The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries. II (Q759917)

This is the second part of the authors' paper in ibid., 335-359 (1983; Zbl 0532.35065). Pointwise estimates for the Green tensor G(x,\(\xi)\) of the linear Stokes system in a piecewise smooth domain in \(R^ 3\) are proved. These estimates explicitly depend on the position of points x and \(\xi\) with respect to edges and vertices. The Miranda-Agmon maximum principle is proved for solutions of the Stokes system. The existence of a strong solution of the Dirichlet problem for the Navier-Stokes equations is proved along with some estimates for the Sobolev and Hölder norms. In particular, it is shown that the solution belongs to the Hölder class \(C^{\alpha}({\bar \Omega})\) if all right-hand sides are smooth. The following assertion is proved. Let (\(\vec v,p)\) be a solution in \(W'_ 2(\Omega)\times L^ 2(\Omega)\) of the boundary value problem \(-\nu \Delta \vec v+\nabla p+v_ k\partial \vec v/\partial x_ k=0,\) \(div \vec v=0\) in \(\Omega\), \(\vec v|_{\partial \Omega}={\vec \phi}.\) If \({\vec \phi}\in C(\partial \Omega)\) then \(\vec v\in C({\bar \Omega})\). The results similar to those proved for the Stokes system are obtained for the Lamé system and the Laplace operator.