Resolvent Estimates for Singularly Perturbed Elliptic Operators in Hölder Spaces

Abstract

AbstractThe norm of the inverse operator of ϵA + B ‐ λI between the Besov spaces B, ∞ (Ω) and Bt∞,∞(Ω) is estimated, where A and B are uniformly elliptic operators with smooth coefficients and Dirichlet boundary conditions, A is of order 2m, B of order 2m, m > m'. The estimate holds for negative t. The Besov space B, ∞ (Ω) reduces to the space of Hölder continuous functions Cs (Ω) if s < 0 is non – integer.In particular, it is shown that Aϵ generates an analytic semigroup in B, ∞ (Ω), s ϵ (‐1,0), if Ω = ℝn or ℝ and A, B are constant coefficient operators without lower order terms.