Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] J. Cea and K. Malanowski, An example of a max-min problem in partial differential equations, SIAM J. Control Optim. 8 (1970), 305-316. MR 0274915 (43:673)
  • [2] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [3] G. H. Hardy, J. F. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1967.
  • [4] F. Murat and L. Tartar, Calcul de variations et homogenization, Cours de l'Ecole d'Ete d'Analyse Numerique, 1984.
  • [5] G. Pólya, Torsional rigidity, principal frequencey, electrostatic capacity and symmetrization, Quart. Appl. Math. 6 (1948), 267-277. MR 0026817 (10:206b)
  • [6] B. de Saint-Venant, Memoire sur la torsion des prismes, Memoires Presentes par Divers Savants a l'Academie des Sciences, t. 14, 1856, pp. 233-560.
  • [7] S. Sternberg, Lectures in differential geometry, Prentice-Hall, Englewod Cliffs, N.J., 1964. MR 0193578 (33:1797)
  • [8] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 697-718. MR 0601601 (58:29170)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 49B22, 35B37, 35J20

Retrieve articles in all journals with MSC: 49B22, 35B37, 35J20

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society