Gradient Flows pp 227-278 | Cite as

Metric Slope and Subdifferential Calculus in Open image in new window(X)

Part of the Lectures in Mathematics ETH Zürich book series (LM)


As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, lower semicontinuous functionals φ : X → (−∞,+∞] defined in a Hilbert space X, the Fréchet Subdifferential ∂φ : X → 2 X of φ is a multivalued operator defined as
$$ \xi \in \partial \varphi \left( \upsilon \right) \Leftrightarrow \upsilon \in D\left( \varphi \right),\mathop {\lim \inf }\limits_{w \to \upsilon } \frac{{\varphi \left( w \right) - \varphi \left( \upsilon \right) - \left\langle {\xi ,w - \upsilon } \right\rangle }} {{\left| {w - \upsilon } \right|}} \geqslant 0, $$
which we will also write in the equivalent form for vD(φ)
$$ \xi \in \partial \varphi \left( \upsilon \right) \Leftrightarrow \varphi \left( w \right) \geqslant \varphi \left( \upsilon \right) + \left\langle {\xi ,w - \upsilon } \right\rangle + o\left( {\left| {w - \upsilon } \right|} \right)asw \to \upsilon . $$
As usual in multivalued analysis, the proper domain D(∂φ) ⊂ D(φ) is defined as the set of all vX such that ∂φ(v) ≠ φ; we will use this convention for all the multivalued operators we will introduce.


Variational Inequality Lower Semicontinuity Doubling Condition Multivalued Operator Minimal Selection 
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© Birkhäuser Verlag AG 2008

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