Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains

We consider a \emph{family} $(P_{\omega }{)}_{\omega \in \mathrm{\Omega }}$ of elliptic second order differential operators on a domain $U_0\subset {\mathbb{R}}^m$ whose coefficients depend on the space variable $x\in U_0$ and on $\omega \in \mathrm{\Omega },$ a probability space. We allow the coefficients $a_{ij}$ of $P_{\omega }$ to have jumps over a fixed interface $\mathrm{\Gamma }\subset U_0$ (independent of $\omega \in \mathrm{\Omega }$). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution $u_{\omega }$ to the equation $P_{\omega }{u}_{\omega }=f$ with transmission and mixed boundary conditions (we consider ``sign-changing’’ problems as well). In particular, we show that, if $f$ and the coefficients $a_{ij}$ are smooth enough and follow a log-normal-type distribution, then the map $\mathrm{\Omega }\ni \omega \to \mathrm{\parallel }{u}_{\omega }{\mathrm{\parallel }}_{H^{k+1}(U_0)}$ is in $L^p(\mathrm{\Omega })$, for all $1\le p<\mathrm{\infty }$. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.