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Proceedings of the American Mathematical Society

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

 
 

 

Multiple integrals of Lipschitz functions in the calculus of variations


Author: Frank H. Clarke
Journal: Proc. Amer. Math. Soc. 64 (1977), 260-264
MSC: Primary 49F99
DOI: https://doi.org/10.1090/S0002-9939-1977-0451156-3
MathSciNet review: 0451156
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Abstract: We consider a multiple integral problem in the calculus of variations in which the integrand is locally Lipschitz but not differentiable, and in which minimization takes place over a Sobolev space. Using a minimax theorem, we derive an analogue of the classical Euler condition for optimality, couched in terms of “generalized gradients". We proceed to indicate how these results may be applied to deduce existence and smoothness properties of solutions to certain Poisson equations.


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Keywords: Multiple integrals, nondifferentiable functions, Euler-Lagrange equation, generalized gradients, Poisson’s equation
Article copyright: © Copyright 1977 American Mathematical Society