Dirichlet and Neumann problems for planar domains with parameter (Q2862128)

Let \(\Omega^\lambda\) \((0\leq\lambda\leq 1)\) be a family of bounded domains in the complex plane of class \(C^{k+1+\alpha}\), where \(k\geq 0\) is an integer and \(\alpha\in (0,1)\). Let also \(f^\lambda,\;g^\lambda\) be \(C^\alpha\) functions on the boundary \(\partial\Omega^\lambda\). The authors study the family of Dirichlet problems NEWLINE\[NEWLINE \Delta u^\lambda =0\text{ in }\Omega^\lambda,\quad u^\lambda=f^\lambda\text{ on }\partial \Omega^\lambda\tag{1}NEWLINE\]NEWLINE and the family of Neumann problems NEWLINE\[NEWLINE \Delta v^\lambda =0\text{ in }\Omega^\lambda,\quad \partial_{\nu^\lambda} v^\lambda=g^\lambda\text{ on }\partial \Omega^\lambda\tag{2}NEWLINE\]NEWLINE together with the compatibility conditions NEWLINE\[NEWLINE \int_{\partial\Omega^\lambda}g^\lambda d\sigma^\lambda=0,\quad \int_{\partial\Omega^\lambda}v^\lambda d\sigma^\lambda=0.\tag{3}NEWLINE\]NEWLINE NEWLINENEWLINEThe first major result of the paper is as follows: Let \(j,k,l\geq 0\) be integers such that \(k\geq j\), \(k+1\geq l\geq j\). Let \(\Omega\subset{\mathbb C}\) be a bounded domain with \(C^{k+1+\alpha}\) boundary. Let \(\Gamma^\lambda\), \(\lambda\in [0,1]\), be a \(C^{k+1+\alpha,j}\) embedding of \(\overline{\Omega}\) into \(\overline{\Omega^\lambda}\). Assume \(f^\lambda\circ \Gamma^\lambda\in C^{l+\alpha,j}(\partial\Omega)\) and \(g^\lambda\circ \Gamma^\lambda\in C^{k+\alpha,j}(\partial \Omega)\). Finally, let \(u^\lambda\in C^{\alpha}(\overline{\Omega^\lambda})\) be the unique solution of (1) and let \(v^\lambda\in C^1(\Omega^\lambda)\) be the unique solution of (2) - (3). Then \(u^\lambda\circ \Gamma^\lambda\in C^{l+\alpha,j}(\overline\Omega)\) and \(v^\lambda\circ \Gamma^\lambda\in C^{k+1+\alpha,j}(\overline\Omega)\). NEWLINENEWLINENEWLINENEWLINE As a consequence, the authors establish the failure of the local Schwarz principle with parameter in the following formulation: There exist embeddings \(\Gamma(\cdot,\lambda)\) from the closed unit disc \(\overline{\mathbb D}\) onto \(\overline{\Omega^\lambda}\) such that \(\Gamma\) is of class \(C^\infty\) on \(E=\overline{\mathbb D}\times[0,1]\) and real analytic at \((1,0)\in E\) such that \(R(\Gamma(z,\lambda),\lambda)\) is not real analytic at \((1,0)\in E\) for every family of Riemann mappings \(R(\cdot,\lambda)\) from \(\overline{\Omega^\lambda}\) onto \(\overline{\mathbb D}\).