Advertisement

Variational boundary value problems

  • B. Daya Reddy
Part of the Texts in Applied Mathematics book series (TAM, volume 27)

Abstract

In the preceding few sections we have built up a theory of regularly elliptic BVPs, in which the typical problem involves finding a function u that satisfies
$$\begin{array}{*{20}{l}} {\quad \,{\text{PDE}}:Au = f{\mkern 1mu} {\text{in}}\:\Omega ,} \\ {\begin{array}{*{20}{c}} {B{C_S}:}&{{B_0}u}& = &{{g_0}} \\ {}&{}& \vdots &{} \\ {}&{{B_{m - 1}}u}& = &{{g_{m - 1}}} \end{array}\} \quad {\text{on}}\:\Gamma ,} \end{array}$$
where A is an elliptic PDE of order 2m in a domain Ω, whereas the boundary conditions are normal, and cover A. The question of well-posedness of solutions to elliptic BVPs has been settled, at least for the case of a smooth domain and homogeneous BCs; provided that certain conditions are met, a unique solution exists. Furthermore, if fH s (Ω), then u is smooth enough to belong to H s+2m (Ω).

Keywords

Bilinear Form Variational Boundary Admissible Function Natural Boundary Condition Essential Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • B. Daya Reddy
    • 1
  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

Personalised recommendations