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Published in:
2016 | OriginalPaper | Chapter
4. Optimality
Author : Pablo Pedregal
Published in: Optimal Design through the Sub-Relaxation Method
Publisher: Springer International Publishing
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Abstract
All of our previous analysis had the final goal of getting to the relaxation
$$\displaystyle{\mbox{ Minimize in }(t,\mathbf{s}):\quad \overline{I}(t,\mathbf{s}) =\int _{\varOmega }f(\mathbf{x})u(\mathbf{x})\,d\mathbf{x}}$$
$$\displaystyle\begin{array}{rcl} & t(\mathbf{x}) \in [0,1],\int _{\varOmega }t(\mathbf{x})\,d\mathbf{x} = r,\quad \vert \mathbf{s}(\mathbf{x})\vert \leq 1, & {}\\ & -\mbox{ div}\left [a(\mathbf{x})\nabla u(\mathbf{x}) + b(\mathbf{x})\vert \nabla u(\mathbf{x})\vert \mathbf{s}(\mathbf{x})\right ] = f(\mathbf{x})\mbox{ in }\varOmega,\quad u = u_{0}\mbox{ on }\partial \varOmega,& {}\\ \end{array}$$