A Remark on the Elliptic Boundary Value Problem in an Angular Domain

1. The general boundary value problems of elliptic equations have been studied quite extensively by several authors (cf., e.g., Ql, 2, 5, 21]). The method is based on the a priori estimates. Most authors have restricted themselves to domains with sufficiently smooth boundary. Their main tool is to map the neighborhood of a point on the boundary onto a semisphere by means of a sufficiently smooth transformation. On the other hand Ladyzhenskaya C13], [114] and others (cf.? e.g., []9, 12]) showed the a priori estimates for domains with piece wise smooth surfaces. Their method is integration by parts. Hence the operators need to be at most of second orders and to be real valued. In this note, assuming a relation between the domain and the elliptic operator, we study L-a priori estimates with a parameter for the second order operators with mixed boundary conditions in an angular domain (see Theorems 1, 2 and 3 in the following section). The a priori estimate with a parameter has been treated in £2], [^4], Q5] and [10]. In addition the mixed boundary value problem has been studied by several authors (cf. [17, 18, 22, 23, 24]). Our proof relies upon mainly their results.