The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries. II
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.
scientific article; zbMATH DE number 3882891