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

¹ Université de Lorraine, CNRS, IECL, 54000 Nancy, France
² Université de Lorraine, CNRS, IECL, 57000 Metz, France

^* Corresponding author: simon.labrunie@univ-lorraine.fr

Received: 25 September 2023
Accepted: 11 July 2024

Abstract

We consider a family (P_ω)_ω∈Ω of elliptic second order differential operators on a domain U₀ ⊂ R^m whose coefficients depend on the space variable x ∈ U₀ and on ω ∈ Ω, a parameter space. We allow the coefficients a_ij of P_ω to have jumps over a fixed interface Γ ⊂ U₀ (independent of ω ∈ Ω). We obtain estimates on the norm of the solution u_ω to the equation P_ωu_ω = f with transmission and mixed boundary conditions that are polynomial in the norms of the coefficients. 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 Ω ∋ ω ↦ ‖u_ω‖_H^k+1(U0) is in L^p(Ω), for all 1 ≤ p < ∞. The same is true for the norms of the inverses of the resulting operators. We also obtain similar integrability results for the parametric Finite Element approximations of the solution. We expect our estimates to be useful in Uncertainty Quantification.