 
Summary: REGULAR INVERSION OF THE DIVERGENCE OPERATOR WITH
DIRICHLET BOUNDARY CONDITIONS ON A POLYGON*
DOUGLAS N. ARNOLD, L. RIDGWAY SCOTT, and MICHAEL VOGELIUS§
Abstract. We consider the existence of regular solutions to the boundary value problem div U = f
on a plane polygonal domain with the Dirichlet boundary condition U = g on . We formulate
simultaneously necessary and sufficient conditions on f and g in order that a solution U exist in the
Sobolev space Ws+1
p (). In addition to the obvious regularity and integral conditions these consist of
at most one compatibility condition at each vertex of the polygon. In the special case of homogeneous
boundary data, it is necessary and sufficient that f belong to Ws
p (), have mean value zero, and vanish at
each vertex. (The latter condition only applies if s is large enough that the point values make sense.) We
construct a solution operator which is independent of s and p. As intermediate results we obtain various
new trace theorems for Sobolev spaces on polygons.
Key words. divergence, trace, Sobolev space
AMS(MOS) subject classifications (1991 revision). 35B65, 46E35
1. Introduction. The constraint of incompressibility arises in many problems of
physical interest. In its simplest form this constraint is modelled by the partial differential
equation
div U = 0 in ,
