INTEGRABILITY OF DERIVATIVES OF CLASSICAL SOLUTIONS OF DIRICHLET’S PROBLEM FOR AN ELLIPTIC EQUATION

The main result concerning the question of whether the classical solution u(x) of Dirichlet’s problem for an elliptic equation Lu=f is in the Sobolev space W~1 is that u∈W~1 provided the coefficients of L areessentially bounded and f∈L~2. Here, the author proves a result showing that u may be in W~1 even when f is not in the class L~2.