2580 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Now let β (t) be a function which satisfies

β′ (t) =

1k (K−1 (t))

for t ̸= τ ≡ K (σ)

and it has a jump equal to b at τ .

τ

β (v)

v

Let v = K (u) . Then for u ̸= σ , equivalently v ̸= τ,

vt = K′ (u)ut = k(K−1 (v)

)ut

and sout =

1k (K−1 (v))

vt =ddt

(β (v))

Also,v,i = K′ (u)u,i = k (u)u,i

and sou,i =

1k (u)

v,i

Hence

(k (u)u,i),i =(

k (u)1

k (u)v,i

),i= ∆v

Thus, off the set S,β (v)t −∆v = f

Now let φ ∈ L2(0,T,H1

0 (U))

with φ (x,T ) = 0. Then assume S is sufficiently smooth thatthings like divergence theorem apply. Also note that u = σ is the same as v = τ .∫

Q(β (v)t −∆v)φ =

∫S+

(β (v)t −∆v)φ +∫

S−(β (v)t −∆v)φ

=∫

S+(β (v)φ)t −β (v)φ t − (v,iφ),i + v,iφ ,i

+∫

S−(β (v)φ)t −β (v)φ t − (v,iφ),i + v,iφ ,i

Now using the divergence theorem, and continuing these formal manipulations, the abovereduces to∫

Sβ (v(+))φnt − (v,i (+)φ)ni +

∫S+−β (v)φ t + v,iφ ,i−

∫U∩S+

β (v(x,0))φ (x,0)

+∫

S−β (v(−))φnt +(v,i (−)φ)ni +

∫S−−β (v)φ t + v,iφ ,i−

∫U∩S−

β (v(x,0))φ (x,0)