Hamilton-Jacobi in metric spaces with a homological term

Ugo Bessi

Hamilton-Jacobi in metric spaces with a homological term

Optimization and Control 2020-02-03 v1 Analysis of PDEs

Authors: Ugo Bessi

Abstract

The Hamilton-Jacobi equation on metric spaces has been studied by several authors; following the approach of Gangbo and Swiech, we show that the final value problem for the Hamilton-Jacobi equation has a unique solution even if we add a homological term to the Hamiltonian. In metric measure spaces which satisfy the $RCD(K,\infty)$ condition one can define a Laplacian which shares many properties with the ordinary Laplacian on $\R^n$ ; in particular, it is possible to formulate a viscous Hamilton-Jacobi equation. We show that, if the homological term is sufficiently regular, the viscous Hamilton-Jacobi equation has a unique solution also in this case.

Keywords

hamilton-jacobi equation metric spaces homological term rcd condition viscous solution uniqueness

Cite

@article{arxiv.2001.11823,
  title  = {Hamilton-Jacobi in metric spaces with a homological term},
  author = {Ugo Bessi},
  journal= {arXiv preprint arXiv:2001.11823},
  year   = {2020}
}

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