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

TL;DR

This paper establishes polynomial bounds on solutions to parametric elliptic transmission problems with jumps over interfaces, providing probabilistic integrability results useful for uncertainty quantification.

Contribution

It introduces polynomial estimates for solutions of parametric elliptic transmission problems with jumps, including probabilistic bounds for smooth coefficient distributions.

Findings

Polynomial bounds on solution norms in terms of coefficient norms

Solutions' norms are in L^p for log-normal coefficient distributions

Estimates applicable to inverse operators in the problem

Abstract

We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ 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 $\omega \in \Omega,$ 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.