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

Abstract

We consider a \emph{family} $(P_{\ddot{I}}‰{)}_{\ddot{I}‰\in \hat{I}\text{c◯}}$ 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 $\ddot{I}‰\in \hat{I}\text{c◯},$ a probability space. We allow the coefficients $a_{ij}$ of $P_{\ddot{I}}‰$ to have jumps over a fixed interface $\hat{I}“\subset U_0$ (independent of $\ddot{I}‰\in \hat{I}\text{c◯}$). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution $u_{\ddot{I}}‰$ to the equation $P_{\ddot{I}}‰u_{\ddot{I}}‰=f$ 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 $\hat{I}\text{c◯}\ni \ddot{I}‰\to \Vert u_{\ddot{I}}‰{\Vert }_{H^{k+1}(U_0)}$ is in $L^p(\hat{I}\text{c◯})$, 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.

We fixed a small .tex problem in the abstract on the site (the manuscript has not changed)