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A Unified Variational Approach to Discontinuous Differential Equations

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Abstract

A direct variational technique involving Clarke generalized gradient is used to treat general boundary value problems with discontinuous nonlinearities. Based on the theory of positive definite symmetric operators it is established the nonsmooth variational form of the regularized inclusions which give the Filippov solutions of the discontinuous problems. These solutions reduce to classical solutions in case that a transversality condition on the set of discontinuities is satisfied. The results apply to a wide class of concrete boundary value problems of different orders. Two illustrative examples are given.

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Acknowledgements

Jorge Rodríguez-López was partially supported by Xunta de Galicia ED431C 2019/02.

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Authors and Affiliations

  1. Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Radu Precup

  2. Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain

    Jorge Rodríguez-López

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  1. Radu Precup
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  2. Jorge Rodríguez-López
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Correspondence to Radu Precup.

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Precup, R., Rodríguez-López, J. A Unified Variational Approach to Discontinuous Differential Equations. Mediterr. J. Math. 18, 62 (2021). https://doi.org/10.1007/s00009-021-01705-9

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