On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation

Abstract

The behavior of solutions of a second-order elliptic equation near a distinguished piece of the boundary is studied. On the remaining part of the boundary, the solutions are assumed to satisfy the homogeneous Dirichlet conditions. A necessary and sufficient condition is established for the existence of an L₂ boundary value on the distinguished part of the boundary. Under the conditions of this criterion, estimates for the nontangential maximal function of the solution hold, the solution belongs to the space of (n − 1)-dimensionally continuous functions, and the boundary value is taken in a much stronger sense.