Unimprovable derivative estimates of a solution of the dirichlet problem for a biharmonic equation in bounded domains

Abstract

We obtain estimates of n-order derivatives (n > 1) of a generalized solution of the Dirichlet problem for a biharmonic equation in nonsmooth two-dimensional domains. We distinguish classes K1 of domains where these estimates are best possible in a certain sense. For wider classes K2 (K1 C K2) of domains analogous estimates of the derivative are established, which strengthen the corresponding estimates for domains of the class K1. Finally, we find classes K3 of domains for which any order derivatives of the solution are bounded. The topicality of the problems considered in the present paper has been emphasized more than once by Fikera. Starting from the classical paper [1], the Dirichlet problem for a biharmonic equation in domains with corner points on boundary, which is important for applications, has been investigated in many papers (see, for example, [2-7]). The methods we use here are similar to those applied in [3-5, 7]. Classes of domains different from those we consider in the present paper are studied in [8]. 1. Notations and definitions. Suppose that f/is a bounded domain in the plane (xl, x2), 0f/is the boundary of f/, and F is a nonempty set of 0f~, F C Of/. We denote by H2(f/,F) a space of functions of f/ obtained by the completion of functions of Ca(f/), equal to zero in a neighborhood of F, with respect to the norm [lull2 = f~ ~ DC~u[ 2 dx, I,~1=2 where = I' 1 = + D'~u =at lu/ax,;,az .. We put