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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Symmetrization and optimal control for elliptic equations

Authors: Charles Voas and Daniel Yaniro
Journal: Proc. Amer. Math. Soc. 99 (1987), 509-514
MSC: Primary 49B22; Secondary 35B37, 35J20
MathSciNet review: 875390
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Abstract: We consider an optimal control problem where $ u(x)$ satisfies $ - \operatorname{div}(H(x)\nabla u) = 1$ in $ \Omega $ and $ H(x)$ is a control. We introduce the functional $ {J_\Omega }(H) = {\vert\Omega\vert^{ - 1}}\int\limits_\Omega {u(x)} dx$ and show using a symmetrization argument that if the distribution function of $ H$ is fixed, then $ {J_\Omega }(H)$ is largest when $ \Omega $ is a ball and $ H$ is radial and decreasing on radii.

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Article copyright: © Copyright 1987 American Mathematical Society