Linear Elliptic Equations

Part of the Applied Mathematical Sciences book series (AMS, volume 115)


The first major topic of this chapter is the Dirichlet problem for the Laplace operator on a compact domain with boundary:
$$\Delta u = 0\text{ on }\Omega,\quad {u\bigr |}_{\partial\Omega } = f.$$
We also consider the nonhomogeneous problem Δu=g and allow for lower-order terms. As in Chap.2, Δ is the Laplace operator determined by a Riemannian metric. In §1 we establish some basic results on existence and regularity of solutions, using the theory of Sobolev spaces. In §2 we establish maximum principles, which are useful for uniqueness theorems and for treating (0.1) for f continuous, among other things.


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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