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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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DOI: https://doi.org/10.1090/S0002-9939-1977-0451156-3
Keywords: Multiple integrals, nondifferentiable functions, Euler-Lagrange equation, generalized gradients, Poisson's equation
Article copyright: © Copyright 1977 American Mathematical Society

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