it satisfies the following two properties:

• For almost every \(\x \in \Omega\), the mapping \(s \mapsto f(\x,s)\) is continuous.

• For all \(s \in \R\), the mapping \(\x \mapsto f(x,s)\) is measurable.

Now, to each Caractheodory function, we associate formally a (non linear) operator \(T_f\) acting on functions \(u : \Omega \to \R\) via the following formula:

\[T_f u (\x) = f(\x, u(\x)). \]

Under suitable conditions on the function \(f\), this defines a mapping between certain Lebesgue spaces, as made more specific by the following statement.

Theorem 6.4

Let \(p,q \geq 1\), and let \(f : \Omega \times \R \to \R\) be a Caratheodory function, satisfying the following growth condition:

\[\text{For a.e. } \x \in \Omega \text{ and all } s \in \R, \quad \lvert f(x,s) \lvert \leq a + b \lvert s \lvert^{\frac{p}{q}}.\]

Then the mapping \(T_f\) maps \(L^p(\Omega)\) into \(L^q(\Omega)\), and it defines a continuous operator between these spaces.

The following statement now gives sufficient conditions for this mapping to be differentiable.

Theorem 6.5 (Differentiability of Nemitski operators, case \(p>2\))

Let \(p > 2\) be given, and let \(f : \Omega \times \R \to \R\) be a Caratheodory function such that:

• The function \(\Omega \ni \x \mapsto f(\x,0)\) is bounded on \(\Omega\);

• It holds, for a.e. \(\x \in \Omega\) and all \(s \in \R\), \(\left\lvert \frac{\partial f}{\partial s} (\x,s) \right\lvert \leq a + b \lvert s \lvert^{p-2}\).

Then the mapping \(T_f\) defines a continuous operator from \(L^p(\Omega)\) into \(L^{p^\prime}(\Omega)\), where \(p^\prime\) is the conjugate exponent of \(p\). Moreover, this mapping is Fréchet differentiable, with derivative:

\[f(\x,u(\x)+v(\x)) = f(\x,u(\x)) + \frac{\partial f}{\partial s} (\x,u(\x)) v(\x) + \o(\lvert\lvert v \lvert\lvert_{L^p(\Omega)}). \]

Note that the operator \(T_f\) is well defined as an operator from \(L^p(\Omega)\) into \(L^{p^\prime}(\Omega)\), since the above condition in particular implies, upon integration, that:

\[\lvert f(\x,s) \lvert \leq a + b \lvert s \lvert^{p-1}, \text{ for different constants } a,b >0.\]

The above theorem does not cover the case where \(p=2\). A similar – albeit a little weaker – statement holds in this case, under slightly stronger assumptions.

Theorem 6.6 (Differentiability of Nemitski operators, case \(p=2\))

Let \(f : \Omega \times \R \to \R\) be a Caratheodory function such that \(\frac{\partial f}{\partial s}\) is also a Caratheodory function. Assume in addition that:

\[\text{There exists a constant } C >0 \text{ s.t. } \lvert f(\x,s) \lvert \leq C \text{ for a.e. } \x \in \Omega \text{ and all } s \in \R. \]

Then \(T_f\) is a continuous operator from \(L^2(\Omega)\) into itself. Moreover, this mapping is Gâteaux differentiable at every \(u \in L^2(\Omega)\), i.e.

\[\forall v \in L^2(\Omega), \quad \left\lvert\left\lvert \frac{f(\x,u(\x)+t(v(\x))) - f(\x,u(\x))}{t} - \frac{\partial f}{\partial s}(\x,u(\x)) v(\x)\right\lvert\right\lvert \xrightarrow{t\to 0} 0.\]

Exercise

Show that the mapping \(T:H^1(\Omega) \to L^2(\Omega)\), defined by:

\[\forall u \in H^1(\Omega), \quad T(u) = \max(0,u)\]

is Fréchet differentiable at any \(u \in H^1(\Omega)\), with derivative:

\[T^\prime(u)(h) = \chi_{\left\{ u \geq 0\right\}} h.\]

Correction

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