Abstract
AbstractLet Ω ⊂ ℝN be (Wiener) regular. For λ > 0 and f ∈ L ∞(ℝN ) there is a unique bounded, continuous function u: ℝN → ℝ solving λu – Ωu = f in 𝔻(Ω)′, u = 0 on ℝN \ Ω. Given open sets Ωn we introduce the notion of regular convergence of Ωn to Ω as n → ∞. It implies that the solutions u n of (P ) converge (locally) uniformly to u on ℝN . Whereas L 2‐convergence has been treated in the literature, our criteria for uniform convergence are new. The notion of regular convergence is very general. For instance the sequence of open sets obtained by cutting into a ball converges regularly. Other examples show that uniform convergence is possible even if the measure of Ωn \ Ω stays larger than a positive constant for all n ∈ ℕ. Applications to spectral theory, parabolic equations and nonlinear equations are given. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Published Version
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