THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS
Juan Luis Vázquez · 2006
Abstract This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann boundary conditions and the problems posed on manifolds. Section 11.1 introduces the problem and concepts of weak solution, proves a uniqueness result, and presents examples. Section 11.2 reviews the theory for the existence and uniqueness of weak solutions and limit solutions. Section 11.3 provides proof of better estimates and boundedness of solutions in the case of the PME. Section 11.4 examines the mixed problems and problems posed in exterior space domains. The second main topic of this chapter is the theory of PME and GPME on Riemannian manifolds, which is in Section 11.5.