Kenneth Kuttler

76.7. OTHER EXAMPLES, INCLUSIONS 2583

Also for some δ > 0⟨Au−Av,u− v⟩ ≥ δ ∥u− v∥p

The nonlinear operator is obviously monotone and hemicontinuous. As for u0, it is onlynecessary to assume u0 ∈ L2 (Ω,W ) and F0 measurable. Then Theorem 76.4.7 gives theexistence of a solution in the sense that for a.e. ω the integral equation holds for all t. Notethat b can be unbounded and may also vanish. Thus the equation can degenerate to the caseof a non stochastic nonlinear elliptic equation.

The existence theorems can easily be extended to include the situation where Φ isreplaced with a function of the unknown function u. This is done by splitting the timeinterval into small sub intervals of length h and retarding the function in the stochasticintegral, like a standard proof of the Peano existence theorem. Then the Ito formula isapplied to obtain estimates and a limit is taken.

Other examples of the usefulness of this theory will result when one considers stochas-tic versions of systems of partial differential equations in which there is a nonlinear cou-pling between a parabolic equation and a nonlinear elliptic equation. These kinds of prob-lems occur, for example as quasistatic damage problems in which the damage parametersatisfies a parabolic equation and the balance of momentum is a nonlinear elliptic equationand the two equations are coupled in a nonlinear way.

76.7 Other Examples, InclusionsThe above general result can also be used as a starting point for evolution inclusions orother situations where one does not have hemicontinuous operators. Assume here that

Φ ∈ L∞

([0,T ]×Ω,L2

(Q1/2U,H

)).

We will use the following simple observation. Let α > 2. Let ∥Φ∥∞

denote the norm inL∞([0,T ]×Ω,L2

(Q1/2U,H

)). By the Burkholder Davis Gundy inequality,∫

Ω

(∣∣∣∣∫ t

sΦdW

∣∣∣∣)α

dP≤

∫Ω

(sup

r∈[s,t]

∣∣∣∣∫ r

sΦdW

∣∣∣∣)α

dP≤C∫

Ω

(∫ t

s∥Φ∥2 dτ

)α/2

≤C∥Φ∥α

∞

∫Ω

(∫ t

sdτ

)α/2

dP =C∥Φ∥α

∞|t− s|α/2

By the Kolmogorov Čentsov theorem, this shows that t →∫ t

0 ΦdW is Holder continuouswith exponent

γ <(α/2)−1

α=

12− 1

Since α > 2 is arbitrary, this shows that for any γ < 1/2, the stochastic integral is Holdercontinuous with exponent γ . This is exactly the same kind of continuity possessed by theWiener process. We state this as the following lemma.

76.7. OTHER EXAMPLES, INCLUSIONS 2583Also for some 6 > 0(Au — Av,u—v) > 6 |lu—v)|f,The nonlinear operator is obviously monotone and hemicontinuous. As for uo, it is onlynecessary to assume ug € L? (Q,W) and Yo measurable. Then Theorem 76.4.7 gives theexistence of a solution in the sense that for a.e. @ the integral equation holds for all t. Notethat b can be unbounded and may also vanish. Thus the equation can degenerate to the caseof a non stochastic nonlinear elliptic equation.The existence theorems can easily be extended to include the situation where ® isreplaced with a function of the unknown function u. This is done by splitting the timeinterval into small sub intervals of length 4 and retarding the function in the stochasticintegral, like a standard proof of the Peano existence theorem. Then the Ito formula isapplied to obtain estimates and a limit is taken.Other examples of the usefulness of this theory will result when one considers stochas-tic versions of systems of partial differential equations in which there is a nonlinear cou-pling between a parabolic equation and a nonlinear elliptic equation. These kinds of prob-lems occur, for example as quasistatic damage problems in which the damage parametersatisfies a parabolic equation and the balance of momentum is a nonlinear elliptic equationand the two equations are coupled in a nonlinear way.76.7 Other Examples, InclusionsThe above general result can also be used as a starting point for evolution inclusions orother situations where one does not have hemicontinuous operators. Assume here thatbel” ((0.7) x2, (0'u,H)) ,We will use the following simple observation. Let @ > 2. Let ||®||,, denote the norm inL” ({0,7] x 2,4 (ol/ ?U,H)). By the Burkholder Davis Gundy inequality,t aL (fe) dP <Q Ssr “ t a/2Lo [o0w|) ap<c | (/ o|? dr) dPQ \ refs] 14s Q \st a/2<cl@ls [ (far) ap=clielgir—sit”Q sBy the Kolmogorov Centsov theorem, this shows that t > Jo @dW is Holder continuouswith exponent(a/2)-1 1 1< os = =Y a 2 @Since @ > 2 is arbitrary, this shows that for any y < 1/2, the stochastic integral is Holdercontinuous with exponent y. This is exactly the same kind of continuity possessed by theWiener process. We state this as the following lemma.