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On the Eikonal equation for degenerate elliptic operators


Author: Paolo Albano
Journal: Proc. Amer. Math. Soc. 140 (2012), 1739-1747
MSC (2010): Primary 35F30, 35F21, 35D40
Published electronically: September 22, 2011
MathSciNet review: 2869158
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Abstract: We consider the nonnegative viscosity solution of the homogeneous Dirichlet problem for an eikonal equation associated to an operator sum of squares of vector fields of Grushin type in a symmetric domain. We show that the solution is locally Lipschitz continuous except at the characteristic boundary point. In the characteristic boundary point the solution has a Hölder regularity with exponent related to the Hörmander bracket condition. Finally, the singular set is an analytic stratification given by the characteristic boundary point and a half line.


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Additional Information

Paolo Albano
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy
Email: albano@dm.unibo.it

DOI: https://doi.org/10.1090/S0002-9939-2011-11132-8
Keywords: Viscosity solutions, eikonal equation, degenerate equations, singularities
Received by editor(s): June 8, 2010
Received by editor(s) in revised form: December 6, 2010, and January 27, 2011
Published electronically: September 22, 2011
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.