Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains
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We consider a family (P_ω)_ω∈Ω of elliptic second order differential operators on a domain U_0 ⊂ℝ^m whose coefficients depend on the space variable x ∈ U_0 and on ω∈Ω, a probability space. We allow the coefficients a_ij of P_ω to have jumps over a fixed interface Γ⊂ U_0 (independent of ω∈Ω). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution u_ω to the equation P_ω u_ω = 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 Ω∋ω→u_ω_H^k+1(U_0) is in L^p(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.
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