Estimation of the continuity constants for Bogovskiĭ and regularized Poincaré integral operators

10/08/2020

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by   Johnny Guzmàn, et al.

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We study the dependence of the continuity constants for the regularized Poincaré and Bogovskiĭ integral operators acting on differential forms defined on a domain Ω of ℝ^n. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) L^2(Ω,Λ^ℓ) to H^1(Ω,Λ^ℓ-1), ℓ∈{1, …, n}. For domains Ω that are star shaped with respect to a ball B we study the dependence of the constants on the ratio diam(Ω)/diam(B). A program on how to develop estimates for higher order Sobolev norms is presented. The results are extended to certain classes of unions of star shaped domains.