Uniformly elliptic affine random diffusion model

Let $D\subset {\mathbb{R}}^m$ be a bounded polyhedral open set and $V=H_0^1(D;\mathbb{R})$ with norm $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D)}$. The zero-boundary Sobolev energy space is a separable Hilbert space supplies the separable Hilbert structure under Countable Choice; the gradient norm is equivalent to its Sobolev norm there.

Let $Y=(Y_1,\dots ,Y_d)$ have the product law $\rho$ of Finite product-law polynomial-chaos input data, and assume its support $\mathrm{\Gamma }$ is bounded, so $\mathrm{\mid }{y}_j\mathrm{\mid }\le R_j<\mathrm{\infty }$ for every $y\in \mathrm{\Gamma }$. For real coefficient functions $a_0,\dots ,a_d\in L^{\mathrm{\infty }}(D)$ define $a(x,y)=a_0(x)+{\sum }_{j=1}^d{y}_j{a}_j(x)\mathrm{.}$ Assume constants $0<a_{\ast }\le a^{\ast }<\mathrm{\infty }$ satisfy, for every $y\in \mathrm{\Gamma }$, $a_{\ast }\le a(x,y)\le a^{\ast }$ for almost every $x\in D$. This explicit all-parameter condition ensures the pointwise variational problem is well-posed for every stated parameter, not merely almost every one. Let $F(y)=f_0+{\sum }_{j=1}^d{y}_j{f}_j$ with $f_j\in V^{\ast }$.

The pointwise weak form is $a_y(u,v)={\int }_{D}a(x,y)\mathrm{\nabla }u(x)\cdot \mathrm{\nabla }v(x)dx=F(y)(v),v\in V\mathrm{.}$ The Bochner weak form uses the bilinear form $\mathcal{A}(U,W)={\int }_{\mathrm{\Gamma }}{a}_y(U(y),W(y))d\rho (y)$ and the functional $\mathcal{F}(W)={\int }_{\mathrm{\Gamma }}F(y)(W(y))d\rho (y)$ on $L_{\rho }^2(V)$ from Bochner L2 space for a finite product-law input.