Viscosity solutions

  • Luigi Ambrosio
  • Alessandro Carlotto
  • Annalisa Massaccesi
Part of the Publications of the Scuola Normale Superiore book series (PSNS, volume 18)


In this section we want to present the notion of viscosity solution for equations having the general form
$$E(x,u(x),\nabla u(x),{\nabla ^2}u(x)) = 0.$$
The idea behind this approach is a second-order comparison principle, which makes it suitable for dealing with both elliptic and parabolic problems. Consistently with this goal, we shall assume u to be defined on some locally compact domain An, so that we require every point in the domain A to have a compact neighborhood. This topological assumption is actually very useful, as it allows to deal at the same time with open and closed domains, as well as with domains of the form n−1 × [0, ∞), which typically occur in the study of parabolic problems.


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Copyright information

© Scuola Normale Superiore Pisa 2018

Authors and Affiliations

  • Luigi Ambrosio
    • 1
  • Alessandro Carlotto
    • 2
  • Annalisa Massaccesi
    • 3
  1. 1.Scuola Normale SuperiorePisaItalia
  2. 2.ETHZürichSwitzerland
  3. 3.Università di VeronaVeronaItalia

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