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On Asymmetric Distances

Andrea C.G. Mennucci — 2013

Analysis and Geometry in Metric Spaces

In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the...

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity

Andrea C.G. Mennucci — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a dimensional manifold , with assumptions of convexity of (x, .) and regularity of (locally in a neighborhood of {=0} in ); we define the “ solution” , a generalized solution; to this end, we view as a . The definition of “ solution” is suited to proving regularity results about ; in particular, we prove in the first part that the closure of the set where is not regular may be covered by a countable...

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