Abstract
We present a research program designed by A. Bressan and some partial results related to it. First, we construct a probability measure supported on the space of solutions to a planar differential inclusion, where the right-hand side is a Lipschitz continuous segment. Such measure assigns probability one to solutions having derivatives a.e. equal to one of the endpoints of the segment. Second, for a class of planar differential inclusions with Hölder continuous right-hand side F, we prove existence of solutions whose derivatives are exposed points of F. Finally, we complete the research program if the right-hand side of the differential inclusion does not depend on the state and prove a result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of Brownian motion.
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Dedicated to Arrigo Cellina on the occasion of his 70th birthday.
The authors are supported by M.I.U.R. project “Viscosity, metric, and control theoretic methods for nonlinear partial differential equations”, GNAMPA of INDAM, Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems” and University of Padova research project “Some analytic and differential geometric aspects in Nonlinear Control Theory, with applications to Mechanics”, the project VARIANT PTDC/MAT/111809/2009 “Variational Analysis and Applications”, funded by the Portuguese institutions FCT, COMPETE, QREN, and the European Regional Development Fund (FEDER).
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Colombo, G., Goncharov, V.V. Brownian motion and exposed solutions of differential inclusions. Nonlinear Differ. Equ. Appl. 20, 323–343 (2013). https://doi.org/10.1007/s00030-012-0168-z
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DOI: https://doi.org/10.1007/s00030-012-0168-z