Calculus of Variations and Critical Points

2018 ◽  
pp. 179-213
Author(s):  
Hervé Le Dret
Keyword(s):  
Calculus Of Variations ◽  
Critical Points

Critical points for multiple integrals of the calculus of variations

1996 ◽  
Vol 134 (3) ◽  
pp. 249-274 ◽  
Author(s):  
David Arcoya ◽  
Lucio Boccardo
Keyword(s):  
Calculus Of Variations ◽  
Critical Points ◽  
Multiple Integrals

scholarly journals Multiple Critical Points for Symmetric Functionals without Upper Growth Condition on the Principal Part

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 898
Author(s):  
Marco Degiovanni ◽  
Marco Marzocchi
Keyword(s):  
Variational Methods ◽  
Growth Condition ◽  
Calculus Of Variations ◽  
Critical Points ◽  
Principal Part ◽  
Dimensional Case

This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For this purpose, nonsmooth variational methods are applied.

Linking sandwich pairs

2009 ◽  
Vol 147 (3) ◽  
pp. 679-700 ◽  
Author(s):  
MARTIN SCHECHTER
Keyword(s):  
Calculus Of Variations ◽  
Critical Points ◽  
The Other ◽  
Distinct Advantage ◽  
Cerami Sequence ◽  
Image Position ◽  
Clear Method

AbstractSince the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice, the solutions are critical points. In searching for critical points, there is a distinct advantage if the functional G is semibounded. In this case one can find a Palais–Smale (PS) sequence or even a Cerami sequence These sequences produce critical points if they have convergent subsequences. However, there is no clear method of finding critical points of functionals which are not semibounded. Linking subsets do provide such a method. They can produce a PS sequence provided they separate the functional. In the present paper we show that there are pairs of subsets that can produce Cerami-like sequences even though they do not separate the functional. All that is required is that the functional be bounded from above on one of the sets and bounded from below on the other, with no relationship needed between the bounds. This provides a distinct advantage in applications. We apply the method to several situations.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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