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Elliptic Operators

Chapter
Part of the Universitext book series (UTX)

Abstract

In previous chapters the emphasis was on the Dirichlet and Neumann Problems associated with the Laplacian for which mixed boundary conditions were allowed. In this chapter the Laplacian is replaced by an elliptic operator of the form
$$\mathbf {L} u(x) = \sum _{i,j=1}^{n}a_{ij}(x)\frac{\partial ^2 u}{\partial x_i \partial x_j}(x) + \sum _{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}(x) + c(x)u(x), x \in \Omega $$
and boundary conditions are replaced by an first order differential operator
$$\mathbf {M}u(x) = \sum _{i=1}^{n}\beta _i(x)\frac{\partial u}{\partial x_i}(x) + \gamma (x)u(x) = 0, x \in \partial \Omega .$$
The Dirichlet and Neumann are is special cases of the problem
$$ \mathbf {L} = f \text{ on } \Omega \subset R^n $$
subject to the boundary conditions \(u = g \text{ on } \partial \Omega \) and \(\mathbf {M} = g\) on \(\partial \Omega \), respectively. The latter is known as the oblique derivative boundary problem. To transfer classical results to a more general setting, a method of continuity is used. Proceeding from an elliptic operator with constant coefficients, classical Dirichlet solutions are morphed into solutions on a ball for the operator \(\mathbf {L}\). The Perron-Brelot-Wiener method is then adapted to extend results to the more general setting.

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

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