On solvability of the Cauchy problem for the Laplace operator (Q1114850)

Let \(\Omega\) be a bounded domain in \(R^ n\). The solvability of the problem \(\Delta u=0\) on \(\Omega\), \(u=v\) on \(\Gamma_ 1\) and \(\partial u/\partial \nu =0\) on \(\Gamma_ 1\) is proved for some class of boundary functions v. \((\Gamma_ 1\subset \partial \Omega\) is a closed \(C^{\infty}\)-manifold.) It is shown that if \(u_{\epsilon}\) minimizes the functional \[ (\epsilon /2)\int_{\Omega}| Du|^ 2 dx+(1/2)\int_{\Gamma_ 1}| v-u| dS \] (v\(\in L_ 2(\Gamma_ 1)\) given function) then \(\| v-u_{\epsilon}\|_{L_ 2(\Gamma_ 1)}\to 0\).