Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Good and viscosity solutions of fully nonlinear elliptic equations

Authors: Robert Jensen, Maciej Kocan and Andrzej Swiech
Journal: Proc. Amer. Math. Soc. 130 (2002), 533-542
MSC (2000): Primary 35J60, 35J65, 35J25, 49L25.
Published electronically: June 6, 2001
MathSciNet review: 1862134
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We introduce the notion of a ``good" solution of a fully nonlinear uniformly elliptic equation. It is proven that ``good" solutions are equivalent to $L^p$-viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic equation and its $L^p$-viscosity solution. The results also extend some results about ``good" solutions of linear equations.

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Additional Information

Robert Jensen
Affiliation: Department of Mathematical and Computer Sciences, Loyola University, Chicago, Illinois 60626

Maciej Kocan
Affiliation: Department of Mathemetics, University of Cologne, Cologne 50923, Germany
Address at time of publication: Maple Partners Bankhaus, Feuerbachstr. 26-32, 60325 Frankfurt, Germany

Andrzej Swiech
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Received by editor(s): July 5, 2000
Published electronically: June 6, 2001
Additional Notes: The first author was supported in part by NSF grants DMS-9532030, DMS-9972043 and DMS-9706760.
The second author was supported by an Alexander von Humboldt Fellowship.
The third author was supported in part by NSF grant DMS-9706760. Part of this work was completed while this author was visiting the University of Cologne, supported by the TMR Network “Viscosity Solutions and their Applications”.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society