Variational boundary value problems

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


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).


Bilinear Form Variational Boundary Admissible Function Natural Boundary Condition Essential Boundary Condition 
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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

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