Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Dirichlet problem for first order quasilinear equations on a manifold

Author: E. Yu. Panov
Journal: Trans. Amer. Math. Soc. 363 (2011), 2393-2446
MSC (2010): Primary 35L60, 35L65; Secondary 58J32, 58J45
Published electronically: December 15, 2010
MathSciNet review: 2763721
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Abstract: We study the Dirichlet problem for a first order quasilinear equation on a smooth manifold with boundary. The existence and uniqueness of a generalized entropy solution are established. The uniqueness is proved under some additional requirement on the field of coefficients. It is shown that generally the uniqueness fails. The nonuniqueness occurs because of the presence of the characteristics not outgoing from the boundary (including closed ones). The existence is proved in a general case. Moreover, we establish that among generalized entropy solutions laying in the ball $ \Vert u\Vert _\infty\le R$ there exist unique maximal and minimal solutions. To prove our results, we use the kinetic formulation similar to the one by C. Imbert and J. Vovelle.

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E. Yu. Panov
Affiliation: Department of Mathematics, Novgorod State University, B. Sankt-Peterburgskaya, 41, 173003 Velikiy Novgorod, Russia

Received by editor(s): December 8, 2008
Published electronically: December 15, 2010
Additional Notes: This work was carried out under partial support of the Russian Foundation for Basic Research (grant RFBR No. 06-01-00289) and the Deutsche Forschungsgemeinschaft (DFG project No 436 RUS 113/895/0-1)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.