42.3. SUBGRADIENTS 1387

φ (ũ(t))≤ ⟨z̃(t) , ũ(t)− ũ(t0)⟩+φ (ũ(t0)) (42.3.21)

Then 42.3.21 shows t→ φ (ũ(t)) is in L1 (0,T ) since z̃ ∈ Lp′ ([0,T ] ;X ′), ũ ∈ Lp ([0,T ] ;X).Also, for t ∈ [0,T −h],⟨

X[0,T−h] (t) z̃(t) ,ũ(t +h)− ũ(t)

h

⟩≤X[0,T−h] (t)

φ (ũ(t +h))−φ (ũ(t))h

≤⟨

X[0,T−h] (t) z̃(t +h) ,ũ(t +h)− ũ(t)

h

⟩Now X[0,T−h] (·) z̃(·+h)→ z(·) in Lp′ (0,T ;X ′) by continuity of translation. Also,

X[0,T−h] (·)ũ(·+h)− ũ(·)

h= X[0,T−h] (·)

u(·+h)−u(·)h

= X[0,T−h] (·)1h

∫ (·)+h

(·)u′ (s)ds

in Lp (0,T ;X) and so by Lemma 42.3.12,

X[0,T−h] (·)φ (ũ(·+h))−φ (ũ(·))

h→⟨z,u′⟩

in L1 (0,T ).It follows from the definition of weak derivatives that in the sense of weak derivatives,

ddt

(φ (u(·))) =⟨z,u′⟩∈ L1 (0,T ).

Note that by Theorem 26.3.3 this implies that for a.e. t ∈ [0,T ], φ (u(t)) is equal to acontinuous function, φ ◦u, and that

(φ ◦u)(t)− (φ ◦u)(0) =∫ t

0

⟨z(s) ,u′ (s)

⟩ds.

There are other rules of calculus which have a generalization to subgradients. Thefollowing theorem is on such a generalization. It generalizes the theorem which states thatthe derivative of a sum equals the sum of the derivatives.

Theorem 42.3.14 Let φ 1 and φ 2 be convex, l.s.c. and proper having values in (−∞,∞].Then

δ (λφ i)(x) = λδφ i (x) , δ (φ 1 +φ 2)(x)⊇ δφ 1 (x)+δφ 2 (x) (42.3.22)

if λ > 0. If there exists x ∈ dom(φ 1)∩ dom(φ 2) and φ 1 is continuous at x then for allx ∈ X,

δ (φ 1 +φ 2)(x) = δφ 1 (x)+δφ 2 (x). (42.3.23)