42.3. SUBGRADIENTS 1379

and so1h

(|y(t +h)− y1 (t +h)|2

2− |y(t)− y1 (t)|2

2

)

+

(1h

∫ t+h

t(z(s)− z1 (s))ds,y(t +h)− y1 (t +h)

)≤ 0

Then taking limh→0 it follows that for a.e. t (Lebesgue points of z− z1 intersected with thepoints where |y− y1|2 has a derivative)

12

ddt|y(t)− y1 (t)|2 +(z(t)− z1 (t) ,y(t)− y1 (t))≤ 0

Thus for a.e. t,ddt|y(t)− y1 (t)|2 ≤ 0

and so|y(t)− y1 (t)|2−|y0− y0|2 =

∫ t

0

ddt|y(s)− y1 (s)|2 ds≤ 0.

Suppose you know that (z,y)≥ δ |y|2−C for all y∈D(A) ,z∈ Ay in addition to A beingmaximal monotone. Then for y the solution to the evolution inclusion such that y′+ z = ffor z ∈ Ay, you could multiply by y and integrate to obtain

12|y(t)|2H −

12|y0|2H +δ

∫ t

0|y(s)|2H ds−CT ≤Cδ

∫ t

0| f (s)|2H ds+

δ

2

∫ t

0|y(s)|2H ds

and so we have an estimate of the form

|y(t)|2H +∫ t

0|y(s)|2H ds≤C

If (z,y)≥ δ ∥y∥2V −C where V is a Banach space contained in H with large norm, then the

same result would hold with H replaced with V .In the above theorem, it was assumed y0 ∈ D(A). This can be generalized in the case

where A is the subgradient of a proper convex function which will be explored in the nextseveral sections.

42.3 Subgradients42.3.1 General ResultsDefinition 42.3.1 Let X be a real locally convex topological vector space. For x ∈ X,δφ (x)⊆ X ′, possibly /0. This subset of X ′ is defined by y∗ ∈ δφ (x) means for all z ∈ X,

y∗ (z− x)≤ φ (z)−φ (x).

Also x ∈ δφ∗ (y∗) means that for all z∗ ∈ X ′,

(z∗− y∗)(x)≤ φ∗ (z∗)−φ

∗ (y∗).

We define dom(δφ)≡ {x : δφ (x) ̸= /0}.