On the Effect of Irregularity of the Domain Boundary on the Solution of a Boundary Value Problem for the Laplace Equation

We consider an inhomogeneous boundary value problem with mixed boundary conditions for the Laplace equation in a domain representing a perturbation <span class="mathjax-tex">\(\Pi _\gamma \)</span> of a rectangle <span class="mathjax-tex">\(\Pi \)</span> where one of its sides is replaced by some curve <span class="mathjax-tex">\(\gamma \)</span> of minimal smoothness. An estimate is obtained for the difference between the solutions of the perturbed and unperturbed problems in the norm of the Sobolev space <span class="mathjax-tex">\( H^1\)</span> on their common domain.