We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ of elliptic second order differential operators on a domain $U_0 \subset \RR^m$ whose coefficients depend on the space variable $x \in U_0$ and on $\omega \in \Omega,$ a probability space. We allow the coefficients $a_{ij}$ of $P_\omega$ to have jumps over a fixed interface \Gamma \subset U_0$ (independent of $\omega \in \Omega$). 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 $\Omega \ni \omega \to \|u_\omega\|_{H^{k+1}(U_0)}$ is in $L^p(\Omega)$, for all $1 \le p < \infty$. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.

翻译：我们考虑定义在区域$U_0 \subset \RR^m$上的椭圆型二阶微分算子族$(P_\omega)_{\omega \in \Omega}$，其系数依赖于空间变量$x \in U_0$和概率空间$\omega \in \Omega$。允许算子$P_\omega$的系数$a_{ij}$在固定界面$\Gamma \subset U_0$（与$\omega \in \Omega$无关）上具有跳跃不连续性。针对具有传输条件和混合边界条件（同时考虑"变号"问题）的方程$P_\omega u_\omega = f$，我们获得了关于解$u_\omega$范数的估计，该估计关于系数范数呈多项式增长。特别地，我们证明：若$f$及系数$a_{ij}$充分光滑且服从对数正态型分布，则对任意$1 \le p < \infty$，映射$\Omega \ni \omega \to \|u_\omega\|_{H^{k+1}(U_0)}$属于$L^p(\Omega)$空间。该结论对所得算子逆的范数同样成立。预期这些估计将在不确定性量化领域具有应用价值。