2582 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Thus a stochastic Stefan problem could be considered in the form(−∆−1)w(t)− (−∆)−1 w0 +

∫ t

0α (w)ds =

(−∆−1)∫ t

0f ds+

(−∆−1)∫ t

0ΦdW

This example can be included in the above general theory because((−∆−1)u−

(−∆−1)v,u− v

)≥ ∥u− v∥2

V ′ , V ′ ≡ H−1,V = H10 (U)

This is seen as follows. V,L2 (U) ,V ′ is a Gelfand triple. Then −∆ is the Riesz map R fromH1

0 (U) to H−1 (U). then you have(y,R−1y

)L2(U)

=⟨RR−1y,R−1y

⟩V ′,V =

∥∥R−1y∥∥2

V = ∥y∥2V ′

Next we give a simple example which is a singular and degenerate equation. This isa model problem which illustrates how the theory can be used. This problem is mixedparabolic and stochastic and nonlinear elliptic. It is a singular equation because the coef-ficient b can be unbounded. The existence of a solution is easy to obtain from the abovetheory but it does not follow readily from other methods. If p = 2 it is an abstract versionof stochastic heat equation which could model a material in which the density becomesvanishingly small in some regions and very large in other regions.

Example 76.6.3 Suppose U is a bounded open set in R3 and b(x)≥ 0, b ∈ Lp (U) , p≥ 4for simplicity. Consider the degenerate stochastic initial boundary value problem

b(·)u(t, ·)−b(·)u0 (·)−∫ t

0∇ ·(|∇u|p−2

∇u)

= b∫ t

0ΦdW

u = 0 on ∂U

where Φ ∈ L2([0,T ]×Ω,L2

(Q1/2U,W

))for W = H1

0 (U) .

To consider this equation and initial condition, it suffices to let W = H10 (U) ,V =

W 1,p0 (U) ,

A : V →V ′, ⟨Au,v⟩=∫

U|∇u|p−2

∇u ·∇vdx,

B : W →W ′,⟨Bu,v⟩=∫

Ub(x)u(x)v(x)dx

Then by the Sobolev embedding theorem, B is obviously self adjoint, bounded and non-negative. This follows from a short computation:∣∣∣∣∫U

b(x)u(x)v(x)dx∣∣∣∣≤ ∥v∥L4(U)

(∫U|b(x)|4/3 |u(x)|4/3 dx

)3/4

≤ ∥v∥H10 (U)

((∫|b(x)|4 dx

)1/3(∫ (|u(x)|4/3

)3/2)2/3

)3/4

= ∥v∥H10 (U) ∥b∥L4(U) ∥u∥L2(U) ≤ ∥b∥L4 ∥u∥H1

0∥v∥H1

0