Calculus of Variations and Geometric Measure Theory
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M. Bardi - A. Goffi

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

created by goffi on 12 Feb 2019
modified on 10 Oct 2019


Published Paper

Inserted: 12 feb 2019
Last Updated: 10 oct 2019

Journal: Calc. Var. Partial Differential Equations
Year: 2019
Doi: 10.1007/s00526-019-1620-2

ArXiv: 1812.09589 PDF


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 H\"ormander 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 H\"ormander vector fields.


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