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In a series of articles[1-4, 6-8, 15, 16, 22-26], Mikhailov, Chabrowski, and others have used energy inequality methods to study the Dirichlet problem with Lp boundary values in C2 domains. Here we show to adapt this method to C1>Dinidomains. Certain important elements of our analysis are present in work of Petrushko [26] for C1-" domains, but our approach was motivated by different considerations. The key idea in the energy method (concerning boundary values) is that the traces of the solution of a suitable ellipticequation on "parallel" surfaces to the boundary should have a limit as these surfaces converge to the boundary. For C2 domains this convergence is readily understood because there is a natural C1 diffeomorphism between a level surface of the distance function (at least near the boundary) and the boundary itself. Petrushko provided special local diffeomorphisms between level surfaces of a regularized distance and the (n―1) dimensional ball which fittogether nicely. Here we use essentially any regularized distance and our conditions on the coefficientsof the equation are weaker than Petrushko's.

The Dirichlet problem with $L^{p}$ boundary data in domains with Dini continuous normal