Example II: The Neumann Problem

Roger Temam

Chapter Chapter 12 in Numerical Analysis, 1973, pp 121-128 from Springer

Abstract: Abstract We are in the situation discussed in Section 1.2; Ω is a bounded open set in R n with boundary Г, we put H=L2(Ω) and V=H1(Ω) and these spaces are provided with their usual Hubert structure (cf. Chapter 7): $$\begin{array}{*{20}{c}} {\left( {f,g} \right) = \int\limits_\Omega {f\left( x \right)g\left( x \right)dx} } \\ {\begin{array}{*{20}{c}} {\left| f \right| = {{\left( {f,f} \right)}^{1/2}},}&{\forall f,g \in H} \end{array}} \\ {\left( {\left( {u,v} \right)} \right) = \left( {u + v} \right) + \sum\limits_{i - 1}^n {\left( {{D_i}u,{D_i}v} \right)} } \\ {\begin{array}{*{20}{c}} {\left\| u \right\| = {{\left( {\left( {u,u} \right)} \right)}^{1/2}},}&{\forall u,v \in V.} \end{array}} \end{array}$$

Date: 1973
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DOI: 10.1007/978-94-010-2565-2_13

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