Uniform convergence for elliptic problems on varying domains
Wolfgang Arendt, Daniel Daners · Mathematische Nachrichten · 2006
Abstract Let Ω ⊂ ℝNbe (Wiener) regular. Forλ> 0 andf∈L∞(ℝN) there is a unique bounded, continuous functionu: ℝN→ ℝ solving λu– Ωu=f in 𝔻(Ω)′, u= 0 on ℝN\ Ω. Given open sets Ωnwe introduce the notion ofregular convergenceof Ωnto Ω asn→ ∞. It implies that the solutionsunof (P ) converge (locally) uniformly touon ℝN. WhereasL2‐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 alln∈ ℕ. Applications to spectral theory, parabolic equations and nonlinear equations are given. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)