Can I read Polynomial Bounds for the Solutions of Parametric Transmission Problems on Smooth, Bounded Domains on EtoBox?

Polynomial Bounds for the Solutions of Parametric Transmission Problems on Smooth, Bounded Domains by Labrunie, Simon; Mohsen, Hassan; Nistor, Victor is a scholarly article available to read on EtoBox.

What is Polynomial Bounds for the Solutions of Parametric Transmission Problems on Smooth, Bounded Domains about?

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.

Author

Labrunie, Simon; Mohsen, Hassan; Nistor, Victor

Published

2023

Language

EN