Capturing Functions in Infinite Dimensions

The following are notes on stochastic and parametric PDEs of the short course in Paris. 1 Lecture 4: Capturing Functions in Infinite Dimensions Finally, we want to give an example where the problem is to recover a function of infinitely many variables. We will first show how such problems occur in the context of stochastic partial differential equations. 1.1 Elliptic equations: general principles We consider the elliptic equation −∇ · (a∇u) = f in D, u|∂D = 0, (1.1) in a bounded Lipschitz domain D ⊂ Rd, where f ∈ L2(D). There is a rich theory for existence and uniqueness for equations of this form which we briefly recall. Central to the theory of elliptic equations is the Sobolev space V := H1 0 (D) (called the energy space) which is a Hilbert space equipped with the energy norm ‖v‖V := ‖∇v‖L2(D). The dual of V is V ∗ = H−1(D). The solution of (1.1) is defined in weak form as a function u ∈ H1 0 (D) which satisfies