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Zeitschrift für Analysis und ihre Anwendungen


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Volume 9, Issue 3, 1990, pp. 235–272
DOI: 10.4171/ZAA/398

Published online: 1990-06-30

Investigation of extremals in a generalized isoperimetric problem by means of higher-order conditions

N.P. Osmolovskii

The problem of a material point going round maximum area on a plane in a fixed period of time, with its velocity vector varying in a compact set $U$, is considered. Under some assumptions on $U$ and with the help of higher-order conditions it is proved that multiply-traversing extremals do not even give a so-called 0-weak maximum (which would at the same time be a weak maximum in the case of continuous velocity). For simply traversing extremals providing an absolute maximum; a non-trivial result on the kind of the maximum is obtained. Furthermore, finite-dimensional and infinite-dimensional quadratic forms connected with the pointed-out generalizationof the isoperimetric problem are studied; their non-negativeness on the corresponding subspaces and some other properties are established.

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Osmolovskii N.P.: Investigation of extremals in a generalized isoperimetric problem by means of higher-order conditions. Z. Anal. Anwend. 9 (1990), 235-272. doi: 10.4171/ZAA/398