Elliptic Equations: An Introductory Course pp 191-202 | Cite as

# Linear Elliptic Systems

Chapter

## Abstract

Sometimes, in many physical situations, one has not only to look for a scalar

*u*but for a vector**u**= (*u*^{ 1 },... ,u^{ m }). (In what follows a bold letter will always denote a vector.) This could be a position in space, a displacement, a velocity... So one is in need to introduce systems of equations. The simplest one is of course the one consisting in*m*copies of the Dirichlet problem, that is to say$$
\left\{ \begin{gathered}
- \Delta u^1 = f^1 in \Omega , \hfill \\
\cdots \cdots \cdots \cdots \hfill \\
- \Delta u^m = f^m in \Omega , \hfill \\
u = \left( {u^1 , \ldots u^m } \right) = 0 on \partial \Omega . \hfill \\
\end{gathered} \right.
$$

(13.1)

## Keywords

Scalar Product Bilinear Form Dirichlet Problem Euclidean Norm Elliptic System
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.

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© Birkhäuser Verlag AG 2009