Boundary value problems for some classes of degenerating second order partial differential equations.
A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.
Let -div be a second order elliptic operator with real, symmetric, bounded measurable coefficients on or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed , a necessary and sufficient condition is obtained for the boundedness of the Riesz transform on the space. As an application, for , we establish the boundedness of Riesz transforms on Lipschitz domains for operators with coefficients. The range of is sharp. The closely related boundedness of ...