2023/08/11 by Simon Labrunie, Labrunie, Simon, Hassan Mohsen +3
Computer Science · Engineering · Mathematics · #35R01 (Primary) 35J75 #46E35 #65N75 (Secondary) #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2308.06215
openalex publication_date 2023/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a family (Pω)ω∈ Ω of elliptic second order differential operators on a domain U0 ⊂ ℝm whose coefficients depend on the space variable x ∈ U0 and on ω∈ Ω, a probability space. We allow the coefficients aij of Pω to have jumps over a fixed interface Γ⊂ U0 (independent of ω∈ Ω). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution uω to the equation Pωuω= f with transmission and mixed boundary conditions (we consider ``sign-changing'' problems as well). In particular, we show that, if f and the coefficients aij are smooth enough and follow a log-normal-type distribution, then the map Ω\ni ω→ ‖uω‖Hk+1(U0) is in Lp(Ω), for all 1 ≤ p < ∞. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.