To the Theory of the Dirichlet and Neumann Problems for Strongly Elliptic Systems in Lipschitz Domains

For strongly elliptic systems with Douglis–Nirenberg structure, we investigate the regularity of variational solutions to the Dirichlet and Neumann problems in a bounded Lipschitz domain. The solutions of the problems with homogeneous boundary conditions are originally defined in the simplest $L_2$-Sobolev spaces $H^\sigma$. The regularity results are obtained in the potential spaces $H^\sigma_p$ and Besov spaces $B^\sigma_p$. In the case of second-order systems, the author's results obtained a year ago are strengthened. The Dirichlet problem with nonhomogeneous boundary conditions is considered using Whitney arrays.