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The Porous Medium EquationMathematical Theory$
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Juan Luis Vazquez

Print publication date: 2006

Print ISBN-13: 9780198569039

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198569039.001.0001

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THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

Chapter:
(p.257) 11 THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS
Source:
The Porous Medium Equation
Author(s):

Juan Luis Vázquez

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0011

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.

Keywords:   Neumann boundary conditions, manifolds, weak solution, limit solutions, PME, GPME, Riemannian manifold

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