Dirichlet and Neumann problems for planar domains with parameter

Abstract

Let Γ ( ⋅ , λ ) \Gamma (\cdot ,\lambda ) be smooth, i.e.  C ∞ \mathcal C^\infty , embeddings from Ω ¯ \overline {\Omega } onto Ω λ ¯ \overline {\Omega ^{\lambda }} , where Ω \Omega and Ω λ \Omega ^\lambda are bounded domains with smooth boundary in the complex plane and λ \lambda varies in I = [ 0 , 1 ] I=[0,1] . Suppose that Γ \Gamma is smooth on Ω ¯ × I \overline \Omega \times I and f f is a smooth function on ∂ Ω × I \partial \Omega \times I . Let u ( ⋅ , λ ) u(\cdot ,\lambda ) be the harmonic functions on Ω λ \Omega ^\lambda with boundary values f ( ⋅ , λ ) f(\cdot ,\lambda ) . We show that u ( Γ ( z , λ ) , λ ) u(\Gamma (z,\lambda ),\lambda ) is smooth on Ω ¯ × I \overline \Omega \times I . Our main result is proved for suitable Hölder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings Γ ( ⋅ , λ ) \Gamma (\cdot ,\lambda ) from D ¯ \overline {\mathbb D} , the closure of the unit disc, onto Ω λ ¯ \overline {\Omega ^\lambda } such that Γ \Gamma is smooth on D ¯ × I \overline {\mathbb D}\times I and real analytic at ( − 1 , 0 ) ∈ D ¯ × I (\sqrt {-1},0)\in \overline {\mathbb D}\times I , but for every family of Riemann mappings R ( ⋅ , λ ) R(\cdot ,\lambda ) from Ω λ ¯ \overline {\Omega ^\lambda } onto D ¯ \overline {\mathbb D} , the function R ( Γ ( z , λ ) , λ ) R(\Gamma (z,\lambda ),\lambda ) is not real analytic at ( − 1 , 0 ) ∈ D ¯ × I (\sqrt {-1},0)\in \overline {\mathbb D}\times I .