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New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

  • Martino BardiEmail author
  • Alessandro Goffi
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Abstract

We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton–Jacobi–Bellman form, such as those involving the Pucci’s extremal operators over Hörmander vector fields.

Mathematics Subject Classification

Primary: 35B50 35J70 35J60 Secondary: 49L25 35H20 
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Notes

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics “T. Levi-Civita”University of PadovaPadovaItaly
  2. 2.Gran Sasso Science InstituteL’AquilaItaly

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