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A remark on the direct method of the calculus of variations

Author: J. P. Penot
Journal: Proc. Amer. Math. Soc. 67 (1977), 135-141
MSC: Primary 49F22; Secondary 49A50
MathSciNet review: 0464022
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Abstract: This note deals with the problem of minimizing a real-valued function f on a weakly closed subset of a reflexive Banach space. We use a mild monotonicity assumption introduced by P. Hess [11] on the derivative f' of f to get the weak lower semicontinuity of f. We show that one can dispense with any continuity assumption on f', so that we get a true generalization of F. E. Browder's results [4]. The relevance of the monotonicity property to the calculus of variations is shown by an example.

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Keywords: Variational problem, direct method, weak lower semicontinuity, operator of monotone type, pseudomonotone operator, pertinent operator, nonlinear eigenvalue problem, multiple integral functional
Article copyright: © Copyright 1977 American Mathematical Society

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